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SEMINAR'ON-COMBINATORIAL TOPOLOGY -
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E: C .-- Zeeman
INSTITUT des MUTES ETUDES SCIEOTIFIQUES 19 6 3
Seminar on Combinatorial Topology by E.G. ZEEMM
INTRODUCTION The purpose of these seminars is to provide an introduction to combinatorial topology. The topics to be covered are : 1. The combinatorial category and subdivision theorems, 2. The polyhedral category, 3. Regular neighbourhoods, 4. Unknotting of spheres, 5- General Position, 6. Engulfing lemmas, 7. Embedding and isotopy theorems. At first sight the unattractive feature of combinatorial theory ao applied to manifolds is the kinkiness and unhomogeneity of a complex as compared with the roundness and homogeneity of a manifold. However this is due to a confusion between the techniques and subject matter. ¥e resolve this confusion by separating into two different categories the tools and objects of study. The tools in the combinatorial category we keep as special as possible, namely finite simplicial complexes embedded in Euclidean space. These possess two crucial properties : i) finiteness, and the use of induction ii) tameness, and niceness of intersection. Meanwhile objects of study we make as general as possible. Our definition of polyhedral category contains not only
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i) polyhedra; ii) manifolds (bounded or not, compact or not), but also the following spaces, which have not been given a combinatorial structure before : iii) non-paracompact manifolds, for example the Long Line; iv) infinite dimensional manifolds, f'^r example the orthogonal group, v) joins of non-compact spaces; for example the suspension of an open interval. vi) f\inction spaces; for example the space of all pieoewise linear embeddings of compact manifold in another manifold. As the examples show, a polyhedral space need not be triangulable, and if it is, it does not have a specific triangulation, but is a set with a structure. The structure is, roughly speaking, a maximal family of subpolyhedra, and the structure determines the topology. Our theory is directed towards the study of manifolds, and in particular of embeddings and isotopies. Recently it has become apparent that combinatorial results differ substantially from differential results; a striking case is S^ in S^ , which knots differentially, and unknots combinatorially. In fact combinatorial theory seems to behave well in, and to have techniques to handle, most situations with codimension
3 . Just as differential theory
behaves well and can handle most situations in the stable range. We shall therefore concentrate on geometry in codimension
3.
This means we shall neglect a number of interesting and allied topics that depend more on algebra, for example
i) codimension 2 ii) immersion theory iii) relations with differential theory.
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Chapter I : TEE COMBINATCEIAL CATEGORY
Simplexes Let E^ denote Euclidean p-space. An n-simplex (n ^ O) A in E^ is the convex hull of n + 1 linearly independent points. We call the points vertices, and say that A spans them. A is closed and compact; A denotes the boundary, A the interior. A simplex B panned by a subset of the vertices is called a face of A , written B < A . Simplexes A,B are .jcinable if their vertices are linearly independent. If A , B are joinable we define the .join AB to be the simplex spanned by the vertices of both; otherwise the join is undefined. Complezes A finite simplicial complex, or complex, K in E^ is a finite collection of simplexes such that (i) if A e K , then all the faces of A are in K , (ii) if A, B £ K , then A n B is empty or a cciEmon face . The star and link of a simplex A K are defined : st(A,K) =|B; A < B j ,
lk(A,K) = [B; AB 6 K ] .
Two complexes K,L in E^ are .joinable provided : (i) if A e K , B ^ L then A,B joinable (ii) if A,A' 6 K and B,B' 6 L , then AB n A'B' is empty or a common face . If K,L are joinable, we define the .join KL = K
Lu
K, B £ L ] ;
otherwise the join is undefined . The underlying point set \ K | of K is called a euclidean polyhedron :
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L is called a subdivision of K if | L | = ] K | , and every simplex of L is contained in some simplex of K . -
Examples,
°
1) Choose a point A
A . Let
mmmmmmmmmmimmmmmmmmm
\
L = (K - st(A,K)) U A A
lk(A,K) .
(
/
.
/
Then L is a subdivision of K , and we say L is obtained from K by A
starring A (at a) . 2) A first derived K^^) of K is obtained by starring all the simplexes of K in some order such that if A > B then A preceedes B (for example in order of decreasing dimension) . Another way of defining K^^^ is to define the subdivision of each simplex, A
•
inductively in order of increasing dimension, by the rule A' = A A' .
(l) Therefore a typical simplex of K
A
.
A
is A'j A •. .A , where A ^^A^...
in K . An r"^^ derived K^^^ is defined inductively as the first derived of an (r-1)"^^ derived. The barycentric first derived is obtained by starring at the barycentres. Convex linear cells A convex linear cell, or cell, A in E^ is a non-empty compact subset given by
I linear equations linear inequalities
f^ = g.
=0
and •
A face B of A is a cell (i.e. non-empty) obtained by replacing some of the inequaties g^
0
by equations g^ = 0 .
The 0-dimensional faces are called vertices. It is easy to deduce the following elementary properties : 1) A is the convex hull of its vertices 2) A is a closed compact topological n-cell, where n + 1 is the maximum number of linearly independent vertices,
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3) A simplex is a cell. 4) The intersection or product of two cells is another. 5) Let X be a vertex of the cell A , and let B be the union of faces of A not containing x . Then A = the cone x B . A convex linear cell complex, or cell complex, K is a finite collection of cells such that (i) if A 6 K , then all the faces of A are in K , (ii) if A, B 6 K , then A n B is empty or a common face .
Lemma 1 : A convex linear cell complex can be subdivided into a simplicial complex without introducing any more vertices. Proof :
Order the vertices of the cell complex K . Write each cell A as a cone A = xB , where x is the first vertex. Subdivide the cells inductively, in order of increasing dimension. The induction begins trivially with the vertices . For the inductive step, we have already defined the subdivision A' of A , and so define A' to be the cone A» = xB' . The definition is compatible with subdivision C' of any face C of A containing x, because since x is the first vertex of A , it is also the first vertex of C . Therefore each cell, and hence K , is subdivided into a simplicial complex .
Corollary 1 : The underlying set of a cell complex is a euclidean polyhedron. Corollary 2 : The intersection or product of two euclidean polyhedra is another. For the intersection or product of simplicial complexes is a cell complex .
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Maps. Suppose K in E^, L in E^ A map f : K -^L is a continuous map | K|
| L^ .
Call f simplicial if it maps vertices to vertices and simplexes linearly to simplexes. Gall f an isomorphism, written f : K = L , if it is a simplicial homeomorphism. The graph I" f of f is defined as usual tl r f = [ (x,fx) ; XfcI K I ] C \ K ! X I L,P+K Call f piecewise linear if either of the two definitions hold : (1) The graph Tf (2)
of f is a euclidean polyhedron
There exist subdivision K' , L' , of K,L with respect to
which f is simplicial . Notice that condition (2) clearly implies condition (1), because the graph of a linear map from a simplex to a simplex is a simplex, and so the graph of a simplicial map K —^ L is a complex isomorphic to K . We shall prove the converse, and therefore the equivalence of the two definitions, in Lemma 7 . Definition (1) is the aesthetically simpler, while definition (2) is the one which is used continually in practice. The reader is warned against the standard mistake of confusing projective maps with piecewise linear maps. For example the projection onto the base of a triangle from the opposite vertex of a line not parallel to the base is not piecewise linear . Infact the graph F f in the square | KI x I L 1 rectangular hyperbola .
K
is part of a
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Lemma 2 : The com"position of two piecewise linear maps is another . Proof :
We use definition (l), Given K r - (Tf X ! Ml)
^ L _£_>. M , let
( ! K| X r g) c E^X
e"" .
Then P consists of all points (x, fx, gfx) , x ^ )K\ Therefore the projection TT : E^ x E^ x E ^ — ^ E^ x E^ maps f" home omorp hie ally onto r~ (g f) . Since f,g are piecewise linear, P is a euclidean polyhedron by Lemma 1 Corollary 2 . The image under the linear projection TT of any complex triangulating P' gives an isomorphic complex triangulating F" (gf) . Hence T" (gf) is a euclidean polyhedron, and gf is piecewise linear . Definition : Lemma 2 enables us to define the combinatorial category Q with objects : finite simplicial complexes maps : piecewise linear maps. We shall also need the subcategory of embeddings
with
the same objects ^maps : injective piecewise linear maps. We proceed to prove some useful subdivision theorems . Lemma 3 : If K 3 L , then (i) any subdivision K' of K induces a subdivision L' of L , and (ii) any subdivision L' of L can be extended to a subdivision K' of K . Proof :
(i) is obvious (ii) subdivide, inductively in order of increasing dimension, those simplexes of K-L that meet L , by the rule A' = A A' A
where A is an interior point .
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Corollary : Given a simplicial embedding f ; K L
, and a subdivision K'
of K . there exists a subdivision L' of L such that f ; K'-->L' is simplicial . Lemma 4 : If IKIDiLI
, then there exists an r"^^ derived K^^^ of K and (r) a subdivision L' of L such that L' is a subcomplex of K .
Proof :
By induction on the number of simplexes of L . The induction starts trivially when L =
. If A is a principal simplex of
L (principal means not the face of another), then by induction (r-1) choose K to contain a subdivision of L-A . (r) (r-l) Choose a derived K , starring each simplex B ^r K at 0
P
B
0
a point in A n B if A meets B , and arbitrarily otherwise (r) Then K contains subdivision of L-A, A and hence of L . Corollary 1. If | K j = j L \ , then K, L have a common subdivision . Corollary 2. If |K|;;>|LJ
,i=
then there exist subdivision K' , L'^
such that all the L'^ are subcomplexes of K' , Corollary 3. The union of two euclidean polyhedra is another . For subdivide a large simplex containing them both, so that each appears as a subcomplex . The union is also a subcomplex . Lemma 5 : Given a simplicial map f ; K—>L , and a subdivision L* of L , then there exists a subdivision K' of K such that f ; K'-->L' is simplicial . Proof :
Let K^ = f "'L' , which is a cell complex subdividing K . By Lemma 1 we can choose a simplicial complex K' subdividing K^ , introducing no new vertices. Then each simplex of K' is
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mapped linearly to a simplex of L' , and so f : K'-^L' is simplicial . Definition : A map f : K-^E*^ is linear if each simplex is mapped linearly Lemma 6 : Let Q be the inclusion L C E . Given a map f ; K-»L , such that ef : K-^E*^ is linear, then there exist subdivisions K', L' of K , L with respect to which f is simplicial .
Proof :
If A. 6 K , let B. = f A. . 1 ' X I By linearity B. is a cell, possibly of lower dimension than A. I Li: By Lemma 4 Corollary 2, choose simplicial and BM 1 subdivisions L', BJ of L , B such that each BJ is a -1
subcomplex of L'
Then for each i, f
^
is a cell complex
subdividing A^ , and the union f ^L' is a cell complex subdividing K . By Lemma 1 choose a simplicial subdivision K' of f "'L' , introducing no new vertices . Then f : K'->L'
IS
simplicial .
Lemma 7 : The two definitions of piecewise linearity are equivalent . Proof :
We have observed (2) =4s(l) trivially. Therefore we shall prove . Suppose K in E^, L in E^ and let f : K —? L be a map whose graph I" f is a euclidean polyhedron. In other words, there exist a complex M in E^^^ such that I M 1 = r f . P The projection E x E
Q
I' 1
p
V E-^ maps M homeomorphically
onto K , and linearly into E^ ; therefore by Lemma 6, there exist subdivisions M', K' with respect to which Tf^ is simplicial . Hence TT^ : M'—^K'
is an isomorphism . Similarly
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yr ^ : E^ X E^—^E^ maps M into L (not necessarily homeomorphieally), and linearly into ES therefore there exist subdivisions M" , L' with respect to which TT^ is simplicia: Let K" be the subdivision of K' isomorphic to M" . Then f is the composition of the simplicial maps ir. K"
Hence f : K K
L'
^ M"
L is piecewise linear by definition (2)
EP
Let T be a finite subset of G
, such that if a map is in T so
is its range and domain . The diagram of T is the 1-ccmplex obtained by replacing each complex by a vertex and each map by an edge. Call T a tree in Q
if its diagram is simply-connected. Call T one-way if each
complex is the domain of at most one map. Therefore in a one-way tree there is exactly one complex that is the domain of no map, and every other complex is the domain of exactly one map . Gall T simplicial if every map of T is simplicial . Gall T' a subdivision of T if it has the same diagram, and each complex of T' is a subdivision of the corresponding complex of T , and each map of T' (qua map between the underlying polyhedra) is the same as the corresponding map of T . Theorem 1 : If T is a one-way tree in O
, or a tree in
« then T has
a simplicial subdivision . Proof by induction on the number of maps in T . Let T be a one-way tree in
. The induction begins trivially with no maps .
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Suppose T has at least one map. Then there exist complex K and a map f : K — i n
T , such that K is not the range
or domain of any other map in T . By Lemma 7 , there exist subdivisions K', L' of K , L with respect to which f is simplicial . Let
be the one-way tree obtained
from T by omitting K and f , and replacing L by L' , By induction there is a simplicial subdivision T^ of T^ . In particular T^ contains a subdivision L" of L' . By Lemma 5 there exists a subdivision K" of K' such that f : K" —>L" is simplicial. Let T' = T^ together with K" ans f . Then T' is a simplicial subdivision of T . Now suppose T is a tree in ^ , not necessarily one-way. There is a complex K which is the range or domain of exactly one map . If K is the domain, proceed as before. If K is the range , let the map be f : L K Proceed as before, except that we can use the Gollorary to Lemma 3 instead of Lemma 5 to form K" , since f is an embedding . The proof of Theorem 1 is complete .
The following two examples show that the hypotheses of Theorem 1 are necessary as well as sufficient . /O Example 1 . If is necessary that a tree in
be one-way ,
otherwise it contains a subtree
We can choose f , g so that there exists no simplicial subdivision as follows :
,
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Let K - L = M = I , the unit interval, and let
f
0
^ 0 1
1/3 f map /
—> 0
1
0,1/3] , f 1/3,1 0
> 0
2/3 g map
1
linearly ,
-
1 ^ 0
*0,2/3J , (2/3,1]
linearly
Suppose there is a simplicial subdivision, containing K' . Let p,q,r be the nianbers of vertices cf K' between, respectively, 0 and 1/3, 1/3 and 2/3, 2/3 and 1 . Since f is simplicial on K' , we have p = q + 1 + r . From g similarly, p + 1 + q = r . Hence q = -1 a contradiction . Therefore there is no simplicial subdivision .
Example 2 . It is necessary that the diagram in H) be a tree, otherwise it contains a circular subdiagram
We can choose the maps so that there is no simplicial subdivision as follows Let all the complexes be I , and all the maps be the identity except f , and let
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f map \
0
0
1/3
2/3
1
1
^0,1/3] |j/3 , lj
linearly
Suppose there is a simplicial subdivision containing f : K ' — L ' . Going round all the other maps we have the identity map simplicial, and so K' = L' , Using the same notation as in Example 1 , since f is simplicial, we deduce p = p + 1 + q. Hence q = -1 , again a contradiction. Therefore T has no simplicial subdivision .
Remark . A "commutative diagram" in S has simplicial subdivision if the maps are determined by a maximal tree. For example :
>-
is determined by N'
but is not determined by a tree.
INSTITDT DES HAUTES ETUDES SCIEmiFIQUES 19 6 3
Seminar on Combinatorial Topology by E.G. zsmm
Chapter 2 : THE POLYHEDIUL CATEGORI
In this chapter we give mainly definitions and examples to describe the category. We omit the proofs to most statements to make the reading easier, and because later chapters do not depend on them. Let X be a set (without as yet any topology). A polyhedron in X is an injective function f : K
X where K is a finite simplicial complex .
By a function we mean, as usual, a function from the set of points of the underlying euclidean polyhedron |K| to the set X . We write dom f = K , Two polyhedron f^ : K^ — X third f^ : K^
im f = f K .
and f^ : K^ —> X are related if there is a
X such that i) ii)
im f^
im f^ r\ im f^
f-^f^ , tff^ e S
.
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A family
of polyhedra in X is a set in which any two are related .
Write i m Y = ^ i m f ; f e 7 } ' A polystiructure. (or more briefly a structure), ^ on X is a family such that i) im ii) im
covers X is a lattice of subsets of X
Ui) y t =
?
The last coniition means that given K ^ ^ L — X a piecewise linear embedding, then fg ^
with f 6
and g
.
A poly space X = (X, 'J-) is a set X together with a polystructure
on X
T 0 p o 1 .0 g y of the structure 'J- is the identification
The topoloisy topology
X = dora ^ / C? . Here dom J means the disjoint xinion of the euclidean polyhedra |dom f ; f
I , and the identification is given by J's dom
—^ X .
V/e can deduce (non-trivially) : i) Each f : K
X is a homeomorphism into
ii) A set U C X is open (or closed) if and only if U '1'f K is open (or closed) in f K , each f
.
If X is a topological space, then a polystructure on X is one with the same topology . Example 1 . The discrete structure on a set X is given by maps of points into X . This gives the discrete topology .
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Example 2 . The natural structure
(E'^) on Euclidean space E^
is the set of all piecewise linear embeddings K
E^ . This gives the
natural topology . Example 3 . The natural structure
(K) on a complex K is the
set of all piecewise linear embeddings L —^ K . The natural structure on the euclidean polyhedron I K) is the same . Example 4 . Suppose f : } K|
X is a homeomorphism from a
euclidean polyhedron onto a topological space X . Then f J- (K) gives a polyhedral structure 'J- (x) on X . ¥e call X , with this structure, a polyhedron . Notice that 'Tp- (x)
contains the triangulation f , and all related
triangulations . Conversely the structure is uniquely determined by any triangulation in it . Remark 1 , We have used the word polyhedron in three ways i) euclidean-polyhedron ii) polyhedron-in-a-set iii) polyhedron . The usage is coherent, because (i) with its natural structure is an example of (iii) , and the image of (ii) vrith its induced (see below) structure is an example of (iii) . Remark 2 . It is possible to have many structures on a set ; more examples are given below . However it can be shown (nonr-trivially) that 1) Any structure is maximal with respect to its topology : the topology of a strictly smaller structure is strictly finer (more open sets) . 2) The natural structures of e'^ and of polyhedra are maximal .
4 -
cu • 13 s p -a c e s Let X = (X j'^) be a polyspace .If Y C X , we define the induced structure on Y to be jimfCY] . It is easy to verify that ¥e call Y - (Y ,
| Y is a polystructure on Y .
I Y) a polysubspace if it has the induced topology : T (g I Y) = TC:?.) I Y .
In general T(1?| Y) has a finer topology . Example 1 . E^ is a polysubspace of s'^ . This is a particularly satisfactory example , because combinatorially it is always a little embarrassing to regard the infinite triangulation of E'^ as a satisfactory "substructure" of the finite triangulation of S^ . We state elementary properties of polysubspaces, leaving the proofs to the reader : i) Any open set of X is a polysub^ace . ii) Any polyhedron in X is a polysubspace . iii) A polysub^ace of a polysubspace is a polysubspace . iv) The intersection of two polysubspaces is a polysubspace . Therefore the notion of polysubspace substantially enlarges the concept of "tame" set to include both polyhedra and open sets . Example 2 . The union of two polysubspaces is not necessarily 2
poly . For example let A = open disk in E B = a boundary point .
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Then A
B , with structure 7 I A u B is locally compact; a compact
neighbourhood of B in A u B is a closed disk D , having B on its boundary , and with D - B C A . But with the induced topology A L>B is not locally-compact, because B has no compact neighbourhood. 2
Example 3 . A circle in E
is not a polysubspace , because the
induced structure is discrete . 2 Example 4 . A closed disk in E
is not a polysubspace . With
the induced structure it is non-compact ; any subset of the boundary being closed . It is like the PrUfer manifold with each attached disk shrunk to a point .
Maps A function f : X —fe Y between two polyspaces is called a polymap if f 7 (X) C
(Y) T2 .
In other words > given g
, then fg can be factored through the
structure of Y , fg = g'f' for some g' 6
(Y) , where F
is piecewise
linear , f' K —^ > L Ig
g' f
X
^Y
It is easy to deduce 1) A polymap is continuous with respect to the structure topologies , and is therefore a map between the underlying topological spaces . 2) f ; K
L is a polymap if and only if it is piecewise linear .
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3) Identities and compositions of polymaps are pol3miaps. Therefore we can define the polyhedral category
to consist of polyspaces
and poljnnaps . Call a polymap a polyhomeomorphism written f : X ~_Y , if f 3r(x) ¥e deduce
. 1) it is a homeomorphism, and 2) f ^ is also a polyhomeomorphism .
Call a polymap a polyembedding , written f : X C Y , if it is an embedding , (i.e. a polyhomeomorphism only a polysubspace of Y) . tfe deduce
3) f : X->Y is a polymap if and only if its graph 1xf:
X~>XxY
is a polyembedding .
Remark It would be natural to call a polymap f : X—>Y in.iective if f 'J(X) = 7(y) I f X . This definition is weaker than polyembedding , because it does not require the image f X to be a polysub^ace of Y . But it is of interest for the following reason . Consider the categories : (1) ^ace and embeddings (2) polyspaces and infective polymaps (3) polyspaces and polyembeddings. Then (I) n (2) = (3) . Wow some constructions such as join and mapping cylinder are fiinctorial in (2) but not in (1) , and therefore not in (3) . For these constructions the polystructure is more natural than its accompanying topology .
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Bases . A base /f"^ for a polystructure on a set X is a family of polyhedra such that im i^i covers X (i.e. only the first structure axiom) . As with structures, the topology T
is the identification topology
X = dom hl/S^ . We say fi is a base for
i)
c
if
y
ii) every set of im ^ is contained in a finite union of sets of im ^ , We can deduce 1) Every structure has a base (trivially) . 2) Every base is the base for a unique structure] and the base and structure have the same topology . Example 1. Any polyspace has a base of simplexes, = ^f
; dom f = simplex '( . Example 2. E^ has a base of all n-simplexes . Example 3. A polyhedron X has a base of one element , namely a
triangulation f : K —> X . 2 Example 4 . The Woven Square . Let X be the square I . Let fj be the base consisting ef all horizontal and vertical intervals, or, more precisely, all horizontal and vertical linear embeddings of I . The resulting structure is smaller than the natural structure, because it contains no 2-dimensional polyhedra . The resulting topology is finer than the natural topology, and is therefore Hausdorff, but is not locally compact , nor simplyconnected . A typical open neighbourhood of a point looks like a maltese cross. Any subset of the diagonal is a closed set . Example 5 . (The pathological Woven Square). We enlarge the structure of the Woven Square by weaving in one more thread so badly, that
-
-
it produces a non-Hausdorff topology . Let d: 1—^1
te the diagonal map ;
and let e : I — b e the function that is the identity on the irrationals, but reflects tho rationals about the mid-point . ¥e add to the base of the structure of the Woven Square one more element , the polyhedron f = de : 2
I — ^ I . The topology of the Woven Square is thereby coarsened , so that the ends of the diagonal cannot be separated by disjoint open sets . (The proof uses measure theory, and depends upon the non-countability of the base) . Definition, ¥e call a base
topological if it is also a base
for the topology T(E) in the following sense : given x 6 X , there exists f G B
, such that im f is a (closed) neighbourhood of x in X in the
topology
. For instance , in Example 2 above , the set of all
n-simplexes in E^ is a topological base , But in Example 4 , the base for the woven square is not topological . The structures for infinite manifolds and function spaces that we give below will not be topological . Triangulable
Spaces
The pathological examples 4 and 5 above indicate some of the consequences of the definitionsof polyspace . However since our interest lies towards manifolds, we do not stress the pathology , but rather use it to obtain insight into the structure of important polyspaces such as function spaces . One of the advantages of polyspace is that it is more general than the triangulable space , even if we use infinite triangulations . In fact we avoid infinite triangulations , because we regard them as alien to the subject , being too diffuse a tool , and defining too restrictive a space . The algebraic elegance of infinite complexes should not be confused with their geometric limitations . However it is worth mentioning the relationship between polyspaces and triangulable spaces . Given a polyspace X , then there are six possibilities : i) X is a polyhedron , i.e. its structure contains finite triangulations .
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ii) X is not a polyhedron , but we can enlarge the structure of X to be a polyhedron - for example the woven square . iii) There is a locally-finite infinite triangulation f : K
X , whose
restriction to any finite subcomplex is in the structure - as for example in E^ . If X is connected , then a necessary and sufficient condition for this is that the structure have a countable topological base . A consequence is that the topology is paracompact , Hausdorff , and locally compact . iv) The structure can be enlarged to give (iii) . v) The structure is maximal , but (i) and (iii) are not true ; for example the Long Line (see below) . vi) The structure is not maximal , but (ii) and (iv) are not true ; for example the pathological woven square, or
-dimensional
manifolds, or function spaces (see below) .
Compactness Qixestion 1 . Is a compact polyspace a polyhedron ? The answer is yes if it has a countable base , or if it ' has a topological base , but is unsolved otherwise . Question 2 . Does the lattice of compact subsets of a polyspace refine the lattice of polyhedra ? The question is important for studying the homotopy structure of function spaces .
Manifolds An n-polyball is a polyhedron triangulated by an n-simplex . An n-polysphere is a polyhedron triangulated by the boundary of an (n+l)-simplex.
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Definition : an n-polymanifold M is a polyspace , each point of which has an n-polyball neighbourhood . More precisely , each point has a closed neighbourhood (with respect to the structure topology) which is a polysubspace , and which, with the induced structure , is an n-polyball . The boundary M is the closed polysub^ace of those points which lie on the boundary of their neighbo\irhoods , and is an (n-l)-polymanifold . The interior M = M - M is the complementary open polysubspace . We call M closed if compact and M = jZi . bounded if compact and M
^.
open if non-compact and M = j^S . If M compact then any triangulation in the structure is a combinatorial manifold (i.e. the link of every vertex is an (n-l)-sphere or ball according as to whether the vertex is in the interior or boundary ; the proof is by verifying that the property is invariant under subdivision, and true in an n-simplex) . Example 1 . The Long Line is obtained by filling in (with unit intervals) all the ordinals up to the first non-countable , and is given the order topology . Then it can be shown (non trivially) that the Long Line has a 1 - polymanifold structure, although it is non-paracompact, and therefore non-triangulable . Example 2 . The Prufer-manifolds are non-triangulable n-manifolds , n^ 2.
Direct
Limits Suppose X^ , n = 0, 1, 2,..., is a sequence of polyspaces , such
that, for each ^ > ^^^ is a polysubspace of
. Define the limit structure
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on X = UXn to be ^(X) = U'^(X^) . The topology of
is the same as the limit topology .
Example 1 Let X^ = e'^ . Assume E^ C e'^"' linearly , Then
U E^^ is Euclidean oO -space . This is not to be confused with ,
nor homeomorphic (in either topology) to , R
, Hilbert ^ace , which is the
product of countable copies of the reals . Example 2 = point ; B^ = S B^"'' , the suspension * Then B^ is an
Let
n-polyball , and B ^ = TJ B
the
with , nor is homeomorphic to , I
-polyball , This is not to be confused , the Hilbert cube .
Example 3 Let S° = two points ; S^ = ss""'' , the suspension . Then S^ is an n-polysphere, and S®'"' == US^ the
-polysphere . It is true that
has S
as a closed subpolyspace , with complementary open subspace
B''^- S
= E .
B^
Nevertheless we do not call these boundary and interior
because it is fairly easy to show polyhomeomorphisms E ^ -g B ^ Therefore B
^ S
.
is homogeneous without boundary , because S
is .
Example 4 Let 0 = U of
be the infinite orthogonal group . Any triangulation
can be extended to a triangulation of
define a polystructure on 0 .
. The resulting structures
- 12 -
Infinite manifolds . The above definition is good for n = o^ . The above examples are all infinite manifolds. Similarly other classical groups , and the infinite Grassman and Stiefel manifolds . We observe that an t?o -manifold has no boundaiy because an co -ball has no boundary .
Products. Let K,L be complexes in E^ , E*^ . Then K x L is a cell , and so the natural structure 'j (K X L) is uniquely
complex in
defined . Given now two poly^aces X, Y , define the product structure . (X x Y)
^(f X g) h ; f $
(X) , g 6 'J (Y), h E J (domf x domg)
f X
h
\
We can deduce : 1) The product is functorial on ^ . 2) A function f : X 1xf:X--)XxY
Y is a polymap if and only if the graph
is a polyembedding .
Joins The topological join X * Y of two spaces X , Y is obtained from X u (X X I X Y) all X
Y by identifying x = (x, 0, y), y =(x, 1, y)
X , y £ Y , and giving the identification topology .
- 13 -
If X , Y are polyspaces we define the .join structure ^ (X * Y) as follows . Given K in E^ , L in E^ , we identify E^ , E^ with E^ X 0 x O , O x E
x1
in E x E
x 1 , C
The images of K, L are joinable in
and we define K
join . The complex K * L has a natural structure 5R (X * Y)
L to ho their
{k * L) . Define
I' (f * g)h ; ffe7- (X) , g E ? (Y) , h S 7 (domf ^ domg) .
¥e can deduce that the topology of
(X * Y) is the same as the topology
of the join X * Y above .
Remark The join * is functorial on the category of maps, but not on the subcategory of emboddings . Give X' C X . Y' C Y then X' * Y' does not always have the topology induced from the inclusion X- * Y' C X * Y , 0
For example let X = I , X' =: I , and Y
Y' = a point , Then the cone on
0
0
I is not a subspace of the cone on I j the cone on I has a finer topology than the irduced topology , and is locally compact at the vertex , whereas the induced topology is not (cf. the polysubspace Example 2) . On the other hand the join i_s functorial in the category of polysubspaces and infective polymaps; the naturaliiy in the category dictates the topology to be chosen on the join , which is then not functorial in the subcategory of polyembeddings . The explanation is that the concept join is essentially a combinatorial idea, and so as we should expect, in this context the polystructure is more basic than the topology .
Function
Spaces
X Let X be a polyhedron , and Y a polyspace ^ Let Y of polymaps X—)Y c We define the function space structure
be the set
^ (Y^) on Y^
- 14 -
as follows . If f : K Y
is an infective function , let f : X x
be the associated function given by f'(x , k) = (fk) x . Define ^ (l^) = I f ; f' is a polymap J. , Lemma (Hudson)
Any two such f
are related , s
Therefore is a family of polyhedra in I
and the three
axioms for a polystructure are easy to verify. ¥e can deduce the following properties 1) The structure of Y^ is functorial on Z,y in words if f:
and g :
. In other
are polymaps, then the induced function
/ = T^-Y^'l is also a polymap . 2) If X,Y are polyhedra and Z a polyspace then there is a natural polyhomeomorphism
Remark 1 . If X not a polyhedron (not compact) then the above 2 definition does not give a polystructure . For example if X = E' , Y - E then Hudson's theorem fails; there exist two f's that are not related . Remark 2 . The topology of the structure is strictly finer than the compact open topology , and is therefore Hausdorff .If Y is a polyhedron or a manifold then both topologies give the same homotopy structure on Y . (Question : is this true for general Y ?)
I s0t 0py Let (X c Y) denote the polyspace of polyembeddings of X in Y , X X with structure induced from Y . (Question : is it a polysubspace of Y ?)
- 15 -
One of the main reasons for the way we have developed the theory is that the following four definitions of isotopy are now trivially equivalent . A polyigotopy of X in I is i) a point of (x cl) ^ ii) a polymap I —J (x c X) iii) a polymap X x
, which is a polyembedding at each level
iv) a level-preserving polyembedding X x I —^ Y x I . If f,g : X — M
are the beginning and end points of the isotopy , we say the
isotopy moves fX onto gX , and that f , g are isotopic . Let H(Y) denote the polyspace of polyhomeomorphisms of Y onto
Y itself, with structxire inducec from Y . An ambient polyisotopy of Y is a polyarc in H(Y) starting at the identity, and finishing at e , say . If X is a polysubspace of Y we say the ambient isotopy moves X onto eX « If f: X-^Y is a polyembedding (or polymap) we say f, ef are ambient isotopic , Later we shall prove a theorem of Hudson , which says that the notions of isotopy and ambient isotopy coincide for manifolds of codimension >/ 3 ' In codimension 2 they are essentially different, because ordinary knots in E^ oan be untied by isotopy , but not by ambient isotopy . Remark If Y = E" , there is another definition of isotopy favoured by some writers , which we call linear isotopy, and it is worthwhile analysing the difference . A linear homotopy of X in E^ is constructed as follows : choose a fixed triangulation K of X , and for each vertex v £ K , a polymap
:I
E^ . For each t , let g^ : K —^ E^ be the linear map
determined by the vertex map v—»f^(t) . Then ^g^}
or g : K x I —>• E"
is the linear homotopy .If g is an embedding at each level we call g a linear isotopy . We make the following observations :
- 16 -
i) Not every linear isotopy is poly, because in general the track g(K X l) left by the linear isotopy is a curvilinear ruled surface rather than a euclidean polyhedron . ii) Not every polyisotopy is linear, as shown by the example below . iii) If two polymaps are linearly isotopic then they are polyisotopic . The converse is also true (non-trivially) if X is a manifold of codimension ^ 3 . iv) Linear isotopy is not functorial. ¥e justify this last statement by defining a polystructure on (XC E'^) that exactly captures linear isotopy ; more precisely we shall construct a polystructure, that linear homotopies are the polymaps I — w i t h
say , on (E^)^ such respect to
, L
and linear isotopies are the polymaps I
(X c E ) with reject to the induced
structure . Define
as follows: if K is a triangulation of X with k L vertices , then the set M^^ of linear maps K - — c a n be given a polystructure M ^ S E ^ . If K» is a subdivision of K , then M^. ^^ M^^, is a polyembedding . Therefore (E ) = iU M , the union taken over all triangulations in the K ^ structure of X , and 'X is defined by the limit polystructure . We shall L show that if E —^E
is a polyhomeomorphism, then the induced function
(E^)^ — ? (E'^)^ , which is a polyhomeomorphism with reject to the function space staructure, is not even continuous with reject to '3' * L 2 Example 1 . Let X = I and Y = E , and consider the isotopies 2 I —?(l C E ) performed by a caterpillar crawling firstly along a straight twig , and secondly along a bent twig . The first isotopy , f say , is linear , and therefore also poly . The second isotopy , g say , is poly but not linear , because we cannot describe it in terms of a fixed triangulation of X = L . This shows
^
. Now suppose the caterpillar
- 17 -
performs g by starting with his nose, and finishing with his tail, at the 0
bend in the twig . Then g I is a closed set in the topology of ^
,
0
whereas fl is not . There is an obvious polyhomeomorphism
of E"^
bending a straight twig into a bent twig , and the induced map of (l c E^) 0
0
into itself maps fl into gl . Therefore it cannot be continuous with respect to the topology of 3^ . L The explanation is that
is functorial on X,Y
^ , whereas
^ is functorial only on X ^ fj^ and Y in the subcategory of euclidean spaces and linear maps . Since our theory is directed towards isotopies of manifolds in manifolds, we favour
and reject
'J' .
L Exanple 2 . Let h" denote the set of polyhomeomorphians of E^ onto itself having compact support . The hypothesis of compact support enables us to define on H^ , as above , both a function ^ace polystructure and a linear polystructure
,
Let H^ , H^ be the resulting topological
spaces, both having H^ as underlying set . Then it appears that H^ , H^ have different homotopy structures . By Alexander's Lemma on isotopy , it is easy to show that H^ is contractible . However Kuiper has used the queer 7 6 differential structures on S to show that either TT° (H„) 0 5 or 0 . This is essentially a phenomen on of codimension zero .
Degeneracy Let f: X-->Y be a polymap . Define the non-degenerate structure (f) of f . by ^(f) Note that in general fg ^ degenerate if
{grr'J(x) 5 fg<£ ^ ( Y ) }
.
bccausc it is not injcctivc . Call f non-
(f) is a base for 7 (x) . Otherwise f is degenerate .
Example 1 . A polyembedding is non-degenerate «
- 18 -
Example 2 . A polyimmersion (local embedding) is non-degenerate . Example 3 . A simplicial map is non-degenerate if and only if it maps each simplex non-degenerately . Example 4 . We shall show that any map of a polyhedron of dimension ^ n to an n-manifold can be put into "general position" where it is nondegenerate .
The
mapping
cylinder
problem
The problem is to define a natural structure on the mapping cylinder G- of a map f : X
Y . We explain why this problem is, in a sense,
insoluble . 1. Topological . The topological mapping cylinder C is obtained from X X I U Y by identifying (x,l) = f x , all x 6; X , and is given the identification topology . Then C
is functorial on the category of maps .
2. Combinatorial . Suppose f : K
L is a simplicial map .
Whitehead gave a rule for defining the simplicial mapping cylinder , G say , of f , which is a triangulation g : G
C of the topological mapping
cylinder . This riile is functorial on the category of simplicial maps , but not on the category of piecewise linear maps . For suppose K', L' are subdivisions of K , L , giving rise to the simplicial cylinder g': G'--^ C . Then , although G,G' are piecewise linearly homeomorphic , g , g' are not in general related . Therefore the identity maps
K , L'-> K induce the
identity G --> C , but only a piecewise projective map G' —^ G . 3. Polyhedral . The inclusion C C X * Y of the mapping cylinder in the join induces a natural polystructure
on C that is functorial
cn the category of polymaps . However ^ (C) gives the wrong topology (too fine a one) .
Example 1 . The identity on I has mapping cylinder a square , and polystructure the Woven Square (of example 4 above) .
- 19 -
Example 2 . The mapping cylinder of a simplicial map of a 2-simplex onto a 1-simplex epitomises the problem , because when embedded
3 in E
it looks like the prow of a ship .
The structure "^(c) has a base consisting of all horizontal sections , and all vertical sections going athwartships , but no 3-dimensional stuff . Example 3 . If f is simplicial , then the simplicial mapping (c) . In other words , ''^(c) can be
cylinder G-:>-C is related to
enlarged (non-naturally) to contain any simplicial cylinder, and is the intersection of all the structures determined by the simplicial cylinders . Example 4 . On the subcategory of non-degenerate polymaps the natural structure
(C) can be enlarged to a structure
that (i)
is functorial on this subcategory (ii) contains all simplicial cylinders , and (iii) gives the correct topology . A base for j'"^ (C) is (B = 7 (Y)
Concluding
U [ (g X 1) h ; g 6
(f) , h 6 7 (dom g x l)j
Remarks
¥e can enlarge or change
by enlarging or changing the tool G- .
Example 1 . Enlarge Q, to contain piecewise projective maps . Example 2 . Further enlarge (3 to contain piecewise algebraic complexes and piecewise algebraic maps. Then algebraic varieties in E^
- 20 -
would become polysubspaces . Example 3 • Replace and differential maps . Then ^
by the category of open subsets of E^ would be the category of differential
manifolds and differential maps. Gabrielle has pointed out that a polyspace is equivalent to a contravariant fuctor from (3 to the category of sets and functions y obeying two axioms of intersection and .union; a polymap is a natural transformation between two such functors .
INSTITUT DES HAUTES ETUDES SCIEKTIFIQUES 19 6 3
Seminar on Combinatorial Topology by E.G. ZEEMM
Chapter 3 : REGULAR NEIGHBOURHOODS
From now on we shall omit the prefix "poly", and whenever we say space, map, manifold, etc., we mean polyspace, poljnnap, polymanifold, etc...
Lemma 8. A convex linear cell is a ball Proof : Given a convex linear cell B we have to exhibit a specific piecewise linear homeomorphism from a simplex
onto B . Since B is in
some Euclidean space, we can choose A 3 B . Let X be a point in § .* Then«radial projection from x gives a homeomorphism A — B , but this is not piecewise linear by the Standard Mistake- We get roimd this difficulty by defining a pseudo radial projection as follows. Let A' »be the cell•subdivision « of A consisting of all cells A ^ O IX B^ , A'
^ A
, B^ £ B . Let A " be a simplicial subdivision of
. Radial projection of A " determines an isomorphic subdivision B" of
B , and radial projection of the vertices determines the simplicial isomorphism, which is of course piecewise linear. Joining to X gives the required homeomorphism A B
.
Corolla3?v . Joins of spheres and balls obey the rules : i) BPB^-BP-^'^-'^
- 2 -
ii) iii) s^s^ssP^^^^ . Proof. Since the structure of a join is functorial, it suffices to prove one example , i) The join of two simplexes is a simplex ii) In
choose B^, B*^^"' to he simplexes crossing at their barycentres.
Then B^B'^"^'' is a convex linear cell. iii) Take the boundary of ii) . We call a complex J a combinatorial n-manifold if the link of each vertex is an (n-l)-sphere or an (n-l)-ball. Lemma 9 . Suppose I J i = M . Then J is a combinatorial manifold if and only if M is a manifold . Proof . One way is trivial; for if J is a combinatorial manifold , then the closed vertex stars of J give a covering of M by balls, such that each point of M has some ball as a neighbourhood . Conversely suppose H is an n-manifold , and let .x he a vertex of J in 8 . By the definition of manifold (polymanifold), there is a pieoewise linear embedding f : A —> J covering a neighbourhood of where A
is an n-simplex , such that f~''oc
f : A ' —^J' A
A
,
. Subdivide so that
is simplicial; we have pieoewise linear homeomorphisms
y lk(f X , A ' ) _^,lk(x,J')
lk(s:,J)
Where the middle arrow is an isomorphism and the other two arrows are pseudo radial projections . Hence lk(x^J) is an (n-l)-sphere . If .X- is a vertex of J in M , there is a similar situation except 1 * that f" t A . and so it follows that lk(x,j) is a "ballf
- 3 -
Corollary 1 . Let I J I = M be an n-manifold . If A is a p-simplex of J , then either lk(A,J) = (n-p-l)-sphere and 1 C M or lk(A,j) zz (n-p-l)-ball and A C M . Proof . We show the link is a sphere or ball by induction on p , the induction starting at p = 0 by the Lemma . If p > 0 , write A = xB , and then lk(A,j) = lk(x,lk(B,j)), which is the link of a vertex in an (n-p)-sphere or ball, by induction, and is therefore an (n-p-l)-sphere or ball by the Lemma . Any point of A has A lk(A',j) as a closed neighbourhood , and so 0
lies in M or M according as to whether it lies in the interior or boundary of this neighbourhood, i.e. according as to whether lk(A,j) is a sphere or ball, 0
0
Therefore if the link is a sphere then A C M , and if the link is a ball then A C M , since M is closed . Corollary 1 justifies the following definition : if J is a combinatorial manifold, define the boundary J to be the subcomplex J = ^A
J I lk(A,j) = ball J 0
and the interior to be the open subcomplex J = J - J , We deduce at once : Corollary 2. If i Jj
M = manifold, then
1 jI ^M .
Definition . If B^ ^ is an (n-l)-ball contained in the boundary M^ of an n-manifold M^ , we call B'^ ^ a face of m"' and write B" " < M
. ¥e are particularly interested when
=: B
a ball also. Let
A ^ denote an n-simplex . Theorem 2 . If B^"^ < b"^ and
A
< A
then anv
homeomorphism B^ ^ — y A ^ ^ can be extended to a homeomorphism
- 4 -
Corollary . If two balls meet in a cogimon face • then their union is a ball . (For by Theorem 2 the union is homeomorphic to the suspension of a simplex) . Theorem 3 . If B^ C S^ then s'^-B^ is a ball Remark 1 The original proofs of Theorem 2 and 3 were given by Newman and Alexander in the 1920's and 30's and used "stellar theory" instead of combinatorial theory . The essential notion of the proof is to replace the finite simplicial structure of a ball by some ordered finite structure, and then use induction on the number of steps in the ordering (the induction starting trivially with a simplex) . Newman and Alexander used an ordering by stellar subdivisions; we give a new proof here, based on ordering by collapsing . The collapsing technique was invented by ¥hithehead in 1939, and is more powerful than stellar theory because it includes the theories of regular neighbourhoods and simple homotopy type . Notice that some concept of an ordered structure seems vital, because without it we cannot prove : Schonflies Conjecture : If
S^ then the closures of each
component of the complement is a ball . The conjecture is true for n ^ 3 , but unsolved for n > 3 • It is known by Morton Brovm's result that they are triangulated topological balls, but not known whether they are polyballs . Our ignorance of whether they are polyballs when n = 4 implies our ignorance of whether they are even polymanifolds when n = 5 (the links of boundary vertices may go hajrwire) . Remark 2 . The proof of Theorem 2 and 3 is done together by induction on n . The induction starts trivially with n = 0 . We shall show first that
- 5 -
The inductive step is achieved
Theorem 2 is equivalent to Theorem 3 -, n ^ n-1 by showing that Theorem 2 , r <: n r
-T'Theorem 3n Theorem 3 , r < n r The inductive step is long, involving Lemmas 10 - 17 and Theorems 4 - 8 , during which we shall often have to make inductive use of Theorems 2 and 3 . However we can avoid going round in a circle by i) assuming everything to be cf dimension^ n ii) avoiding the use of Theorem 3^ until Theorem 3^ is proved . To emphasise which statements are involved in the induction, and at the same time avoid repetition, we put a star against all those lemmas or theorems which depend upon Theorem r ^ n , and Theorem 3^ , r
and its Corollary ,
n.
Lemma 10 . Any homeomorphism between the boundaries of two balls can be extended to the interiors . Proof . We are given f : B^—^ B^ . Choose triangulations g^ : A
^ B^ . Define h : A — b y
the commutative
diagram A
B, •l
h
A
^
2
Extend h conewise to a homeomorphism h': A
^A •
Then the required homeomorphism f : B^ —^-B^ is given by the commutative
- 6 -
diagram
A
^A ^
gV
B.
— > B,
Theorem 2 is equivalent to Theorem 3 -, n ^ n-1 Proof . Assume Theorem to a point x , we have B^^ ^
. Given B"^^ C
, then joining
xs"^^ . Let A ^ be a simplex with face
A ^ ^ and opposite vertex y . Choose a homeomorphism B'^ ^ — ^ extend it to ball y A n-1
? A ^ . Therefore
Conversely assume Theorem
, and
B'^"^ is homeomorphic to the
^ • Given B^ ^
B^ , then we know
b'^- B^^ is a ball . Therefore given a homeomorphism B^ ^—y A ^ ^ , we can , - • n-1 • n-1 , , , . -n „n-l x v t ^a extend B to a homeomorphism B -B —>. y A > by Lemma 10 . Therefore we have defined B ' ^ — , and can extend to B^—^ A"' , again by Lemma 10 .
Stellar
subdivision Recall from Chapter 1 that an elementary stellar subdivision of K
is given by K' = (K - st(A,K)) u a A lk(A,K)
where a <£ A , Afr K .
A stellar subdivision of K , written cr K , is the resiolt of a finite number of elementary ones . Examples
til i) An r derived is a stellar . ii) If K D L , then any stellar subdivision of K
detemines a unique stellar subdivision of L , and conversely .
- 7 -
iii)
is not a stellar subdivision of a triangle .
Collapsing If K D L , we say there is an elementary simplicial collapse from K to L if K - L consists of a principal simplex A of K together with a free face . Therefore if A = a B , then B L
K=L U A aB=L n A
¥e describe the elementary simplicial collapse by saying collapse A onto a B , 01" collapse A from B . We say K simplicially collapses to L , written iC^ iL , if there is a sequence of elementary simplicial collapses going from K to L . If L is a point we call K simplicially collapsible, and write K
0.
Examples . i) A cone simplicially collapses onto any subcone . For just collapse all the other simplexes in towards the vertex .
/
More precisely let a K be the cone on K , and a L be the subcone on L , where L c K . Then order the simplexes
of K - L in order of
decreasing dimension , and collapse a B^ from B^ , i = l,...,r . ii) A cone is simplicially collapsible . iii) A simplex is simplicially collapsible . Both these are special cases of i) .
- 8 -
¥e now repeat the definition for polyhedra . If X 3 Y are polyhedra , we say there is an elementary collapse from X to Y if there exists such that
X=YU
X
B^-^ . Y n B^
¥e describe the elementary collapse by saying collapse B
onto B
, or collapse B^' from B' - B ^ " .
We say X collapses to Y , written X ^ Y , if there is a sequence of elementary collapses going from X to Y . If Y is a point we call X collapsible , and write X
, For example a ball is collapsible .
¥e now investigate the relationship between simplicial collapsing and collapsing . We write K---^.L if
. The significance of this
last definition is that the balls across which the collapse takes place may not be subcomplexes of K . It is trivially true that K
L
K
L,
but the converse is unknown . Ifhat we can prove is * Theorem 4 • If K of K . L such that K'
, then there exists a subdivision K', L'
L' .
* Corollary 1 . If X "-^^.Y then there exists a triangulation such that K --^i. L . * Corollary 2 . If X is ccllapsible, then there exists a triangulation that is simplicially collapsible . Before proving Theorem 4 we digress a little to indicate the consequences of the definition of collapsing .
- 9 -
Simple
homotopy
type
The relation X' ^ Y between X and Y is ordered . If we forget the ordering, then we generate an equivalence relation between polyhedra called simple homotopy type . Since a collapse is a homotopy equivalence, this is a finer equivalence relation than homotopy tjrpe . It is strictly finer, because, for example, the lens spaces L(7,1, L(7,2) are of the same homotopy type , but not of the same simple homotopy type . But for simply-connected spaces homotopy type = simply homotopy type , and there are simply-connected nonhomeomorphic manifolds of the same homotopy type .
The
Dunce
Hat If we preserve the order X
then the relation between X , Y
is much sharper . Trivially if X is collapsible then X is contractible (homotopywise) . But the converse is not true. For example consider the Dunce Hat D which is defined to be a triangle with its sides identified ab = ac = be . Then D is contractible (although the contraction is hard to visualise) , and so D is the same simple homotopy type as a pointj but D is not collapsible because there is nowhere to start. Although D
, it can be shown that D x I ^ ^ 0 .
Con,iecture . If K'^ is a contractible 2-complex then K x I'-^.O This conjecture is interesting because it implies the 3-dimensional Poincare Conjecture, as follows . Let M^ be a compact contractible 3-manifold; it is sufficient to show that M^ is a ball . Call X is a spine of M if M
. Now M^ has a contractible spine E^ . By the conjecture
M^ X I '^K^ X I , and we shall show in Theorem 8 Corollary 1 that this 3 A 3 • 4. 5 implies M x I = B . Hence M C B = S , and by the Sch'dnflies Theorem M^ = B^ , a ball.
- 10 -
In particular the conjecture is true for the Dunce Hat , and so any M^ , n - 3 , having D as a spine is a ball . This is also true for n ^ 5 because I unknots in ^ 5 dimensions . However it is not true for n = 4 / fact Tf (/,' 4\ 0)\ having D as a spine . because there is an M4 ijfc B 4 (in M) 4 ^ The construction of M is due to Mazur, and defined by attaching a 2-handle to S' X B^ , by a curve in the boundary that is homotopic , but not isotopic , to the first factor :
Lemma 11 . If
L , then we can reorder the elementary
collapses so that they are in order of decreasing dimension . Proof . Suppose K^
K^ Si K^ are consecutive elementary collapses,
the first being across A^ from B^
, and the second across C^ from
.
V/e shall show that if p < q. then we can interchange the order of the collapses (which is not true if P JJ; q) • The lemma follows by performing a finite number of such interchanges . or B,P-1 '
Therefore cS q-1 which is principal in K , remains principal in K, . Also D vfc A or B , 1 because A , B do not lie in K^ , and so D q-1 cannot be a face of A or Since p < q. j C
is not a face of A
- 11 -
B
(again since p ^ q-l) . Therefore D remains a free face of C in K^ .
Therefore , if K* =
- (C L> D) , then there is an elementary collapse
K^ ^ K^ across C from D . Meanwhile A remains principal in K^ , and B remains a free face . Therefore there is an elementary collapse K* N. K^ across A from B , The lemma is proved . Remark . Although Lemma 11 indicates a certain freedom to rearrange the order of collapses, we cannot rearrange arbitrarily . For example if B^ is a simplicially collapsible 3-ball , if we start collapsing B^ carelessly we may get stuck before reaching a point - for instance the dunce hat is a spine of B , so that by mistake we might get stuck at the dimoe hat . This problem is the reason why the methods which classified 2-manifolds failed to classify 3-inanifolds . s Again , if K then trivially
L and K' is an arbitrary subdivision of K , s but we do not know if K'
L' . However we can
prove a more limited result : s Lemma 12 . If K
s L then
Q- K of K . Proof . By induction we may assume both the simplicial collapse and stellar subdivision to be elementary . Suppose K =LUA a B = L n A , and suppose Cr K is obtained by starring C at c . There are three cases (i) If C
A , then the lemma is trivial
(ii) If C B
, then the cone a(
(iii) If C ^ A , but C ^ B , let G = a B^ , B = B^B^ . Then
cT K ^ cT L u cone a(c B^B^) f * \ (y- L (J subcone a(o
\' =
.
- 12 -
*Lemma_13. . If K
L is an elementary collapse , then there exists
a subdivision such that K'
L' and L' is stellar (but K' may not be) .
Proof . Let A = K - L , and B n-ball land B a face . Let By Theorem
A , P
be an n-simplex and an (n-l)-face .
choose a homeomorphism h : A,B — A
K
A n L . Then A is an
, r .
/
'A
Choose subdivisions so that h is a simplicial isomorphism h :
p'
Let IT : A — > T be the linear projection , mapping the vertex opposite ^ to the barycentre of I"" . Choose subdivisions A " , P " of A ' , P ' so that Tf
A"
r"
is simplicial . Call such a subdivision of A cylindrical . Let A" , B" be the isomorphic subdivisions of A' , B' . Let B"' be an r subdividing B" , and let
derived of B ,
P "' the corre^onding subdivision of P " . By
Lemma 5 , choose a subdivision
A
of A " such that
TT : A " '
^^
simplicial , and let A'" be the corresponding subdivision of A" . Then B'" is a stellar subdivision of B , and induces a stellar subdivision L' of L . Define K' =A"'uL' . Since A'"
is cylindrical , A'"
cylinderwise , in decreasing order of dimension . Hence A"' K'
s
B'
and so
- 13 -
Proof
of
Theorem
4
¥e are given a collapse X "iii Y ; that is to say a sequence of .elementary collapses I
T ^ r-1
'
By Theorem 1 we can find a subdivision K^ of K , such that , for each i , there is a subcomplex K^ covering X^ . Therefore we may write the elementary collapses
If r = 1 the result follows by Lemma 13 . If r > 1 we show the result by induction . Assume we have found a subdivision K' ^ of K ^ such that s K' ^ K' . B y Lemma 3 extend K' ^ to a subdivision K' of K . r-1 o r-1 r r Apply Lemma 13 to the elementary collapse ^^ » obtain a simplicial collapse K" ^ ^ K" , , where K" ^ is a stellar subdivision r ^ r-1 r-1 of K' ^ . The latter fact enables us to appeal to Lemma 12 to deduce r-1 K" K" , and so K" K" . r-1 ^ 0 r 0
Full
subcomplexes If K C J
are complexes , we say K is full in J if no simplex
of J - K has all its vertices in K . ¥e can deduce the elementary properties of fullness : (i) If K C J , and J' a first derived complex of J then K' is full in J' . (ii) If K full in J , and J* any subdivision of J , then K* full in J* . (iii) If K full in J , and A a simplex of J , then A H K is empty or a face of A .
- 14 -
(ir) If K full in J , then there is a unique simplicial map f : J — I (the unit interval) such that f =K.
Neighb 0urh oods Let J he a complex and let X C | J | , The simplicial neighbourhood N (X,J) is the smallest subcomplex of J containing a topological neighboirrhood of X . It consists of all (closed) simplexes of J meeting X , together with their faces . Now suppose X is a polyhedron in an n-manifold M . We construct derived neighbourhoods of X in M as follows . If M is compact choose a triangulation J , K of N , X . If M is not compact choose a triangulation J , K of M^ , X where M^ is a subpolyhedron containing a topological neighbourhood of X in M . Now in general M^ will not be a manifold round the edges , but it will be a manifold near X , which is all that matters . AC More precisely , if A £ N (X , J) then Ik (A,J) = . ACM For simplicity of exposition we identify M^ =
I J/ » X = |K( .
Choose now an r^^ derived complex J^^^ of J . Call N = N
an r^^ derived neighbourhood of X in M .
If r = 1 and K full in J we call N a derived neighbourhood of X in M . A fortiori if r > 2 , then any r"''^ derived neighbourhood is a derived neigh(r-l) (r-l) bourhood, because K^ ' full in J^ ' . If J' , J" denote first and second deriveds, it is easy to show that (i) N (X , J') = U
St (x,J'), the union taken over all vertices
X £ K, (ii) N (x , J") = LJ st (A , J " ) , the union taken over all simplexes A A K , where A denotes the point at which A is starred in J'
- 15 -
Lemma 14- . Any two derived neighbourhoods of X in M are homeomorphiG , keeping X fixed . Proof . Let
N (X , J'^) ,
= N ( X , J'^) be the two given
neighbourhoods . If M is compact , let J^ be a common subdivision of '
' ^^
compact choose subdivisions of J^ , J^ that intersect
in a common subcomplex , and let J^ be this subcomplex . Choose a first derived
J'
0
of
J
0
and l e t
W = N (X , J ' ) • 0
'
o
Let f : J^—i, I be the unique simplicial map such that f 0 = X , which exists by the hypothesis of fullness . Choose S < f X , for all vertices x
> o and such that
J^ , x ^ X . Let J^ (i = o , l) denote
a first derived of J. obtained by starring Aer J. on
f ^^
and arbitrarily otherwise . Then i N (X , J^- ) I = f"^ [ 0 ,
if f A = I ,i= 0 ,1.
Therefore N^ ^ W (X , jy , isomorphic K N (X , J^) , homeomorphic by identity map N^ , isomorphic s
» similarly .
Remark . Lemma 14 fails for first derived neighbourhoods without the fullness condition, which indicates the reason for having to pass to the second derived in general to obtain a derived neighbourhood. For example suppose X is the boundary of a 1-simplex in J . Then the first derived neighbourhood is connected , but the second derived is not .
- 16 -
Corollary . Any derived neighbourhood of X in M collapses to X « Proof . By Lemma 14 it suffices to prove for one particular derived neighbourhood. Therefore choose a triangulation J , K of M , X such that K is full in J , and'let K = N (X , J^" ) where J^ is defined as in the proof of Lemma 14 • Order the simplexes A^, ..., A^ of J - K that meet K in order of decreasing dimension . Each A^ meets N in a convex cell B. , with a face C. -A. ^ f^^ . There is
1'
l i e
an elementary collapse of B^ from C^ , and the sequence of collapses i = 1 ,
r determines the collapses
N ^ X. Lemma 15 . Let h ; K -»K be a homeomorphism of a complex that maps each simplex onto itself^ and keeps a subcomplex L fixed . Then h is ambient isotopic to the identity keeping L fixed , Proof . The obvious isotopy moving along straight paths is not piecewise linear by the standard mistake . However it is easy to construct a piecewise linear isotopy H ; K x l — > K x I
inductively on the prisms
A X I , A ; K , in order of increasing dimension . For each prism , (H| A x I is given by induction, H I A x 0 is the identity , and H (a,l) = ha . Therefore H ( (A x l)* is already defined, so map the centre of the prism to itself and join linearly . By construction H keeps L fixed .
Corollary
1 . The isomorphism between any two first derived
complexes is ambient isotopic to the identity . Corollary 2 . Any two derived neighbourhoods of X in M are o ambient isotopic . keeping X fixed . If X C M , the isotopy can be chosen to keep M fixed .
- 17 -
For in the proof of Lemma 14 the homeomorphism was achieved by two isomorphisms between first deriveds , both keeping X fixed . The first deriveds can be chosen to agree outside the neighbourhoods, and so the isotopy keeps M fixed if X c S . theorem 5 . A derived neighbourhood of a collapsible polyhedron is an n-manifold is an n-ball . Proof . By induction on n , starting trivially with n = 0 . By Lemma 14 it suffices to prove the theorem for one particular derived neighbourhood , and so we choose a second derived neighbourhood N = N (X,J"), where X = I K| , K C J , and J" is the second barycentric derived complex g
of J . Since X is collapsible, we can choose K such that K ^^^ 0 by Theorem 4 . Let r be the number of elementary simplicial collapses involved g
xn K
. We show N is a ball by induction on r . The induction starts
trivially with r = 0 , for then K is a point , and N its closed star, which is a ball by Lemuia 9 . For the inductive step , let K^^L be the first elementary simplicial collapse, collapsing a simplex A from B , say , ^
A
A
where A = a B . Let A , B denote the barycentres of A , B , Now N where P = N (L , J") ,
N (K , J") = P u Q
R ,
Q = N (A , J") , R = N (B , J") .
Now P is a ball by iniuction , and Q , R are balls since they are closed stars of vertices . If we can show that Q is glued onto P by a common face , then P U Q is a ball by the Corollary to Theorem
j similarly if
R is glued onto P U Q by a common face then N is a ball . Therefore the proof is reduced to showing the P n Q , and
(P o Q) n R are (n-l)-balls
because if they are balls then they must be common faces , since the interiors of P , Q , R are disjoint .
- 18 -
B / '
Now P O Q C Q = Ik (A , J") . Let J^ = Ik (AS J') = A' (lk(A,j)): , where the prime thought this proof always denotes the barycentric first derived complex . There is an isomorphism
^ V
^
determined by the vertex map A C — C
- for all C
J^ . Under this
isomorphism P n Q
(a B
, J'^ ) .
Now a B is collapsible , being a cone j and (a B)' is full in J^ , which is an (n-l)-sphere or ball , by Lemma 9 . Therefore N (a B , J'^) is a derived neighbourhood of a collapsible polyhedron , and is an (n-l)-ball by induction on n . Hence P n Q is an (n-l)-ball c Similarly (PuQ)ri R c R , and if we now choose J^ = Ik (B , J') ,
- 19 -
then there is an isomorphism R _ J ' ^ , throwing (P U Q) n R onto A
,
N (A B , J'^) . For the same reason as before we deduce (P U Q) D R is an (n-l)-ball . This completes the proof of Theorem 5 • theorem 6 . Suppose the manifold M^ and the ball b" meet in a common face . Let X be a closed subset of M" not meeting B" . Then there is a homeomorphism m'^—M^ u b" keeping X fixed . Proof . Since X is closed , M^ - X is a manifold . Let B'^"^ be the common face , and let A^ be a derived neighbourhood of B'^ ^ in n ®n n-1 M -X , which is a ball by Theorem 5 . Since A C M , B does not meet
i".
and so B^ ^ is a face of A^ . Since A^ , B^ meet in the common face B^^ their union is a ball by the Corollary to Theorem 2n . Let B^ ^ = b"' - B^ ^ , which is a ball by Theorem
^ .
We now construct the homeomorphism h . Define h to be the identity on (M'^ - A^) U (A^ -
. In particular h is the identity on X . Extend
h = 1 : B^ ^—^ B^^ to a homeomorphism b'^ ^ — ^ ^
Lemma 10 .
Similarly extend h : A'^—^ (A^ U B'^)* to the interiors . Then h has the desired properties . Lemma 16 . Any homeomorphism of a ball onto Itself keeping the boundary fixed is isotopic to the identity keeping the boundary fixed .
- 20 -
Proof . It suffices to prove for a simplex . Given h : A we construct the isotopy f : A x I
y A ,
A x l a s follows . Let
This gives f level preserving on ( A x l)' . Define f level preserving on A X I by mapping the centre of the prism to itself, and joining to the boundary linearly . Then f is the desired isotopy . ^Lemma 17 . Suppose M^C Q,^ are manifolds . and that m" is a closed subset of
. Then Q"" - M"' is a manifold .
Proof . Let M^ = q"' - m"' . We have to show that every point 'XI M^ has a ball neighbourhood in M^ . If x G Q. - M^ , then -x has a ball neighbourhood in Q^ that is contained in M^ , because m'^ is 0
closed in Q . If , on the other hand , x (S M A M^ , then x
Q
by
hypothesis , and so x lies in the interior of a ball in Q^ . Triangulate this ball so that x subcomplex . If S
is a vertex , and so that its meets M^ in a is the link of x , then S
n M
is a ball by
Lemma 9 • Therefore the closure of the complement , s'^ ^ ft M^ , is a ball by Theorem 3 , . Hence x has a ball neighbourhood in M^ , n-1 1 theorem 7 . Suppose C Q are manifolds , and that M is a ^n n ^n n closed subset of Q. . Let B be an n-ball in Q. meeting M in a common face . Let X be a closed subset of Q^ not meeting B^ . Then there is an ambient isotopy of Q" moving M^ onto M^ U B^ , and keeping X V Q^ fixed .
- 21 -
Proof .
Let B n-1 be the common face , and let B,n-l . B^ - Bn:r-l which is a ball by Theorem 3
. Let M^ = Q,^^ - (M^ U B^) , which is
a manifold by Lemma 17 , since M vj B Let D
is a manifold by Theorem 6 .
be
of BA" =inD^^n the manifold Then D^ isaaderived ball byneighbourhood Theorem 5 . Let ^^ ,, A^^ Q - Q' - X . A^ D^'n M M^ If when constructing I) we choose a triangulation that meets , B in subcomplexes , this ensures that A.n , A.n ^^ are respectively derived neighbourhoods of b"-"^ , B^""^ in M^ , M^ and therefore are balls
.n meets
B^ in the common face Bb'^'^ , and A^ , meets B^ in the coirmon face B1 ,n U.„n A^ U B^ are balls by Corollaiy to Theorem Therefore A^ B^ , .n Next we construct a homeomorphism h of Dn onto itself as • r—1 n-1 to follows . Define h = 1 on - B'n-lv Extend h : B ' —^ B^ the interiors by Lemma 10 . Similarly extend ' A ^ — u B^) and (B^t^ A^)'—» A^ to the interiors . By Lemma 16 the identity is isotopic *n n to h , keeping D
fixed . Extend this to an ambient isotopy of Q
keeping fixed q" - D^ (in particular X U ic^) . By construction this isotopy moves M^ onto M^ U b'^ .
•
- 22 -
Regular neighbourhoods . The definition of regular neighbourhood is more powerful than that of derived neighbourhood because it is intrinsic , and leads at once to an existence and uniqueness theorem . Let X be a polyhedron in a manifold M . A regular neighbourhood N of X in M is a polyhedron such that i) N is a neighbourhood of X in
M.
ii) N is an n-manifold (n = dim M) iii) K \J X . theorem 8 (1) Any derived neighbourhood of X in M is regular . (2) Any two regular neighbourhoods of X in M are homeOfflwiuhie , keeping X fixed , 0 0 ,. (3) If X C M t then any two regular neighbourhoods of X in M are ambient isotonic keeping X U M fixed . Remark . Clearly (3) is stronger than (2) . However it is valuable to have (2) in cases where (3) does not apply . For example suppose X is a spine of M in the interior of M ; then by (2) M is homeomorphic to any regular neighbourhood N of X in M . But obviously M and N are not ambient isctopic . Proof of Theorem 8 . Part (1) . Let N = H (x , J') be a derived neighbourhood of X in M . ¥e have to verify the three conditions for regularity . Condition (i) follows from the definition, and (iii) from the Corollary to Lemma 14 . To
- 23 -
verify (ii) we check the link of each vertex DC : N . Let L = 11: (x, J') . If X 6 X , then Ik (x , N) = L , which is a ^here or ball . If sz ^ X , o then X 6- ^ y where A is a unique simplex in J - K , K being the subcomplex of J convering X . By the fullness of K in J , A r") K = B , a face of A . Now L
A'S , where S is isomorphic to (ik (A,J))', and so is
a ball or gjhere . Since S lies in the interior of st (A,K) it does not meet X , and therefore L n X = A'n X = B' . Therefore Ik (x,N) = N (B',L) = N (B',A'S) = N (B',A')S which is a ball , because N(B',A') is a ball by Theorem 5 > being a derived e
neighbourhood of B in A . The proof of part (l) is complete . For part (2) it suffices by Lemma H to show that any regular neighbourhood is homeomorphic to a derived neighbourhood , keeping X fixed . If N is the regular neighbourhood , use Theorem 4 to choose a triangulation J , K of N , X such that J collapses simplicially to K J =K\ K r r-1
o
=K
Let J" be the barycentric second derived of J , and let
- N (K^ , J") .
Then N^ is a derived neighbourhood of X in M , and N^ = N . As in the proof of Theorem 5 , N^ is obtained from Neither of these balls meets X , because
^ by glueing on two balls . is a neighbourhood of X , and
so by Theorem 6 there is a homeomorphism Composing these , we have the desired homeomorphism
^ fixed , —^ N .
For part (3) we make the same construction as for part (2) , and instead of Lemma 14 and Theorem 6 we use Corollary 2 to Lemma 15 and Theorem 7 to show that the two neighbourhoods are ambient isotopic keeping X U M fixed . The proof of Theorem 8 is complete .
- 24 -
Proof of Theorem 3 Iji At last we come to the end of our mammoth induction . ¥e recall that in the proofs of Theorem 4 - 8 we have used Theorem
,r
n and Theorem
,
r < n , but not Theorem 3 . We now use Theorem 8 to prove Theorem 3 . This n n will make Theorem 2 - 8 and the accompanying lemmas valid for all n . Given B C S a homeomorphism f : A
we have to show that S - B
is a ball . Choose
^ S^ throwing a vertex x of A
onto a
. Then the balls A^ , B" are
point y t B^ . Let A^ = f (st(x, A
both regular neighbourhoods of y in S ' , and so by Theorem 8 Part 3 are ambient isotopic . Therefore the closures of their complements are , . the ,, face „ homeomorphic . But S^ - A^ = f A n ^ where A. n is of„ A, n+1 opposite X . Hence S'^ - B^ is a ball . We conclude the chapter with some useful corollaries to Theorem 8 , Corollary 1 . A manifold is collapsible if and only if it is a ball . For if it is collapsible , then it is a regular neighbourhood (in itself) of any point , and therefore a ball by Theorem 5 . 0
Corollary 2 . If X C K , and N , N^ are regular neighbourhoods of X in M , such that N ^ : N , then N - N^ = N x I . Proof . Construct two derived neighbourhoods as in the proof of Lemma 14 . N* = f"^ [ 0 ,6 J , N* ^ f"^ [O , S where 0 <
6< 1 .
1
Then
/ - N* = f"^ [ S ,
= f'^ t X I = ii* x I .
Therefore the result is true for N* , N* . By Theorem 8 (2) choose a
- 25 -
homeomorphism h ! N*—^, N keeping X fixed . Now h N* , N 0 1 J-
are both
regular neighbourhoods of X in N , and so by Theorem 8 (3) we can ambient isotope h N* onto N^ keeping N fixed . Therefore N-
^ N - N* = N X I = N X I .
Corollary 3 . The combinatorial annulus theorem . If A , B are 0 0 n-1 two nr-balls such that A DB , then A - B g S x I . Proof by Corollary 2. 0
Corollary 4 . Suppose X . Y C M , and suppose X is a spine of M (i.e. M
X) . If X ^ Y or Y
Proof . If X
X then Y is also a spine of M .
Y the result is trivial , because then M
X
Y.
If Y Si X , let N be a regular neighbourhood of Y in S . Then N bs. Y "isi X , and so N is also a regular neighbourhood of X , By Corollary 2, M - N 3 M X I , and so M
N . Therefore M
N
Y , and so Y is a spine
of M . Remark 1 . Corollary 4 is a form of factorization of the collapsing process . However such factorization is only true for manifolds , and not true for polyhedra in general . For instance X \\ 0 Y
)
X D Y Consider the following example . Let x y z be a triangle , and let y', z' be two interior points not concurrent with x . Let X be the space obtained by identifying the intervals x y = xy' , xz of y z in X .
xz' , and let Y be the image
- 26 -
X
Then X
0 conewise , axid Y
0 because Y is an arc . But X
Y because
any initial elementary simplicial collapse of any triangulation of X must have its free face in Y , and so must remove part of B . Similarly can build examples to show that X Y
^
/v
XuY
0
XnYRemark 2 . Corollary 4 is useful for simplifying ^ines . For example the spine of a bounded 3-manifold can be normalised in the following sense : we can find a spine , which is a 2-dimensional cell complex in which every edge bounds exactly 3 faces , and every vertex bounds exactly 4 edges and 6 faces . For choose a spine in the interior; expand each edge like a banana and collapse from one side; then expand each vertex like a pineapple and collapse from one face . By Corollary 4 any sequence of expansions and collapsesleaves us with a spine, and the process descTdbed makes it normal .
INSTITDT DES HAUTES ETUDES SCIEKTIFIClUES 19 6 3
Seminar on Combinatorial Topology by E.G. ZEEMAN
Chapter 4 : UMNOTTING BALLS AND SPHERES
Suppose if
M
are manifoldsj we say the embedding is proper
M^ and ITC M^ . A (G.m)-manifold pair m'^'^^CmSm"^) is a pair
such that M^ properly • The codimension of the pair is c = q - m . The boundary
=
is a pair of the same codimension . ¥e write
In this chapter we are interested in sphere pairs pairs
and ball
. The boundary of a ball pair is a ^here pair. If
B^' C M^ '
we call B^'
The standard (q,m)-ball pair is
a face of M^ ' A
^ ( £
A ™ ,
A
where A "
is the standard mr-simp lex, and denotes (q^-mj-fold suspension . The standard (q,m)-sphere pair is A
. fe say a sphere or ball pair
isffiknottedif it is homeomorphic to a standard pair . The cone on an unknotted (q,m) ball or ^here pair gives an unknotted (q+l,m+l) ball pair . Theorem 9 . Any sphere or ball pair of codimension
3 is unknotted.
Remark 1 . In codimension 2 the theorem fails for both spheres and balls. The (3,1) sphere pairs give classical knot theory, and in higher dimensions knots can be tied for example by suspending and spinning (3,1) knots .
- 2 -
Con.iecture 1 . The sphere pair
is unknotted if S -S^
is a homotopy S^ . If q = 3 the result is true hy a theorem of Papakyriakopoulos . If q, 5 an analogous topological theorem of Stallings says that if the sphere is topologically locally unknotted then it is topological^unknotted. Con.iecture 2 . Sphere and ball pairs unknot in codimension 1 . This is the Schbnfiles conjecture which is true for q. 3 , and unsolved for q ^ 4 . Conjecture 3 • If B is a ball pair contained in an unknotted sphere pair of the same dimension, then B is unknotted . This is true for codimension ^ 3 by Theorem 9 . It is true for codimension 2 when q = 3 by the unique factorization of classical knot theory (an unknotted curve is not the sum of two knots) . It is true for codimension 1 when q ^ 3 by the Sch'dnflies Theorem . But otherwise in codimensions 1 and 2 is unsolved . A modified result is that B C S are both unknotted then the complementary ball pair S - B is also unknotted . This proved by Theorem 8 part 3 generalised to relative regular neighbourhoods . Remark 2 . In differential theory Theorem 9 is no longer true because 4k—1 6k Haefliger has knotted S differentially in S . Above this critical dimension, in the stable range, he has unknotted all sphere pairs .
Plan
of
the
proof
of
Theorem
9 .
Most of this chapter is devoted to proving Theorem 9 . The proof is by induction on m keeping the codimension c - q - m fixed . We eventually show that Theorem 9 -, ^ q-l,m-l
S Theorem 9 ^ q,m
- 5 -
The induction starts trivially with m = 0 ; for, given c , then a(c,0) ball 0
pair is a ball B
0
with an interior point B , which is homeomorphic to a
standard pair . Next we observe that : Unknotting of (q,m)-ball pairs implies unknotting of (q,m)-sphere pairs . Proof . Given vertex x
If A
, triangulate the pair and choose a
s"^ , Let
=7
A
is the standard simplex, then
A
= A
u y
A
By hypothesis choose an unknotting homeomorphism B^'"^ X to y and extend linearly to an unknotting
. 5,. A
; then map A
Lemma 18 . Let (b'^. b"^) and (c'^.
.
be two unknotted ball pairs
Then any homeomorphisms f : B^ —» C^ and . . can be extended to a homeomorphism h ;
that agree on B
.
Proof , Extend f conewise to f: B ' ^ — a s in the proof of Lemma 10 . Let e i C ^ — b e the composition c^
'm Then e keeps C
B"^
c"^
•m fixed , since f , g agree on B . By the unknottedness
we can suspend e to a homeomorphism e: c'^—^ c'^ fixed on C^ . Then h = ef : B ^ — a g r e e s with both f and g , and proves the Lemma .
- 4 -
Corollary . Any homeomorphism between the boundaries of two unknotted ball pairs can be extended to the interiors . Lemma 19 . Assume Theorem 9 ^ . Then if two unknotted (q.m) q-l,m-l ball pairs meet in a common face their union is a unknotted ball pair . Proof . Let Let ^
A
ball pairs meeting in the face F .
be the suspension of the standard (q-l,m-l) ball pair A , with
suspension points x , x. say . Choose an unknotting F — A Extend F — > A
to unknottings B^ - F — > x^
A
by hypothesis.
by the above corollary .
Similarly extend B^ — > (x^ A )' to the interiors . Then B^ u B^ is unknotted by the homeomorphism onto ^ A , Lemma 20 . If (B'^.B'^) is a ball pair of codimension ^ 3 then
Remark .
Lemma 20 fails in codimension 2 ; for example a knotted
arc properly embedded in B^ is not a spine of B^ . The proof of Lemma 20 involves some geometrical construction, and we postpone it until after Lemma 23, which is the crux of the matter . First let us show how Lemma 20 implies Theorem 9 .
Proof
of
Theorem
9
assuming
Lemma
20
¥e assume Theorem 9 -, , > where q-m X 3 . By the observation q_-l,m-l that unknotting balls implies unknotting spheres, it suffices to show that a given ball pair B := (b'^jB™) is unknotted . Choose a triangulation J , K of
such that K is simplioially
collapsible K .. K r
K r-1 ^
... \ ^
K = point . 0 ^
- 5 -
Let J" be the second barycentric derived of J , and let B. be the ball pair B. ={W(K. , J") , N ( K. , K")) ¥e show inductively that B^ is unknotted . The induction starts with i i 0 , because B^ is a cone on the ball or sphere pair (lk(K^,J"),lk(K^,K")) which is unknotted by Theorem 9
For the q.-l,m-l As in inductive step assume B. i-1 unknotted Theorem 5 , we notice that B^ is obtained by glueing on two more small ball pairs, each by a common face , and each of which being unknotted like
Hence B. is unknotted by Lemma 19 . At the end of the induction B 1 r is unknotted
B
Now B^ =
, where
is a regular neighbourhood of B"^
in B^ . But by Lemma 20 , B^ itself is another regular neighbourhood . Therefore by Theorem 8 Part 2 there is a homeomorphism B^ —j-N*^ keeping fixed , or , in other words , a homeomorphism B
, shewing B
unknotted
Conical
subdivisions ¥e shall need a lemma about subdividing cones . Let C = vX be a
cone on a polyhedron X , with vertex v . If Y c G , the subcone through Y is the smallest subset of C containing Y of the form vZ , Z c X . For example a subcone through a point is a generator of the cone . A triangulation of G is called conical if the subcone through each simplex is a subcomplex . Lemma 21 . Any triangulation of G has a conical subdivision
- 6 -
Proof . Let G also denote the given triangulation . Let f: C -^-I denote the piecewise linear map such that f ^(o) = v , f ^(l) = x , and such that f maps each generator linearly . Choose 6 > 0 ? and such that i.
fx
for every vertex x S: C , oc
v . Choose a first derived C' of G such Then that each simplex of C meeting f-1, is starred on
f"" [6 ,ll , f~ (S) are subcomplexes , K , L say , of C' , and C = K (J V L . Let g : K-vL be radial projection , which is a projective map and not piecewise linear . Then f x g : K —^ [ 8 .l3 x L is a projective homeomorphism ,
-rr /K
that maps K protectively onto an isomorphic complex , K^ say , triangulating 1J
X L . The projection TP : K^—» L onto the second factor is
piecewise linear, and so there are subdivisions such that IT : K' —> L' is simplicial . Let K' = (f x g)~
. Then K' is a subdivision of K ,
containing L' as a subcomplex , because T!'(f x g) : L
L is the identity „
Let G" = K' u V L' . Then C" is a subdivision of C , and is conical because
is cylindrical .
Shadows Let
be the q-cube . ¥e single out the last coordinate for
special reference and write
=
^xI
Intuitively we regard I as
- 7 -
vertical , and
as horizontal , and identify
cube . Let X be a polyhedron in
^ with the base of the
. Imagine the sun vertically overhead ,
causing X to cast a shadow; a point of
lies in the shadow of X if it
is vertically below some point of X . *
Definition . Let X
be the closure of the set of points of X that
lie in the same vertical line as some other point of X (i.e. the set of points of X that either overshadow , or else are overshadowed by , some other point \ . Then X* is a subpolyhedron of X . of X) Lemma 22 . Given a ball pair there is a homeomorphism b'^.b'^ — »
,B ) of codimension ^ 5 > then X such that
i) X does not meet the base of the cube ii) X meets each vertical line finitelv \ dim X* iii)
m - 2.
Proof . First choose the homeomorphism to satisfy i) , which is easy . Now triangulate I'^jX . Then shift all the vertices of this triangulation by arbitrary small moves into general position , in such a way that any vertex in the interior of
remaihs in the interior, and any vertex in a face of
remains inside that face . If the moves are sufficiently small, the new positions of vertices determine an isomorphic triangulation , and a homeomorphism of I^ onto itself . The general position ensures that conditions (ii) and (iii) are satisfied , because m ^ q - 3 ^ and so
dim X*^(m + l ) + m - q ^ m - 2
,
Remark . The "general position" of the above proof may be analysed more rigorously as follows . Each vertex is in the interior of some face , and has coordinates in that face . The set of all such coordinates of all vertices determine a point OC some high dimensional euclidean space , and
8 -
the sufficient smallness means that Cc is permitted to vary in an open set , U say . To satisfy the conditions (ii) and (iii) we merely have to choose or. £ U , so as to avoid a certain finite set of proper linear subspaces .
Sunny
collapsing Suppose we are given polyhedra
D X 3 Y , such that X H I
is
an elementary collapse . We call this collapse sunny if no point of X - I lies in the shadow of X . We call a sequence of elementary sunny collapses I
a sunny collapse , and if I is a point v/e call X sunnv collapsible . Corollary to Theorem 4 . If X is sunny collapsible then some triangulation is simpliciallv sunny collapsible . For each elementary sunny collapse is factored in a sequence of elementary simplicial simny collapses . Lemma 23 . If (I'^.X) is (q,m)-ball pair of codimension ^ 3 satisfying the conditions of Lemma 22 then X is sunny collapsible , Remark .
Lemma 23 fails with codimension 2 . The classical example 3 of a knotted arc in I gives a good intuitive feeling for the obstruction \
t
\
(\f
to a sunny collapse : looking down from above it is possible to start collapsing away until we hit underpasses , which are in shadow and so prevent any further progress .
^I
I
Definition . A principal k~Gomplex is a complex in which every principal simplex is k-dimensional .
- 9 -
Proof of Lemma 23 . ¥e shall construct inductively a decreasing sequence of subpolyhedra X=X
0
1
D ...yx = a point , m
and , for each i , a homeomorphism f. : x . - ^ vie^-^-^ 1 1 onto a cone on a principal (m-i-l)-coraplex, satisfying the three conditions : , * Ij \ does not contain the vertex of the cone , and meets each generator of the cone finitely . *
2) dim X^ 4 m-i-2 . 3) There is a sunny collapse X^ ^
X^ .
The induction starts with X = X . 0
Condition (2) is by hypothesis and (3) is vacuous . Choose a homeomorphian f
0
: X —> A , where A is the standard m-simplex . Since f X* is a sub0
0
polyhedron of dimension ^ m-2 , we can choose a point v *
°
in general position relative to f^X . Starring A the cone v A
U ^ A - f X , and
at v makes A into
on ^ , which is principal . Condition (2) is satisfied by
our choice of v . The induction finishes with X = a point , and so we shall have a sunny collapse m X ^ X^
X^
... \ X^
which will prove the lemma . The hard part is the inductive step . Suppose we are given f
:X
^v
, satisfying the three conditions,
- 10 -
we have to construct f. , X. ,
^ ^ and prove the three conditions .
Let C , L triangulate v K"^ ^ , f^ ^ X^ ^ . By Lemma 21 we can choose C to be conical . In particular C contains a subdivision • of K^-^ . Define = the (ni-i-l)-skeleton of (K™"^) ' , which is a principal complex since
was principal . Let C
of C triangulating the subcone v
be the subcomplex
. Let e^ : C^ —> C be the
inclusion map . We shall construct another embedding e : G
C
that differs slightly , but significantly , from e^ . Having chosen e , then there is a unique subpolyhedron X^ , and homeomorphism f^ , such that the diagram X.
1-1
f. 1
^i-1 G
is commutative . It is no good choosing e = e , because then X. = X_. which i-1 would be of too high a dimension . In fact this is the crux of the matter : we must arrange some device for collapsing away the top-dimensional shadows of X* , . The first thing to observe is that the triangulation L of x-1 f. , X* ^ is in no way related to the embedding of X^ ^ in the cube I 1-1 1-1 The inverse images of simplexes of L may wrap around and overshadow each other hopelessly, so our next task is to take a subdivision that remedies
- 11 -
this confusion . We have piecewise linear maps f-1
where the first is a homeomorphism , and Tf is vertical projection onto the base of the cube (l^""^) ' of
. By Theorem 1 we can choose subdivisions L' of L , and a triangulation M of X* ^ such that the maps
L.
M
(I^^)'
are simplicial .. Recall that dim M ^ m-i-1 , by induction on i . Let A^ A ,A ,...,A
be the (m-i-l)-simplexes of M . Each A. is projected non-
X
J
degenerately by TT" , because of Lemma 22 (ii) . If j k there are two possibilities : either TT A. / Tf A, or TT A. = "T A, , in the first case 0 0 J ^ J o ^ o IJ maps A. , A disjointly and so no point of A. U A, overshadows ary J k 0 00 ^ other . In the second case either A^ overshadows A^ or vice versa . Consequently overshadowing induces a partial ordering amongst the A' , s and we choose the ordering A^,
to be compatible with this partial
ordering . ¥e summarise the conclusion : 0
Sublemma . All points of X that overshadow A^ are contained in
U
A. We now pass to L' . Let B. = f. A.6L' , The next step in the J 1-1 a
proof is to construct a little (m-i+l)-dimensional blister Z. about each B. in the cone C . The blisters are the device that enable us to make the sunny collapse, and the fact that there is just sufficient room to construct them is an indication of why codimension for unknotting .
3 is a necessary and sufficient condition
- 12 -
Fix j. Let B. be the barycentre of B. cone is principal , there is simplex D
6 (K^")
Since the base of the such that B. is
contained in the subcone vD There are two cases depending on whether or < not B, lies in D , in the base of the cone cone . If B.C D , let b. be a J 0 /y J J point in D near B. , and let a. be a point in the generator v B. near J 0 ^ B. . If B.c^ D , let b. be a point in v D near B. , and let a. be a pair of points on the generator through B_G, , near B^ and either side of In either case define the blister B. J Z. = a.b .B. 3 J0 ¥e choose the points sufficiently near to the barycentres so that no two blisters meet more than is necessary (i.e. Z. n Z = ^ j = B. r> B, ) . The bottom of the blister J ^ is a.B. and the top is a.b.B. . Let J J —^ JJ J e^ ne the map e,:a.B. — > a.b.B. 3 0J D a3 that raises the blister, and is given by mapping B .—^ b . . J 3 Now G
0
meets each blister in its bottom . Therefore we can define
the embedding e ; C^
C by choosing e = e. on the intersection with each
blister , and e = 1 otherwise . In other words e is a map that raises all the blisters . Having defined e , we have completed the definition of X. and f. X. 1 1 There remains to verify the three conditions . Condition (2) holds because by our constmction
\ i
\
= the (m-i-2)-skeleton of M .
- 13 -
Condition (l) holds , because *
*
f. X. = f. , X. 1 1 1-1 1 *
c f. , 1-1
X. ^ 1-1
,
for which the condition holds by induction Finally we come• to condition (3) . Let Z = O Z. . ^ ^^ J For each (m-i)-simplex D 6 )' , V D - Z is a ball because it is a simplex with a few blisters pushed in round the edge, and D - Z is a face * Therefore collapse each v D - Z from D - Z . We have collapsed C \ eC U Z . ^ 0 and the inverse image under f^ ^ determines a suiiny collapse
*
sunny because we have not yet removed any point of
]_ •
¥e now collapse the blisters as follows . Each blister meets e C^ in its top , and so by collapsing each blister onto its top in turn, j=l,...,r, we effect a collapse e C u Z o
eC o
If I. = f"^^ Z. , then the inverse image of this collapse determines a J 1-1 J sequence of elementary collapses r X. U
r ^
I. ^
X. 'J u
Y. ... X. . J 1 is sunny 2 ^ the Sublemma, because by the X 1 Each of these elementary collapses by time we come to collapse Y , say , the only points that might have been ir 0 ^ shadow are those in A^ , but these are simny for we have already removed
- 14 -
everything that overshadows them . We have demonstrated the sunny collapse 1
X^ , which completes the proof of Lemma 23 . Proof of Lemma 20 . We can now return to the proof of Lemma 20 , which will conclude
the proof of Theorem 9 . Given a ball pair
of codimension
3,
we have to show B*^ \i b"^ . By Lemma 22 it suffices to show for a ball pair (l*^, X) satisfying the conditions of Lemma 22 . By Lemma 23 and the Corollary to Theorem 4 we can choose a triangulation K of X that is simplicially suniiy collapsible t K = K^ ^^ K^
.... \
Let L^ be the polyhedron consisting of
K^ = a point . X together with all points
in the shadow of K. . We shall show that 1 ^
L^ ^
L^ \
...
L^ \4 X .
The first step is as follows . Choose a cylindrical subdivision of
containing a subdivision L' of L . Then collapse (l*^) ' ^ L' 0
0
prismwise from the top, in order of decreasing dimension of the prisms . The last step is easy , because L^ consists of by a single arc . Collapse
onto the bottom of this arc, and then
collapse the arc . There remain the intermediate steps L i =: 1 , ... , r .
[j X joined L. ,
- 15 -
i-1
Fix i , and suppose the elementary simplicial sunny collapse K.
Nj K. collapses A from B , when A = a B , Choose a point b ^
A
A
below the barycentre B of B , and sufficiently close to B for bA O X = A (this is possibly by Lemma 22 (ii)) . Let T = L. together with all points in the shadow of ab B . Since the collapse is sunny, no points of K. 0 0 / s• ^ overshadow A U B , and so T n b A = a(b B) , T u b A = L^ ^ . In other words coll^sing b A from b B gives an elementary collapse
Finally collapse T
L^ prismwise downwards from a b B , as in the first
case . This completes the proof of Lemma 20 and Theorem 9 .
Isctopies
of
balls
and
spheres
Recall that Lemma 16 proved that any homeomorphism of a ball onto itself keeping the boundary fixed is isotopic to the identity keeping the boundary fixed .
- 16 -
Corollary 1 to Theorem 9 . If q-m >> 3 . then any two proper embeddings
•q 1 :
C b'^ that agree on b"^ are ambient isotopic keeping b'^ fixed.
Proof . Let f,g be the embeddings • By Lemma 18 we can extend •q — j Q TTi B^ and gf
:fB
g B' , which agree on B , to a homeomorphism
cetween the ball pairs h : (B^f B-")
.
By construction hf = g , and, by Lemma 16, h is ambient isotopic to the •q identity keeping B^ fixed . Theorem 10 . Any orientation preserving homeomorphism of S^ is isotopic to the identity . Proof : by induction on n , starting trivially with n = 0 , Let f be the given homeomorphism . Choose a point X 6 S^ , and ambient isotopic f X to jc . This moves f to f^ , say , where f^ Jr. = X- . Choose a ball B containing ex, in its interior . Then B,f^B are regular neighbourhoods of X. , and so by Theorem 8 ambient isotope f^B onto B , This moves f^ to f^ , say , where f2B = B . The restriction f^ I B preserves orientation , and is therefore isotopic to the identity by induction . Extend the isotopy conewise to B and S^^ - B , making it into an ambient isotopy , that moves f^ to f^ , say where f^ B = 1 . Apply Lemma 16 to each of B , S^ - B to ambient isotcpe f^ into the identity . Corollary 2 to Theorem 9 . If q-m >x 3 . then any two embeddings s'^C s'^ are airibient isotopic . Proof . If (s^js'^), q> m , is an unknotted sphere pair, then S^ is the (q-m)-fold suspension of s"^ , and so there is (l) an orientation reversing homeomorphism of s'^, throvring s'^
- 17 -
onto itself, and (2) an orientation preserving homeomorphism of S*^ , throwing s''^ onto itself with reversed orientation . Let f , g :
s'^ be the two given embeddings . Since q - m ^ 3 ,
the unknotting gives a homeomorphism
, which we can 2
choose to be orientation preserving on S
>7(1) , and which is therefore
isotopic to the identity by Theorem 10 . Therefore f is ambient isotopic to f^ , say , such that f^S® = gS® . Let h
gf"! :
f^s"^ . By
(2) above and Theorem 10 , we can choose f^ so that h is orientation preserving . Now apply Theorem 10 to the smaller sphere s''^ , to obtain an isotopy from the identity to h ; suspend this isotopy of s"^ into an ambient isotopy of s'^ moving f^ into g . Remark . The above two corollaries are also true for unknotted ball and sphere pairs of codimension 1 and 2 . The aim of the next four chapters is to obtain similar results for arbitrary manifolds .
INSTITDT DES HAUTES ETUDES SCIEKTIFIQUES 19 6 5
Seminar on Combinatorial Topology by E.G. ZEEMAN
Chapter 5 : I30T0PY
The natural way to classify embeddings of one manifold in another is by means of isotopy. But there are several definitions of isotopy, and the purpose of this chapter is to prove three of the definitions equivalent. The three that we consider (of which the first two were mentioned in Chapter 2) are : (0 Isotopy. sliding the smaller manifold in the larger through a family of embeddings ; (2) Ambient isotopy. rotating the larger manifold on itself, carrying the smaller with it ; (3) Isotopy by moves, making a finite number of local moves, each inside a ball in the larger manifold, analogous to moving a complex in Euclidean space by shifting the vertices, like the moves of classical knot theory . Since any homeomoiphism of a ball keeping the boundary fixed is isotopic to the identity it follows at once that isotopy by moves
-T-JT-^ ambient isotopy
isotopy .
In Theorems 11 and 12 we shall show that these arrows can be reversed . To reverse the second arrow, that is to cover an isotopy by an ambient isotopy, it is necessary to impose a local unknottedness condition on the isotopy *
- 2 -
For otherwise the knots of classical knot theory give counterexamples of embeddings that are mutually isotopic but not ambient isotopic. However the results of Chapter 4 show that this phenomenon occurs only in codimension 2, and possibly codimension 1 , Throughout this chapter we shall be considering embeddings of a compact m-manifold M in a q-manifold Q , which may or may not be compact . We restrict attention to proper embeddings f : M —^Q | recall that f is -1' * proper provided f Q = M ; in particular if M is closed , then any embedding of M in Q is proper . By a homeomorphism of Q , we mean a homeomorphism of Q onto itself
Definitions
of
in particular a homeomorphism is a proper embedding .
Isotopy
Recall definitions that have been given in previous chapters . (1) A homeomorphism h of M is a homeomorphism of M onto itself . If YC M and h I Y = the identity we say h keeps Y fixed . (2) An isotopy of M in Q is a proper level preserving embedding F:MxI—>QxI
.
Denote by F^ the proper embedding M.->>Q defined by F (x,t) = (F^x,t) , all X
M . The sub^ace U t I
F M of Q is called the track left by the ^
isotopy . If X<1M , we say F keeps X fixed if F(x,t) =F(X,0), all x
X
and
t6 I. (3) The embeddings f , g : M-->.Q are isotopic if there exists an isotopy F of M in Q with F^ = f , F^ = g .
- 3 -
(4) An ambient isotopy of Q is a level preserving homeomorphism H : Q X I —> Q X I such that H = the identity, where as above H, is defi: 0 t by H(x,t) rr (H^x,t), all x
Q . Wa say that H covers the isotopy P if
the diagram Q XI H F XI o
XI
M XI
is commutative; in other words F, = H F , all t I . t tu (5) The embeddings f,g : M-^Q are ambient isotopic if there is an ambient isotopy H of Q such that H^f = g . Remark . If M = Q , then a proper embedding M x I —j Q x I is the same as a homeomorphism Q x I — Q x I . Therefore since we have restricted attention to proper embeddings , the only difference between an isotopy of Q in Q , and an ambient isotopy of Q , is that the latter has to start with the identity | consequently two homeomorphisms of Q are isotopic if and only if they are ambient isotopic . (6) A homeomorphism or ambient isotopy of Q is said to be supported by X if it keeps Q - X fixed . By continuity the frontier X
Q - X of X in Q must also be kept fixed . (7) An interior move of Q is a homeomorphism of Q supported by a
ball keeping the boundary of the ball fixed . A boundary move of Q is a
- 4 -
homeomoiphism of Q supported by a ball that meets Q in a face ; the complementary face is the frontier of the ball that is kept fixed by continuity. (s) The embeddings f,g are isotopic by moves if there is a finite sequence
Locally
unknotted
of moves of Q such that ii^ ^^ ... h^f = g .
embeddings
Let f : M — b e
a proper embedding . Let Q,^ be a regular
neighbourhood of fM in Q . Let K,L be triangulations of M, Q^ such that f : K —> L is simplicial . We say that f is a locally unknotted embedding if, for each vertex v 6 K , the pair flk (fv,L),f(lk(v,K) ^ is unknotted . Notice that since the embedding is proper, the pair is either 0 • a ^here or ball pair according to whether v 6 M or v $ M . Corollary 3 to Lemma 9 . Any proper embedding of codimension ^ 3 is locally unknotted . Therefore then we say "locally unknotted" in future we refer only to the cases of codimension 1 or 2 . Remark 1 . The definition is independent of Q^ , and the triangulations, because if all the links are unknotted, then the same is true for any subdivisions of K , L , and hence also true for any other triangulations . Remark 2 . An equivalent condition is to say that the closed stars of vertices are unknotted ball pairs, but in codimensions 1 and 2 the equivalence, for a boundary vertex , depends upon a result that we have
- 5 -
quoted, but not proved, that if an unknotted ball pair has an unknotted face then the complementary face is also unknotted . Remark 3 . If f : M-»Q is locally unknotted embedding, then so • t is the restriction to the boundaries f : M —y Q . Remark 4 . We say a ball pair (B^^jB™) is locally unknotted if the inclusion is so | for example this always happens in codimension
5 or in
the classical case (q,m) = (3,1) . Suppose (B^jB'^) is locally unknotted , and let n'^ be a regular neighbourhood of B™ in b'^ . Then although (B'^jB'") may be (globally) knotted, it can be shown that (N'^jB'^) is unknotted , by adapting the proofs of Lemma 19 and Theorem 9 .
Locally
unknotted
isotopies
We say an isotopy F : M x I — ^ Q x I is locally unknotted if (i) each level
: M—^Q is a locally unknotted embedding , and
(ii) for each subinterval J C I , the restriction F t M x J — Q x J is a locally unknotted embedding . Remark 1 . If F is a locally unknotted isotopy, then so is the restriction to the boundaries F : M x I —^ Q x I . The proof is non-trivial (as in Remark 2 above) and is omitted . As we need to use the fact in Corollary 1 to Theorem 12 below , we should either accept it without proof , or else add it as an additional condition in the definition of locally unknotted isotopy . Remark 2 . Any isotopy of codimension
3 is locally unknotted .
Remark 3 . The above definition is tailored to our needs . There is an alternative definition as follows j we say an isotopy is locally
- 6 -
trivial if , for each (x,t)
M x I , there exists an m-ball neighbourhood A
of X in M , and an interval neighbourhood J of t in I , and a commutative diagram
A
X
£ A XJ
J
C F MXI where
QXI
denotes (q-m)-fold suspension , and G is a level preserving
embedding onto a neighbourhood of F(x,t) . It is easy to verify that F is a locally trivial isotopy <
F is a locally -unknotted isotopy I 1 F
is an isotopy and a locally unknotted embedding .
¥e shall prove in Corollary to Theorem 12 that the top arrow can be reversed . Therefore a locally trivial isotopy is the same as a locally unknotted isotopy . We conjecture the bottom arrow can also be reversed - it is a problem depending upon the Schbnflies problem, and the unique factorisation of sphere knots . ¥e now state the theorems , and then prove them in the order stated . Theorem 11 . Let H be an ambient isotopy of Q with compact support keeping Y fixed . Then H^ can be expressed as the product of a finite number of moves keeping Y fixed .
- 7 -
Addendum . Given any triangnlation of a neighbourhood of X . then the moves can be chosen to be supported by the vertex stars » Therefore the moves can be made arbitrarily small .
Corollary . Let M be compact . let f ; M — b e
a proper locally
unknotted embedding • and let g be a homeomorphism of M that is isotopic to the identity keeping M fixed . Then g can be covered by a homeomorphism h of Q keeping Q. fixed ; in other words the diagram is commutative :
h Q /s
Q /S
M
M
Remark . In fact the corollary is improved by Theorem 12 below , to the extent of covering not only the homeomorphism but the whole isotopy . However we need to use the corollary in the proof of Theorem 14 , in the course of proving Tteorem 12 . Theorem 12 . (Covering isotopy theorem) . Let F ; M X I •—» Q x I be a locally unknotted isotopy keeping M fixed , and let N be a neighbourhood of the track left by the isotopy . Then F can be covered by an ambient isotopy supported by N keeping Q fixed . Addendum . Let X be a compact subset of Q and W a neighbourhood of X in Q . Then an ambient isotopy of ft supported by X can be extended to an ambient isotopy of ft supported by N .
8 -
Corollary 1 . Theorem 2 remains true if we omit "keeping M fixed" from the hypothesis and "keeping Q fixed" from the thesis . Corollary 2 . Let f,g : M —^Q. be two proper locally unknotted embeddings . Then the following four conditions are equivalent : (1) f,g are isotopic by a locally unknotted isotopy (2)
f,g are ambient isotopic
(3) f,g are ambient isotopic by an ambient isotopy with compact support (4) f,g are isotopic by moves . Corollary 3 . An isotopy is locally trivial if and only if it is locally unknotted . Proof of Theorem 11 We are given an ambient isotopy H : Q x l — ^ r Q x I
with compact
support, and have to show that H^ is a composition of moves . We first prove the theorem for the case when Q is a combinatorial manifold , namely a simplicial complex in E^, say . Then Q x I is a cell complex in E^ x I . We regard E^ as horizontal and I as vertical. Let K 5 L be subdivisions of Q x I such that H : K —» L is simplicial (in fact a simplicial isomorphism) . Let A be a principal simplex of L , and B a vertical line element in A . Define ^ angle between H~^(B) and the vertical . Since H : K — L
(A) to be the is simplicial, this
does not depend upon the choice of B , Since H is level preserving , 0 {k) ^IL . Define
^ - max & (A) , the maximum taken over all principal
simplexes of L . Then 8 <;" _ii . 2
9 -
Now let ¥ denote the set of all linear maps Q — ( i . e . maps that map each simplex of Q linearly into l) . Let W<
=
If f g ¥ , denote by f
( f e w ; max f = min f < the graph of f , given by
f* = 1 X f : Q—> Q X I . Then f* maps each simplex of Q linearly into E" x I , Let
(f) be the
maximm angle that any simplex of f Q makes with the horizontal . Given 6 > 0 , there exists S > 0 , such that if f 6
then
(f) < ^ >
for choose S sufficiently small compared with the 1-simplexes of Choose 8 < H - S f and choose accordingly . 2 Now let f be a map in intersections of the arc
.
, and q a point of Q . Consider the
x I) with f* Q ; we claim there is exactly
one intersection .
Q XI
f^ Q
For since f
QXI
^ ^ ( q x I)
fT^
H
qXI
P Q
<1
is a graph , f Q separates the compleir-.nt (Q x I) - f Q into
points above and below the graph . If there were no intersection , then the arc would connect the below-point H ^(q,O) to the above-point H ^(q,1) , contradicting their separation . At each intersection, since
(f) +
—^ 2
- 10 -
the arc , oriented by I , passes from below to above . Hence there can be at most one intersection . Let p : Q X I — ^ Q denote the projection onto the first factor . Then k = p H f* : Q
^Q
is a 1 - 1 m ^ by the above claim , and so is a (piecewise linear) homeomorphism of Q . By the compactness of Q and I , choose a sequence of maps in W^ , such that f^(Q) = 0 , f^(Q) = 1 , and for each , f. and f. agree on all but one , v. say , of the vertices of Q . Define 3!T!i.
1
X
k. = p H ft . Then k. = H. = the identity, and k = H . Define h. = k.kT^, X I 0 0 n l 1 1 1-1 Then h^ is a homeomorphism of Q supported by the ball k^ (st(v^,Q)) , keeping k^ (lk(v^,Q,)) fixed , and so is a move . Therefore H^ = h^h^ i ' " ^ ' a composition of moves . If H keeps Y fixed , then k^ I I = k^ | Y , for each i , and so each move h^ keeps Y fixed . In particular the moves keep Q - X fixed , and are supported by X . Suppose now that Q is a compact manifold ; let T — Q
be a
triangulation in the structure . We have proved the theorem for T and so it also follows for Q . Suppose now that Q is non-compact , Let N be a regular neighbourhood of X in Q . Then N is a compact submanifold , and N n(Q - N) C Y . Therefore H | N x I is an ambient isotopy of W keeping N HY
fixed , and by the compact case H^ N is a composition of moves
supported by X keeping N 0 Y fixed . The moves can be extended by the identity to moves of Q, keeping Y fixed , and so H^ is composition of moves of Q . The proof of Theorem 11 is complete .
- 11 -
Proof of the Addendum to Theorem 11 We are given a triangulation T-^N of a neighbourhood of X , and have to show that the moves chosen to be supported by the vertex stars of T . Without loss of generality we can assume N is a regular neighbourhood , because any neighbourhood contains a regular neighbourhood . Therefore T is a combinatorial manifold . Let ^
denote the open covering of K x I ;
i'st(w,T)
xI;w6T|
L where w runs over the vertices of T . Let X be the Lebesgue number of the covering
of N x I . Choose a subdivision T' of T such that
the mesh of the star covering of T' is less than
X/2 . In the above
proof of Theorem 11 use T' instead of Q , and choose b with the additional restriction that
S ^ A/2 .
H
\\\\ \
\\\\\
St (vi, T')
st (w^, T)
Continuing with the same notation as in the proof of The'^-^em 11 , for each i the ball f* (st(v^, T')) , is of diameter less than X contained in H~^(st(w^, T) X I) for some vertex support h^ C. k^ ("^(v^, T')) C as desired .
, and so is
T , Therefore st(w^, T)
- 12 -
Proof of the Corollary to Theorem 11 . g
f
Given M >M ^ Q , where g is isotopic to the identity keeping M fixed , we leave to cover g by a homeomorphism h of Q . Let N be a regular neighbourhood of fM in Q , and choose triangulations of M , N - call them by the same names - such that f : M-^N is simplicial . By the Addendum we can write
where g^ is supported by the ball B™ = st(v^,M) , for some vertex v^ ^ M . Let B^ = st(fv^, Q) . Then the ball pair (B^ , f B™) is unknotted , because f is locally unknotted , and therefore the homeomorphism fg^
of the
smaller ball can be suspended to a homeomorphism , h. say, of the larger ball . • 'HI •n Since g keeps M fixed , g^ keeps B^ fixed , and so h^ keeps B^ fixed. Therefore h^ extends to a move of Q keeping Q fixed . The composition h
h^ h^.-.h^ covers g .
Collars Before proving Theorem 12, it is necessary to prove a couple of theorems about collars of compact manifolds . The theorems can be generalised to non-compact manifolds, but we shall only need the compact case . Define a collar of M to be an embedding c :M X I —
>M
such that c(x,0) = x, all x 6 M . Let f ; M — b e
a proper locally-unknotted embedding between two compact
manifolds, and let c , d be collars of M , Q , We say c , d are
- 13 -
compatible with f if the diagram
QXI f'
fXI M XI is commutative and i m d n i m f
= imfc.
Lemma 24 . Given a proper locally unknotted embedding between compact manifolds, there exist compatible collars . Corollary . Any compact manifold has a collar . (For in the lemma choose the smaller manifold to be a point) , Proof of Lemma 24 . Let M
denote the mapping cylinder of M C M . Then
M = M X I U M , with the identification (%, l) = % , and the induced structure . Then ST*" has a natural collar . The given proper embedding +
+
+
f : M —> Q induces a proper embedding f : M — > Q, with which the natural collars are compatible . Let
denote the retraction maps of the mapping cylinders ,
shrinking the collars ; then the diagram
M
- 14 -
is commutative . We shall produce homeomorphisms c,d (not merely maps) such that Q
M is commutative, and such that c,d agree with ^ on the boundaries . The restrictions of c,d to the natural collars will prove the lemma . Choose triangalations of M,Q - call them by the same names such that f is simplicial, and let M' , Q* denote the barycentric derived complexes . For each p-simplex A £ M , let A
denote its dual in M j
more precisely A* is the (m-l-p)-ball in M' given by A* = O st (v,M') . vf A Using the linear structure of the prisms A x I , A ^ M , define the m-ball A <1 M X I to be the join A"^ = (A X 0) (A* X l) . The • set of all such balls cover M x I and determine a triangulation of M X I ; the latter agrees with M' on the overlap , and so together with M' determines a triangulation of M"^ . Order the simplexes
of K in an order of locally
increasing dimension (i.e. of k. ^ k. then i j) . Similarly order the 1 J simplexes B. , B„,...,B of L such that fA. = B, , 1 < i / r . L d 5 Define inductively ,
- 15 -
M^ - M', 0 '
M. = M. o A. 1 1-1 1
We have ascending sequences of subcomplexes 0
1
Q' = Qq C
r
r+ 1
s
C
such that f^M. C Q. , for each i .We shall show inductively there exist 1 1 homeomorphisms such that
n
M. 1
is commutative, and such that
- M agree with f on the boundaries -
The induction begins trivially with c^ = d^ = identities, and ends with what we want . For the inductive step, fix i , and assume
,
to be
defined . There are two cases . Case (i), i ^ r . For j = 0,1 let a. denote the barycentre , 0 of A. X j , and let b. = f a. . Let P denote the (q - 1, m - l) ball i J 3 pair P = ^Ik (b^ ,
, lk(b^, f^
»
which is unknotted because by hypothesis f ; M — ^ Q is locally unknotted ,
- 16 -
and so by induction f • ^^^
\ ]_ is also . Then b^P is the cone
pair on P , and b^ b^ P the cone pair on ^^ ^ •
Sublemma . There exists a homeomorphigm h ! b b,PUb.P-»b.P that — — — — — — — — — .—0 1 1 1 maps b —yb , is the identity on P , and maps b P b P linearly . Proof . If P were a standard ball pair , and b^P a cone on P , and b^ the barycentre of (the smaller of the pair) b^P , then the proof would be trivial by linear projection . An unknotting homeomorphism from P onto a standard pair maps the given set-up onto the standard set-up , and the sublemma follows by composition . Returning to the proof of the lemma, notice that b^b^P is none other than the ball pair (B^ , f"^ A^) , and so h extends by the identity to a homeomorphism of manifold pairs
- 17 -
Define c^ = c^
and d^ ~
'
>
homeomorphisms
satisfying the commutativity condition . Finally we have to sho¥ that
agree with f on the boundaries.
For points outside A^ , B^ this follows by induction . For points in A't , B^ it follows from the diagram
b^(fP) which is commutative by the linearity of the sublemma . Case (ii) i > r . This case is simplex, because only the larger manifold Q is concerned . In case (i) ignore the smaller ball ; the proof gives a homeomoiphism h : Q ^ — ^
keeping ^^^ =
fixed «
Define c. = c. ^ and d. = d. ^ h . The proof of Lemma 24 is complete . X x-1 1 1-1 Our next task is to improve Lemma 24 in Theorem 14 to the extent of moving the smaller collar from thesis to hypothesis i First it is necessary to ^ow , in Theorem 13 , that any two collars of the same manifold are ambient isotopic , and for this we need three lemmas . Lemma 24 is about shortening a collar ; Lemma 25 is about isotoping a homeomorphism which is not level preserving into one which is level preserving over a small subinterval ; and Lemma 26 is about isotoping an isotopy . In each lemma an isotopy is constructed, and we must be careful to avoid the standard mistake and make sure that it is a polymap (i.e. piecewise linear) . Notation . Suppose 0 < b ^ 0,6
1 . Let I ^ denote the interval
. Given a collar c of M , define the shortened collar .
- 18 -
c ^ :M X I
^M
by Cp (x,t) = c (x, £t) , all x 6 M , t S I .
Lemma 25 .
The collars c , c^
are ambient isotopic keeping M
fixed . Proof . First lengthen the collar c as follows . The image of c is a submanifold of M , and so the closure of the complement is also a submanifold (by Lemma 17), with boundary C(M X l) . Therefore the latter has a collar , which we can add to c to give a collar , d say , of M such that c = d^^^ • Then c ^^ = d
.
By Lemma 16 there is an ambient isotopy G of I , keeping I fixed , and finishing with the homeomorphisffl that maps 1 0, 1/2
, 11/2,
1j
linearly onto 0, , £/2, 1 . Let 1 x G be the ambient isotopy of • M• X I , and let H be the image of I x G under d . Since 1 x G keeps M X I fixed , we can extend H by the identity to an ambient isotopy H of M keeping M fixed . Then H^c = c ^ , proving the lemma . Lemma 26 . Let X be a polyhedron, and f; X x l ^ — » X x I an embeddinis: such that f I X x 0 is the identity . Then there exists S, 0 < ^ < t , and an embedding g : X x I
> X x I such that :
(i)
g is level preserving in I^ ; <3 (ii) g is ambient isotopic to f keeping X x I fixed ; (iii) If Y C X , and f I Y x I^
is already level preserving, then
we can choose g to agree with f on Y x I^ , and the ambient isotopy to keep f (Y X
) fixed .
Proof * Let K , L be triangulations of X x l g
,XxI
such
that f : K —i; L is simplicial (in fact a simplicial embedding) . Choose 5 ,
- 19 -
0 < ^ <. 6 , so small that no vertices of K or L lie in the interval 0
and b , and any point in between tloese two levels lies on
a unique interval that is mapped linearly onto another interval , both intervals beginning (at the same point) in X x 0 and ending in X X ^ . To prove property (ii) define another first derived L" of L by the rule : if a simplex lies in f K then star it so that f : K'__jL" is simplicial: otherwise star it barycentrically . The isomorphism L" — is isotopic to the identity by Lemma 15 Corollary 1 , and so f , g are ambient isotopic . The isotopy keeps fixed any subcomplex of L on which *
L' and L" agree , and in particular keeps X x I fixed . To prove property (iii) we put extra conditions on the choices of K and L' . Choose K so as to contain Y x
as a subcomplex . Having
chosen K , K' , and therefore L" , then choose L' so as to agree with L" on f(Y X
) , this being compatible with the condition of starring on the S-
levd. because f|Y x
is already level preserving . Therefore H keeps
f(Y X Ig ) fixed .
Lemma 27 • Let g ; X x I — » X x I be an ambient isotopy of X . Let h be the ambient isotopy of X that consists of the identity for half the time followed by g at twice the speed . Then g , h are ambient isotopic keeping X x I fixed .
- 20 -
Proof . Triangulate the 2 square I as shown, and let
2 u : I — ^ I be the simplicial map determined by mapping the vertices to 0 or 1 as shown . Define G : (X x l) x I — ^ (X x l) x I by
Then (i) G is a level preserving homeomorphism, and (ii) G is piecewise linear, because the graph T G of G is the 22 intersection of two subpolyhedra of (X X : TG 2
where (l x u)
= ((lxu)^)"Vg^(x^xri) , 2 2
2
denotes the map (X x I ) — > (X x l) , where P g
is the
graph of g , and i 1 is the graph of the identity on I « Therefore G is an isotopy of X x I in itself . By the construction of u , G moves g to h keeping X x I fixed . Therefore g , h are ambient isotopic keeping X x I fixed . Theorem 15 . If M is compact, then anv two collars of M are ambient isotopic keeping M fixed . Proof . Given two collars , the idea is to (i) ambient isotope one of them until it is level preserving relative to the other on a small interval, (ii) isotope it further until it agrees with the other on a smaller interval, and then (iii) isotope both onto this common shortened collar .
-
Let c,d : M x I M
21
-
be the two given collars . Since each maps
t
onto a neighbourhood of M in M , we can choose £ > 0 , such that C(M X
)C d(M X L) . Since c,d are embeddings, we can factor c = d f ,
where f is an embedding such that the diagram M X I, M M XI is commutative, and f j M x 0 is the identity .
S 2S
oh
r '
2i
h 1 •'
/ L 1- -- -VZZTl^
By « Lemma 25 there exists S , 0 <• 2 S • < ^ , and an ambient isotopy F of M X I moving f to g keeping M x I fixed , and such that g is level
- 22
preserving for 0 ^ t ^ 2 S . The reason for making g level preserving is that we can now apply Lemma 27 and obtain an ambient isotopy G of M x moving g | M x I.
to h keeping M x
fixed , and such that h is the
identity for 0 ^ t ^ o . Extend h to an embedding h : M x I^j- — M x I by by making it agree with g outside M x I-
, and extend G by the identity
to an ambient isotopy of M x I . «
•
Then G F is an ambient isotopy moving f to h keeping M x I fixed . Let H be the image of G F under d . Since G F keeps M x 1 fixed , we can extend H by the identity to an ambient isotopy H of M keeping M fixed • Let e = HjC . Then e is a collar ambient isotopic to »
c , and agreeing with the beginning of d , because of x t M and t G. I then e^ (x,t) = e(x,^ t) = H^c(x, S t) = d G^F^d-lc(x, t) = d G^F^f(x, i t) = d h (x,
S
t)
= d(xjt) = dg (x, t) . Therefore e^ =
>
^y Lemma 25 there is a sequence of ambient
isotopic collars c, e, e,- , d . The proof of Theorem 13 is complete . 0
Theorem 14 . Given a proper locally unknotted embedding f :M
Q between compact manifolds, and a collar c of M . then there
exists a compatible collar d of Q . Proof . Lemma 24 furnishes compatible collars , c* , d* say , of • M , Q . By Theorem 13 ^ there is an ambient isotopy G of M keeping M fixed , such that G c = c . By Corollary to Theorem 11 we can cover by a homeomorphism h of Q keeping Q fixed . Let d = hd .
- 23 -
Then the conunutativity of the diagram
M XI and the fact that *
imdoimf
= imhd
0
imhf
= h (im d* a im f) = h (im f c*) = im fc , ensure that the collars c , d are compatible with f . The proof of Theorem H is complete . We now prove the critical lemma for the covering isotopy theorem , Theorem 12 . Lemma 28 . Let M., Q be compact, and F a locally unknotted isotopy of M in Q keeping M fixed . Then there exists £ > 0 . and a short ambient isotopy H ;
x
—
^
of Q, that keeps Q fixed and
covers the beginning of F . In other words the diagram X I, H Q X I,
F X1 o MXI
F
- 24 -
is commutative . Proof. For the convenience of the proof of this lemma we assiane *
that F = F, o 1
For , if not , replace F by F
t
where F
1/2
*
*
Then, since ^q -
>
proof below gives H covering the beginning of
F^ , which is the same as the beginning of F if J: ^ I/2 • Therefore assume embeddings F , F ^ x l
=
of M x l
' in Q x l
means that the two proper agree on the boundary (M x l) ,
because F keeps M fixed . Choose a collar c of M x I , and then by Theorem 14 choose collars d , d^ of Q x l
such that c , d are compatible
with F , and 0 , d^ are compatible with ^^ x 1 .We have a commutative diagram of embeddings ( Q X I)'x I
Qxl
Qxl
Mxl Notice that both the collars d , d maps (Q X 0) X 0 to Q X 0 . Therefore 0 x 0 in Q x l , and so contains Q x Ip^ , im d contains a neighbourhood of Q for some
> 0 , Similarly d^ d~^(Q x
0. X 0 , and so contains Q x
) contains a neighbourhood of
, for some (X , 0 < p< ^
.
- 25 -
F^M
Let G = d d"^ : Q x I , — ^ Q x 0 A jj
. Then G has the properties :
•
(i) (ii)
•
G I Q X I - identity, because d , d^ agree on (Q x l) x 0 j G j Q X 0 = identity .
(iii) G covers the beginning of F in the sense that the diagram QX I o<
G QX I
F X1 0
is commutative . For if x (F^ X, t)
M
and
im (F^ X l) n
Therefore for some y(S(Mxl)
t <£
, then by compatibility
im d^ = im (F^ X l) C .
xl,
(F^ X, t) = (F^ X l) cy = d^ (F X l) y .
- 26 -
Therefore G (F^xl)(x,t) = (dd^^) d^ (F X 1) y = d(F X l) y - F cy = F(F^X1)"^ (F^XI) cy = F(F^xl)"^ (F^x,t) =F(x,t) . In other words G(F^X1) = F , which proves (iii) By Lemma 26 there i s a n £ , 0 < § - < l ' ^ H: Q x l ^ — ^ Q x l ^ .
and an embedding
ambient isotopic to G , such that H | Q x 0 = identity ,
and H is level preserving in I c
• Further, since G is already level
preserving on (Q U F^ M) X I ^^ , we can by Lemma 26 (iii) choose H to agree with G on this subpolyhedron . In other words , the restriction H : Qxlp._> Q x l ^
is a short ambient isotopic covering the beginning of
F and keeping Q fixed . Proof of Theorem 12 . the covering isotopv theorem . ¥e are given a locally unknotted isotopy F : I x I —> Q, x I keeping M fixed , and a subdivision N of the track of F , and we have to cover F by an ambient isotopy H of Q, supported by N keeping Q fixed . ¥e are given that M is compact , and we first consider the case when Q is also compact and N = Q . If 0 < t < 1 5 the definition of locally unknotted isotopy ensures that the restrictions of F to
0 ,t
and
t ,1
are locally
unknotted embeddings , and therefore we can apply Lemma 7 to both sides of the level t , and cover F in the neighbourhood of t . More precisely, for each t ^ 1 , there exists a neighbourhood
of t in I , and a
- 27 -
level preserving homeomorphism H^ ^ of Q x J (t) Q fixed , H = 1 , and such that the diagram u
F^xl
such that H
QXJ
keeps
(t)
MXJ is commutative . By compactness of I we can cover I by a finite niimber of such intervals J^^^ . Therefore we can find values t^jt^j'.-jt^ and 0 = s^ < S2 < ••• Write Hi =
= 1 » such that for each i , "s^ ,
.
.
We now define H by induction on i , as follows . Define H^ = 1 . Suppose H : Q—>Q has been defined for 0 "t Then define H = t t s. 1
s. 1
,
t ^ s. such that H F = F . 1 1 / 0 0
for s_< t / s. , 1 x + 1
Therefore H^ F = to t s. 1
H F s. o 1
= eHE^ r ^ F
= 4
^t. 1
=F t • At the end of the induction we have H u
defined and H, F = F , all t X 0 X
I
- 28 -
Moreover H is piecewise linear, because it is composed of a finite number of • i piecewise linear pieces, and H keeps Q fixed because H does . Tlierefore the proof is complete for the case when Q is compact and N = Q , ¥e now extend the proof to the general case , when Q is not necessarily ccmpact , and N c Q . We may assume that N is a regular neighbourhood of the trad , because any neighbourhood contains a regular neighbourhood . Therefore N is a compact submanifold of Q . By the compact case , cover F by an ambient isotopy of Q covering F supported by N and keeping Q fixed . The proof of Theorem 12 is complete . Proof of Addendum to Theorem 12 . ¥e have to extend a given ambient isotopy H of Q with compact support X to an ambient isotopy H
of Q supported by a given neighbourhood
N of X in Q . (H is not a corollary because the embedding Q x I
QxI
induced by H is not proper) . Without loss of generally we may assume the neighbourhood N of X to be regular , and therefore a compact manifold . Restrict H to X and extend by the identity to an ambient isotopy , G say , of N keeping N - X fixed . 2 Triangulate the square I as 2
shown, and let n : I — b e
the simplicial
map determined by mapping the vertices to 0 or 1 as shown. * Define G : (N x l) x I — ^ (N x l) x I by
As in the proof of Lemma 27, it follows that G N X I keeping (N x I) U (N - X) x I fixed .
is an ambient isotopy of
- 29 •
*
Choose a collar o :•W x I —^ N and *let H be the image of G * under c . Since G keeps N x 1 fixed , H can be extended by the
*
identity to an ambient isotopy of N ; and since G keeps (N - X) x 0 , * H keeps the frontier of N fixed , and so can be further extended to an ambient isotopy H H , as desired
of Q supported by N . By construction H
extends
Proof of Corollary 1 to Theorem 12 . Corollary 1 is concerned with the case when the isotopy F of M in Q does not keep M fixed . Let T be the track of F in Q , which •
•
•
is compact since M is compact . Let F : M x I — x I denote the restriction of F to the boundary , which is locally unknotted because F •
•
•
is . Let X be a regular neighbourhood of the track T 0 Q of F in Q , and let N^ be a regular neighbourhood of X in Q . By choosing X , sufficiently small , we can ensure that the given neighbourhood N of T is also a reighbourhood of N^ . Use Theorem 12 to cover F by an ambient isotopy of Q supported by X , and by the Addendum extend the latter to an ambient isotopy , G say , of Q supported by N^ . Then G~^F is an isotopy of M in Q keeping M fixed , whose track is contained in T U neighbourhood of T o
. But N is a
, and so we can again use Theorem 12 to cover G"^ F
by an ambient isotopy , H say , of Q supported by N . Therefore GH covers F and is supported by N . Recall
Lemma 16 . Any homeomorphism of a ball keeping the boundary fixed
is isotopic to the identity keeping the boundaiv fixed . Corollary . Any homeomorphism of a ball keeping a face fixed is isotopic to the identity keeping the face fixed . For by Theorem 2 the ball
- 30 -
is homeomorphic to a cone on the complementaiy face . First isotope the complementary face back into position, and extend the isotopy conewise to the ball ; then isotope the ball . Proof of Corollary 2 to Theorem 12 . We have to show the equivalence of (1) isotopic, (2) ambient isotopic, (3)
ambient isotopic by an ambient isotopy with compact support,
(4)
isotopic by moves .
and
(1) implies (3) by Theorem 12 and Corollaiy 1, because we can choose the neighbourhood N to be compact. (3) implies (4) by Theorem 11 . (4) implies (2) by Lemma 16 and Corollary, because then each move is ambient isotopic to the identity , Finally (2) implies (I) trivially . Proof of Corollary 3 to Theorem 12 . If P is a locally trivial isotopy , then by definition each point in M X I has a neighbourhood which is locally unknotted 1 therefore F is locally unknotted . Conversely if P is a locally unknotted isotopy , then the level P
is a locally \mknotted embedding , and so the constant isotopy
P^ X 1 is locally trivial . By Theorem 12 cover P by II ; then P = H(P^ X l) is locally trivial , because the homeomorphism H preserves local triviality . This completes the proofs of the theorems and corollaries stated at the beginning of the chapter .
- 31 -
Remarks
on
combinatorial
isotopy .
We have framed the definitions of isotopy and proved the theorems in the polyhedral category , because that is the spirit of these seminars . In other words , there is no reference to any specific triangulations of either of the manifolds concerned . However there is a definition of isotopy in the combinatorial category when the receiving manifold Q happens to be Euclidean space , by virtue of the linear structure of Euclidean space . The manifold M is given a fixed triangulation , K say , and the isotopy is defined by moving the vertices of K . At each moment the embedding of M is uniquely determined by the positions of the vertices , and by the linear structure of Euclidean space . But a general polyhedral manifold Q has only a piecewise linear structure , not a linear structure, and so the positions of the vertices of K do not determine a unique embedding of M . It is no good picking a fixed triangulation L of Q , and considering linear embeddings K —> L , because this has the effect of trapping M locally , and preventing the movement of any simplex of K across the boundary of ary simplex of Q . Therefore to obtain any useful form of isotopy it is essential to retain the polyhedral structure of Q , even though we may descend to the combinajorial structure of M . We now give a definition in these terms , which looks at first sight much more special than the definitions of isotopy above , but in fact turns out to be equivalent | we state the theorem without proof • The moral of the story is : stick to the polyhedral category and don't tinker about with the combinatorial category ; keep the latter out of definitions and theorems , and use it only as it oizght to be used , as an inductive tool for proofs .
- 32 -
L i n e ar
moves Let
with
respect
to
a
trian
be the standard q-simplex , and
face , q ^ m . Let x be the barycentre of and the barycentre of
gulation an m-dimensional
A ^ . and y a point between x
^ ^ . Let (j- : A ^ ^
^ be the homeomorphism
throwing x to y , moping the boundary by the identity , and joining linearly . Let M be closed , K a triangulation of M , and let f , g : be proper embeddings . ¥e say there is a move from f to g linear with respect to K if the following occurs : There is a closed vertex star of K , A = st (v,K) say , and a q-ball B c. Q , and a homeomorphism h : B —^ ^ ^ such that (i)
o f,g agree on K - A ,
(ii) A = f-^B = g"^B , (iii) hf maps lk(v,K) -—> A V
X
A
X
, homeomorphically ,
m , by joining linearly
(iv) g I A = h"^ n- h(f I A) .
- 33 -
Roughly g)eaking, h is a local coordinate system , chosen so that the move from f to g looks as simple as possible , just moving one vertex of K linearly in the most harmless fashion, like a move of classical knot theory . Addendum
(stated without^proof) . Let M be closed , and K an
arbitrary fixed triangulation of M . Let f,g ; M — » Q, be proper embeddings that are locally unknotted and ambient isotopic . If codimension > 0 , then f . g are isotopic by moves linear with respect to K . The addendum becomes surprising if we imagine embeddings of a 2-^here in a manifold , and choose K to be the boundary of a 3-simplex , with exactly 4 vertices . Then we can move from any embedding to any other isotopic embedding by assiduously shifting just those 4 vertices linearly back and forth . All the work is secretly done by judicious choice of the balls , or local coordinate systems in the receiving manifold , in which the moves are made . Remark . Notice the restriction codimension > 0 that occurs in the addendum (but not in Theorem 11 for example) . It is an open question as to whether or not the restriction is necessary . In particular we have the problem ; is a homeomorphism of a ball that keeps the boundary fixed isotopic to the identity by linear moves ?
1 INSTITUT DES HAUTES ETUDE~ SCIENTIFIQUES
Seminar on Combinatorial Topology by E. C. ZEEMAN. - .......•..•. ....--..-.....--.---
The University of Warwick, Coventry.
l
J
INSTITUT des HAUTBS ETUDES SCIENTIPIQUES 1963
Seminar on ComlDinatorial Topology By
E.G. ZEEMAN
Chapter 6; GENERAL POSITION General position is a technique applied to (poly) maps from polyhedra into manifolds.
The idea is to use the
homogeneity of the manifold to minimise the dimension of Intersections.
Throughout this chapter
X,Y will denote
polyhedra and M a compact manifold.
The small letters
will always denote the dimensions of
X, Y, M
we shall assume
x, y, m
respectively, and
x,y<m . In particular we tackle the following
two situations. Situation (1)
let
f; S
M
he an embedding and let Y he a
suhpolyhedron of M , In Theorem 15 we show it is possible to move f to another embedding g such that dimension, namely
gX A Y is of minimal
< x + y - m . We describe the move
f
g by
saying ambient isotope f into general position with respect to Y. There are refinements such as keeping a stibpolyhedron Xq of fixed, and moving Situation (2) embedding.
let
f|X-XQ fs X
into general position« M
be a map^n6t^iiecessarily an
First we show in lemma 32 that f
homotopic to
X
-2a non-degenerate map, g say, where non-degenerate means that in any triangulation with respect to which g is simplicial each simplex is mapped non^degenerately (although of course many simplexes of X may be mapped onto one simplex of M). Next we show in Theorem 17 that g is homotopic to a map, h say, for which the self-intersections are of minimal dimension, namely
s 2x - m , Not only the double points but
also the sets of triple points, etc«, we \«ish to make minimal. We describe the composite homotopy move f into general position. keeping Xq fixed if
f|XQ
(x^]
g -> h
by saying
There are refinements such as
happens already to be in general
position, and arranging also for for a finite family
f
f|X^
to be in general position
of subpolyhedra (notice that the
general position of f does not imply the general position of f|X^
unless
X = Xj^ ) „
We observe that situation, (2) is part of a general programme of "Improving" maps, and is an essential step in passing from algebra to geometry.
For example suppose that an
algebraic hypothesis tells us there exists a continuous-map X -> M
(we use the hyphenated "continuous-map" to avoid
confusion with our normal usage map - polymap), and that we want to deduce as a geometrical thesis the existence of a homotopic embedding . X C M . Then the essential steps are;
-3-
continuous map simplicial approximation N/
map local hoinotopy
<
general position*^ > Chapter 6
non-degenerate map general position NI/ \
gloToal homotopy
^map in general position
embedding Remark on homotopy The general programme is to investigate criteria for (1) an arbitrary continuous-map to be homotopic to a polyembeddingj and (2) for two polyembeddings to be polyisotopiCo
Therefore although
we are very careful to make our isotopies piecewise linear (in situation (1)) we are not particularly interested in making our homotopies piecewise linear (in situation (2)), We regard isotopy as geometric, and homotopy as algebraictopologicals Invariant definition
If M is Euclidean space, then general
position is easy because of linearity;
it suffices to move
the vertices of some triangulation of X into "general position", and then the. simplexes automatically intersect
-4mlnimally.
However in a manifold we only have piecewise
linearity, and the prohlem is complicated Toy the fact that the positions of the vertices do not uniquely determine the maps of the simplexes?
therefore the moving of the vertices into
"general position" does not guarantee that the simplexes intersect minimally.
In fact defining general position in
terms of a particular triangulation of X leads to difficulties. Notice that the definitions of general position we have given ahove depend only on dimension, and so are invariant in the sense that they do not depend upon any particular triangulation of X or M « The advantages of an invariant definition are considerable in practice,
For example, having
moved a map f into general position, we can then triangulate f
so that f is hoth simplicial and in general position (a
convenient state of affairs that was not possible in the more naive Euclidean space approach).
The closures of the sets of
double points, triple points, etc. will then turn out to be a descending sequence of subcomplexes, Transversality
In differential theory the corresponding
transversality theorems of I'hitney and Thorn purpose, because they assume
X,Y to be manifolds.
in our theory it is essential that polyhedra than manifolds.
serve a different Whereas
X,Y be more general
For general polyhedra the concept
of "transversality" is not defined, and so our theorems aim at minimising dimension rather than achieving transversality.
-5^'i^.en S,Y
are manifolds then transversality is -well defined in
comlDinatorial theory, hut the general position techniques given "below are not sufficiently delicate to achieve transversality, except in Theorem 16 for the special case of O-dimensional intersections (x + y = m) , ^ITien X + y > Q follows„
the difficulty can be pinpointed as
The basic idea of the techniques below is to reduce the
intersection dimension of two cones in euclidean space by moving their vertices slightly apart„
However this is no good
for transversality, because if two spheres cut combinatorially transversally in E^ , then the two cones on them in
, with
vertices in general position, do not in general cut transversally; there is trouble at the boundary. The use of cones is a primitive tool compared V'Sith the function space techniques used in differential topology, but is sufficient for our purposes because the problems are finite.
It
might be more elegant, but probably no easier, to work in the combinatorial function space. Wild embeddings
Without any condition of local niceness, such
as piecewise linearity or differentiability, then it is not possible to appeal to general position to reduce the dimension of intersectionso
For consider the following example.
It is
possible to embed an arc and a disk in E'^ (and also in E^ , n > 4 ) intersecting at one point in the interior of each, and to choose
C > 0 , such that is is impossible to ^-shift the disk off
the arc (although it is possible to shift the arc off the disk). The construction is as followss
Let A be a wild.arc in E^ ,
and let D be a disk cutting A once at an interior point of each, such that 1) is essential in E^ - A . If we shrink A to a point X ,,and then multiply by a line, the result is 4-space, ( E V A ) X E = E'^
(by a theorem of Andrews and Curtis).
denotes the iaage of D in in one point
X K O , and
e V A , then D'XIO
D'X 0
If D'
aeets
is essential in
E^ - (x xR) .
Therefore if £ is less than the distance bet\s?oen ^D'X 0 X ;!< R , it is impossible to £-shift the disk Compactness
D'XO
xxR
and
off the arc xxR,
We restrict ourselves to the case when X is a
polyhedron and therefore compact.
Consequently we can assume
that M is also compact, for, if not, replace M by a regular neighbourhood W of
fX
in M .
Then N is a compact manifold
of the same dimension, and moving f into general position in N
a fortiori moves f into general position in M .
General position of points in Euclidean space
Before we can
move maps into general position we need = precise definition of the general position of a point in Euclidean spape E^ with respect to other points, as follows.
Let X be a countable
(finite or denumerable) subset of E^ . Each point is, trivially, a linear subspace of E^ , and the set X generates a countable sublattice,
IJ(X) say, of the lattice of all linear subspaces of
E^ , . Let fl{X)
be the set union of all proper linear subspaces
in L(X) 0 Since l(X) is countable, the complement
E^-IKx)
is every?>/here denser
Define
with respect to X if
y C E^
to be in general position
y ^ A(X) »
Now let ^ be an n-sim.plex and X a finite set of points in A „ We say
y
A is in general position with respect to X
if the same is true for some linear embedding ^ c E ^
(the
definition being independent of the embedding).
Let
A' be a
subdivision of A , with vertices
say.
x^jXgj^.^jX^
We define
an ordered sequence (y.,o„«,y_)C A to be in general position with I
S
respect to A' if, for each i,
1 $ iS s , y^^ is in general
position with respect to the set (x^ ,. o» yX^ , y^ ,, . <, Lemma 29 Given a subdivision A' of A and a sequence
).
of vertices of A" (not necessarily distinct), then it is possible to choose a sequence position with respect to ^
, such that
y^
) C A
in general
is arbitrarily
v^ , 1 $ i £ s .
close to Proof
(y-j ,». =
Inductively, the coinpleGient A
dense at
,..«
) is
v^ ^ enabling us to choose y^ arbitrarily near
Remark 1
Notice that all the
Remark 2
This is the first time we have used the reals;
v^ .
y^^ have to be interior to A .
previously all our theory would work over the rationals, and even now it would suffice to use smaller field, like the algebraic number field. Remark 3
There is an intrinsic inelegance in our definition of
a sequence of points being in general position^ because if the
-7-
order is changed they may. no longer "be soo
To construct a
counter-example in E^ , choose 4 points X such that jfKx) contains all rationals on the real axis (regarding E
as the
complex n u m b e r s a n d then add IT , / f
in that order.
To get
rid of this inelegance, and at the same time preserve the lattice property would be more trouble than it is v\;orth; because all we need is some gadget to make Lemmas 30 and 34 worko •ISOTOPING 5I.1EEDDINGS INTO GENERAL POSITION (1) of the introductiono let M be a manifold. Let
gs X
M
Let
Let
be a map«
X^C X , Y C M
We consider situation be polyhedra, and
x = dim X ~ X q , y = dim Y , m = dim S We say that
gjX-X^
is in general
position with respect to Y if dim igiX-X^) n Y) S x + y - m . Theorem such that
Given
XqC X
^
fCX-X^) C M ,
YCM,
and an embedding
f ; X -^ M
then we can ambient isotope f into g
by an arbitrarily small ambient isotopy keeping M and the image of Xq fixed, such that
g|X - Xq
is in general position with
respect to Y . Remark 1
In the theorem we say nothing about
general position.
^I^Q
being in
In fact in many applications for engulfing
in the next two chapters,
definitely not be in
general position with respect to Y . The intuitive idea is to think of Xq and Y as large high-dimensional blocks, and as a little low-dimensional feeler attached to Xq by its
frontier
^q
^~
°
theoreai says we can aoalDient
Isotope the feeler keeping its- frontier fixed so th-^.t the interior of the feeler meets Y minitnally (although its frontier may not). f Xq
In other applications we may already have
in general position, as in the follov^ing three corollaries
Gorollary 1 fXQ
If
does not meet
f|Xq
is already in general position, or if
fCX-X^) , then Theorem
is true for maps
as well as embeddings„ Proof
Apply the theorem to the embedding of the image
fx C B/I , and ambient isotope fX into general position with respect to Y keeping
fXQ
fixedo
(Notice the extra
hypothesis is necessarj/', otherwise having to keep may prevent us from moving awkward pieces of
fX^
" ^Q
fixed
that
overlap Xq ) . Gorollary 2 Y C M
(interior Case)
Given a map
f; X
it and
then we can ambient isotope f into general position
with respect to Y keeping M fixed. For put
Xq = 0
Gorollary 3 let S
in Corollary 1„
(Bounded Case)
XQ = f"^M, Yq = Y A M . such that
g|Xq
Given a map
f ; X -> M
and
Y C
Then we can ambient isotope f
J
^
is in general position in M with respect o
to Yq , and
g X - Xq
in general position in M with respect
12 Y „ Proof
First apply Corollary 2 to the boundary, and extend the
ambient isotopy of M' to M by Theorem 12 Addendum; Corollary
then apply
-9-
Por the proof of Theorem 15 we shall use a sequence of special moves which we call t-shifts, and which we construct beloWo
The parameter t concerns diniensionj, v
The construction involves choices of local coordinate systems . (i.e. replacing the piecewise lineir structure by local linear structures) and choices of points in general position., The t-shift of an embedding of
X , Xq
and
siaplicial=
By Theorem 1 choose triangulations
M, Y with respect to which
Let
f; X
M
K, L denote the triangulations of
is
X,M.
Let
K' 5 L' denote the barjrcentric derived complexes modulo the (t-1)-skeletons of dimension f ° K'
E, 1 (obtained by starring all simplexes of
> t , in some order of decreasing dimension).
L'
Then
remains simplicial because f is non-degenerate (it
is an embedding)o Let A be a t-simplex of K , and t-simplex of 1 = Let fa = b)«
the image
3, b be the barycentres of
A,B (with
Then st(a,K') = aAP
where
B=fA
st(b,L')
P, Q are subcomplexes of
K',L' . If
' b BQ A a' Xq , then
dim P < X - t - 1 J and Q is an (m-t-1 )-sphere because
fA C S „
Let f ^ s a AP
b BQ
denote the restriction of f <, Then
f^
is the join of three
-100
maps 0
AP
a
a b 5 A ->B and
?->Q , and therefore eaibeds the frontier
0
of
aAP
9
in the boundary
O
BQ
of the Le-ball bBQ .
The idea is to construct another efabedding o
o
g.^ I aAP that agrees with
f^^
bBQ AP , and is ambient
on the frontier
isotopic to f,i. "! keeping the boundary BQ fixed„ ?/e shall call the move f. g, a local shift, and give the explicit constructA
ion belov/o
_____
Prom the construction it will be apparent that g^
can be chosen to be arbitrarily close to f^ , and the ambient isotopy,be made arbitrarily small. Now let A run over all t-simplexes of K ° for each A C X - Xq A C Xq X
construct a local shift
define
°
f^
g^^ , and for each
closed stars
cover
and overlap only in their frontiers, on which the
with f , and therefore with each other.
to f „ Also since the stars jst(b,L')^
agree
Therefore the |gj\J
g o X
combine to give a global embedding
{s^
M
arbitrarily close
overlap only in their
boundaries which the local ambient isotopies keep fixed, the latter combine to give an arbitrarily small global ambient isotopy from f to go
Moreover the ambient isotopy is
supported by the simplicial neighbourhood of
fCX-X^)
in L' ,
0
•and so in particular keeps We caU. the mov.e keeping Xq fixed.
fXQ U M
f g
fixed.
a t-shift with respect to Y
Notice that Y entered into the construction
-11-
^A/hen choosing the trisngulation L of M so as to have Y a subcomplexo Local shift of. an eahedding
We are given a simplicial eaibedding
f s aAP ^ bBQ «
which is the join of the three maps
0
a.-^h , A-^B and P->Q ,
and we want to construct o
o
g S aAP (We drop the subscript A from
TDBQ . f,
and
g. o)
J.1
ii
Now Q is an (a-t-1 )-sphere, and "by construction a suhcomplex of Q , and hy hypothesis both lower dimension than Q = Therefore, if ^
YAQ
Y jH Q is
and fP are of
is an (m-t)-simplex
with an (m-t-l) face T , we can choose a hoQeoniorphistn h IQ ^ throwing
A
fP U ( Y A Q ) into the face P „
of A , and extend
h :Q
and' joining linearly.
to
h ; bQ
by mapping
b-^v
A'
Choose v^ near v in A in general position with
respect to A' . Then in particular of A .
A
Choose subdivisions such that h ; (bQ)'.
is simplicialo
Let v be the barycentre
because v is a vertex
Define the homeomorphism k^ ° A
A
to be the join of the identity on A to the map k ; bBQ
v-^v^ <, Define
bEQ
to be the join of the identity on B to the homeomorphism ; bQ
bQ .
-12o
Then
1: is a ho.x-oaorphisQ of the To-11 "bBQ keeping the
boundnrj?- fixed.
Define o
g = kf s aAP
•
bBQ
.
Then g is am"bient isotopic to f keeping the "boundary fixed "by Lemnia 15.
We can make g arbitrarily near f , and the isotopy
arbitrarily small, by choosing v^ sufficiently near v . This completes the definition of the local shifts
Since , f, k are joins, it follows that
Remark
is the join of
g ; aP
bQ' to
f s A
g s aAP
B . However
bBQ
g s aP
bQ
is not a join with respect to the simplicial structure of bQ , as the, diagram shows, but is a join V\/ith respect to the linear structure induced on bQ from A by h Lemma 30
G-iven the hypothesis of Theorem
let
f-^g
be a
t-shift with respect to Y keeping Xq fixed. (i) If f is in general position with respect to Y , then so is g . (ii) On the other hand if f is not in general position, and if di;;..i(f(A-Xq) /^Y) = t > x + y - Q
then
dim(g(X - Xq) A Y) = t - 1
-13Proof«
(i) « It suffices to examine the local shift from f to
g ; aAP
ToBQ , for a t-simplex
A C X - Xq . Since g agrees
with f on the frontier AP , we have dim(g(X--XQ) A Y ABQ) £ dim(f(X-XQ) AY) ^ x + y - m „ Therefore it suffices to examine the intersection of with the interior of the hall
bBQ .
Since Y is a suhcomplex of L , Y "bBQ only if
g(X-XQ) A Y
meets the interior of
B C Y , and so we assume this to be the casBo
Therefore g(X-XQ) A Y ^^ bBQ
=
Bh~^ (v^hfP n vh(Q rsY))
and so dim(g(X-XQ) n Y A int(bBQ)) = t + max dim(v^O^nvC) where the maximum is taken over all pairs of simplexes of
such that
v^C^ n vC
and such that
meets the interior of ^ »
Since hf , h
G^ c hfP , C c h(QAY)
C^,C
BC^ , BC
are in the images of
respectively 5 we have
dim
X - Xq , Y
under
< x - t - 1 , dim C ^ y - t - 1 »
vC n v^C^
m-t-1
-14Reg^.rcl ^ as eciloedded. in
, and let
[cQ
denote the linear
subspace spanned "by C . There are two possibilities, according as to whether or not Case (a)
and
span
[A]
[c] and C
[c^]
span
"1
span [rj . 0 Therefore
[vC] and
and so
dini(v^C^ n vC) < dim v^G^ + dim vC ~ dim. A - (x-t) + (y-t) - (m-t) „ Therefore t +
Case (b) and
n vC) <
[cj and
x+y-m.
.
[c^] do not span [ P i . Therefore
tc'-jj span a proper subspace of [A}?
contain
{vC]
v\'hich does not
, by our choice of v^ in, and the definition of,
general position in A with respect to A ' » (Note that this application was the reason for our definition of the general position of a point off the lattice of subspaces generated by the vertices of A' 5 a more complicated application of the same kind occurs in Lemma 34 below,) v^C^ A vC = C^ A
c
which does not meet the interior of A assumption that it dido
Therefore
, contradicting our
Therefore case (b) does not apply, and
the proof of part (i) of Lemma 30 is complete„
-15(ii) We are given
diEL(f(X-XQ) A Y ) = t > x + y - m ,
and have to show that the t-shift drops this diniension hy onBo Again it suffices to examine the local shifts f(X-XQ)
Y
Since
is contained in the t-skeleton of L , and since
bBQ A (t~skeleton of L ) = B
,
we have g(Z~XQ)AYABQ which is of - dimension for some B , and so sho"w
t - 1 o Moreover
,
f(X-XQ) A Y => B ,
dim(g(X-XQ) A Y) > t - 1
Conversely, to
dim(g(XrXQ) A Y) < t - 1 , it suffices to show that
GIX-X^) A Y If
= f(X-Xq) A Y A BQ C B
B<^.Y
does not meet the interior of
TDBQ , for any B .
this is trivially true, and so assume
B = fA C Y «
Again there are two cases, and, as above, only case (a) applies. In case (a), dim(v^C^ A vG) ^ (x+y - m) - t < 0 , and so
v^C^ A vC
is eaipty.
Therefore
gCX-X^) A Y
does
9
not meet the interior of
loBQ , and the proof of Lemma 30 is
complete. Proof of Theorem 15 and assume
Given
f? X
M , let
s = dim(f (X - XQ) A Y)
s > x + y - m , otherwise the theorem is trivial.,
Perform t-shifts for
t = s, s~1
,x+y-m+1 , in that order;;
by Lemma 30(ii) each t-shift knocks the dimension of the intersection down by 1 .. until we are left with an embedding in general position A'v'ith respect to Y
Each t-shift, and
-16therefore also their cociposition, can Toe realised Toy an arbitrarily small atntient isotopy keeping
fX^ U M
fixed.
The proof of Theorem 15 is coEiplete<, O-DinENSIQML TRANSVERSALITY that
x + y = in , and let
Let
fs X
X, Y, 1 M
and
be manifolds such
Y c M be proper
eQibeddings such that f is in general position with respect to Y o Therefore M . Given
fX rv Y
v f. fX
is a finite set of points interior to we say
f
is transversal to Y at v
if 5 for soDie (and hence for any) triangulation of X Y
^ ^
containing v as vertex,/ there is a homeociorphisni st(v,M) throv>/ing
st(v,fX) , st(v5Y)
= E^XE^ onto
E^, E^
respectively»
We
say f is transversal to Y if it is transversal at each point of
fX r\ Y .
Example
Let M be a 4-ball with boundary S^ , and let
X, Y
be two locally unknotted disks in M formed by joining the centre to two unknotted curves in S^ that link more than once. f s X C M
Then
is in general position with respect to Y , but not
transversal»
-17Tlieorem 16 and let
Let
X, Y, M
f; S
M
and
"be oianifolds such that
Y C M
x+y = m ,
""oe proper embeddings,
Then we
can amlDient Isotope f into g "by an arbitrarily small amlpient isotopy such that g is transversal to Y o Proof . First aDi"bient isotope f into general position, and then perforni a 0-shift with respect to Y o Since
fX O Y
does not
meet the "boundaries of each local shift it suffices to examine the interior of one local shift "because
g^ s AP
BQ
( A5B = points
t = 0 ) „ As in the proof of Lemma 30 ^ only case (a)
applies, and the intersection of the two cones in
consists
of a finite num"ber of points where an x-simplex crosses a y-simplex at point interior to "both; Therefore
g^^ is transversal to
such a crossing is transversal,
BQ A Y , and so g is transversal
to Y „ SINGULAR SETS
We now pass onto situation (2) of the
introduction, to the homotoping of maps into general position. f; X
Let
M
be a map between .polyhedra (for this definition
it is not necessary that M be a manifold). singular set
S(f)
of f is defined by;
S(f) = closure [ X £ X ; f ^ ^ f x ^ x ] « Then
S(f) = 0
The branch set Br(f) =
if and only if f is an embedding. Br(f) |x X;
of f is a subset of S(f) defined bys no neighbourhood of x is embedded by f J <.
-18We deduce that
Br(f)
is closed, and that
Br(f)=^
if and
only if g is an immersion. The r"^^ singular set
S^(f)
S^'(f) = ^ x X ; S^(f)
of f is defined Toy;
f"^fx
contains at least r points] „
= closure S^'(f) ..
Thus S2(f) is the closure of the double points, triple points, etc„.
the
We deduce
Z = S^(f) => S^Cf)
^ S^(f) .
S(f) = S2(f) = S2'(f) U Br(f) „ To prove the last statem.ent it suffices to show therefore suppose
x
S^ - SgV»
Sg - Sg' C Br ;
Then there is a sequence
X n -^-x, that is identified with a disjoint sequence • which tends to a limit fX = fy , and so
x=y
"because X is compact., because
Hence
x
'
S^ - S2' C Br., in general
S2 - S2' ,
S^ - S^' ff, Br
some n , and so it is
Br .
Notice that although Br ^
Therefore
x ^ S^ . Consequently any
neighbourhood of x contains not embedded.
{y^ »• n 1 4,
and
, for
r> 2 .
The singular sets have been defined invariantly, without reference to any triangulationo
Now choose triangulations
K, I of X, ¥1 such that
L
f; K
is simplicialo
-19
Lemma 31
(i) There is an integer s , and a decreasing
sequence of subcooiplexes K = K^
such that —^——
K
r
K^ ^ ...
Kg = Kg^^
=
... = K,
oo
= S (f) „ r
(ii) S^(f) = 0
if and only if f niaps every
simplex of K non-degenerately. (iii) There is a subcotaplex L , K g ^ L ^ K ^ , such that
1l| = Br(f) ^
dim (L-K^) < dim K^ „
(i) Let A he a p-simplex of K . We shall show th^.t
Proof
if 1 meets
S^'(f) then
ACS^'(f) „
For
f"'^fA is the
disjoint union of open sitnplexes of K , and must contain either a simplex of dimension > p , or at least r simplexes of o
dimension p.
In either.case, each point of A is identified
under f with at least Therefore closure
r-1
other points, and so
A d S^(f) .
S^'(f) is the union of open simplexes, and so the S^(f)
is a subcooiplex, K^ say«
Let n "be the nuQiToer of simplexes of K . If
A
K^ -
,
o
then A is identified with at least so
r < n . Therefore
least r such that (ii)
K^ = K^
for
r - 1 other simplexes, and r > n . Define s to "be the
K^ = K^ .
If a simplex is mapped degenerately then a continuum
is shrunk to a point and
S^^(f)
0 . Conversely if every
simplex is mapped non-degenerately then at most n points can he identified, and so
S^(f) = 0 >
-20(iii)
If
A 6 K^ , then A faces some simplex mapped
degenerately and so
mapped non-degenerately, and either according as to ^.whether or not Br(f)
A ^ K^ , then
A C Br(f) « If
AC.X-Br(f)
lk(A,K)
st(A,K) is or
A C Br(f)
is embedded„
Therefore
is the union of closed Bimplexes, and is therefore a
suhcomplex, L say. If so ''xA
L - K^ , there is a vertex
A
K^ =
Therefore
x
S2(f|lk(A,K)) 5 and
dim A < dim Kp , and so
dim (L-K^) < dim K2 « NON-DEGENERiiOY
Define
f? X
M
to he non-degenerate if
Soo(f) = 0 - The justification for the definition is Lemma 31 (ii) . We recall that this is equivalent to the more general definition given in Chapter 2» Lemma 32
Given a map
manifold M , with
f; X
M
from a polyhedron X to a
x < m , then we can homotope f to a non-
degenerate map g "by an arbitrarily small homotopyo Xq C X
and
f]Xq
If, further,
is already non-degenerate, we can keep Xq
fixed during the homotopy. Proof
Choose triangulations of
is simplicial;
X^M
with respect, to which f
let E he a first derived complex of the
triangulation of X , and let
B^,«.»^B^
denote the open vertex
stars of the triangulation of M (each B^ is either an open m-cell or a half-open m-cell, according to whether vertex lies in the interior or boundary of M)„
The set
[b | is an open
-21covering of M , and f has the property; (P) for each
ASK,
f(it(A,K)) C sotne B^ o Any map s
X
M
sufficiently close to f also satisfies (P) , Now order the sioiplexes
A^^^o.^A^
of K in some
order of locally increasing dimension (i,eo if A.< A. then i 1 J i£ D ) J and let K. = k^A. « We construct, by induction on i , 1
starting with
1 ^
fq ~ ^ '
(i)
sequence of maps
f^ ;
M
such that
arbitrarily small homotopy keeping
fixed, (ti) f^ (P) , 1 satisfies 1 The end of the induction g= f^ proves the lemmao (iii) f. IK. is non-degenerate For the case when Xq C X and fjXQ is already nondegenerate, we choose the original triangulation so as to contain Xq as a subcomplex, choose the ordering so that Xq=
, some j , and then start the induction at j , with
f, = f . 0
We must now prove the inductive step. defined, and let the
B^
such that
S
h ° B
A=
Assume
, L=st(A,K) , and let B denote
f. .L C B » Choose a homeomorphisn I
onto a simplex and define
g=h f ;
in other words
the diagram ^i-1
L
^ B A'
is commutative, We do not claim that f^ ; -> M is siaplicial, nor do we claim that f^ embeds end simplex of K^ in M , but only that
-22Choose subdivisions L'
A'
L',A' of
such that
is siiiiplicial, and such that L' has at least one
vertex in 1 » Let
denote the vertices of L'
6
contained in A , and let
,».»
denote the remaining
vertices of L' . Choose a sequence of points
(y^
^Jp)
in general position with respect to A' ? such that arbitrarily close to
^ 1 s n £ p <,
Define
y^^ is
g' sL'
A
to be the linear uiap determined by the vertex map g'x^
=
fy^ '
1 S n< p
^gx^ 5 Then so
p< n c q
s' A = g A 5 v;hich is non-degenerate by induction, and
g|A
is non-degenerate by our choice of y's because
dim A < dim A . Now define outside L , and on L
f^ ; X
M
so that f^ agrees with
the diagram f. i
L
> B
A
is commutative„
Having defined f^ we must verify the three
inductive properties. Firstly
g'
g
by straight line paths in A , keeping —1
the frontier
FrCl^K)
extended to a homotopy
fixed. f^
choice of ordering of A's , keeps
fixed.
Therefore
h
—1
g'
h
suppoited by 1 . C K
g
can be
By the
, and so the homotopy
Secondly f^ satisfies (P) provided the
-23honiotopy is sufficiently siiiall« "because
Thirdly
is non-degenerate
U A <, and f^ is non-degenerate on
K^^ =
induction and on A by construction.,
by
The proof of LeoiGia 32 is
complete. GENEML POSITION OF MAPS
Consider maps of Z^ into M^"^ , where
X < m. . Define the codimenslon c = m - X Define the double point dlaenslon d
=d2
= x- c =
2x-m.
More generally define the r-fold point dimension d^ = x -
Define
g; X
M
(r-l)co
to be in general position if dira S^(g) < d^ , each r .
Our principal aim is nov>/ to show that any m.ap is homotopic to a map in general position. Remark 1
The dimensions are the best possible, as can be seen
from linear intersections in euclidean space Remark 2
If f is in general position then f is non-degenerate
and dim Br(f) < dg = The first follows from Lemma 31 (ii) , because we are assuming X < m 5 and so
d^ < 0
from Lemma 31 (iii) .
for r larger the second then follows
Recnark 3
We shall confine ourselves to the interior of rrianifolds
for simplicityo the lDOundary«
The engulfing theorems are especially tricky at In applications the boundary problern can generally
te treated independently and more el gantly bj?" the Addendum to Theoreta 12. Repiark 4
In applications we frequently have the relative
situation of A^v-anting to keep a map fixed on a sub polyhedron Xq of X which already happens to be in general position (and is often embedded).
Therefore before stating the main theorem we
introduce a relative definition. Suppose
Xq C X o Define
g? X
M
to be in general
position for the pair (X,Xq) if (i) (ii) (iii) Remark 1
g is in general positiono g|XQ is in general position, if If
Xq < X J then
diEi(S^(g) H Xq) < d^ , each r.
x^ = x , then (i) implies (ii), and (iii) is vacuous
and so then general position of g implies general position for (XjXq)
. But if
Remark 2
Xq<
X , then (i) does not imply (ii) or (iii).
Condition (iii) is, surprisingly, the best possible.
At first sight it would seem that we ought to be able to make dim (S^(g)nXQ) < d^ - (X-Xq) instead of merely
d^ - 1 . But if X is not a manifold,' then
the non-homogeneity of X may cause certain points of X always to lie in S^(g) independent of g . It is not the r-fold points
-25S^'(g) theaselves 5 but the liait points in the branch set that cause the trouble. Example
Let X be the join of a p-sinplex Xq to an n-diaensional
polyhedron not embeddable in 2n~space5 and let X
in = 2n + 1 . Then
cannot be locally eiabedded at any point of .X^ , and so
XQ C Br(f) /for all GO
If g is in general position for (XJXQ)
then dim. S2(g) =
= P + "I
dim (S2(g)AXQ) = p but
d^-Cx-Xg) < 0
Remark 3
for n largeo
In applications we shall be primarily concerned with
the whole singular set
S(f) „ But in critical cases we shall
want to "pipe av^ay" the middles of the top dimensional simplexes of S(f) , and in order to do this it will be important that the interiors of such simplexes consist of pure double points,.and should avoid the triple point set^ the branch set, and a certain subpolyhedron Xq , We summarise this information in a useful form" Theorem 17
Let
(X5Xq) 5 where
fs X
M
be in general position for the pair
Xq < x < m „
Denote the double point dimension
by
d = 2x - m . Let K be a triangulation of X , that contains
Xq
as a subcomplex, and such that
f ^ K -> M
is simplicial
for some tx'ian^ulation of M . (i) Then the singularities S(f) of f forai a subcomplex of K of dimension
< d >,
-26(ii)
If A i8 a d-simplex of S(f) , then there is exactly
one other d-slaplex A^ of S(f) such that denote the open stars of In
A, A^
K , then
X - Xq , and the restrictions
the images
fU
fU^
fA-fA^ „
f|U, fjU^
Intersect in
C, U^
U, U^ are contained are eabeddin^s;
fA = fA^ , and contain no
other points of fX o Proof
(i) By LeaiEiia 31 and the definition of general position
of f . (ii) Dim S^(f) < d by definition of general position, and so
A
S^Cf) . Also
A ^ B r ( f ) , Therefore
dim Br(f) < d
hy Lemma 31 and so
lcS2'(f) because
A C S(f)
Sg' (f) u Br(f) «
a
In other \'!.!ords A consists of exactly double points, and so fA= fA^
for exactly one other simplex A^ «
Now
A ^ Xq , because
dim(S(f)A Xq) < d , by definition
of general position for the pair
(X,Xq)
. Therefore
U = st(A5K) C X - Xq , because Xq is a subcomplex»
Since f is
non-degenerate, if
A C Br(f) ,
3 contradiction.
flu
was not an embedding, then
Similarly for U^ . Finally if a
and
ui^^u^ , then
u
fU , fU^ meet only in
S(f) A U = A , and so
u
IT , fu == fu^
o
u^ £ A^ . Therefore
fA=fA^ , and contain no other points of
fX . The proof of Theorem 1? is completeo (The rest of the chapter is devoted to showing that any map can be moved into general positiono
-27o
Theoreci 18
Let
fs X
M
"be a Eiap froa a polyhedron X to the
interior of a manifold M , where
x < in « Suppose
f | Xq
is in
general position where Xq is a sLibpolyhedron of X » Then
f g
hy an arbitrarily small hoinotopy keeping XQ fixed, such that g is In general position for the pair (X,Xq) . . (Notice that the theoreoi is trivial if x = m ) o Corollary 1
ilny map
X M
is homotopic to a map in general
position. Proof
First homotope X into the interior, and then put Xq = 0
in the theorem,, Corollary 2
With the hypothesis of Theorem 18, let
a finite family of polyhedra such that
(X.1
^
Xq ci X^^' C X , each i ,
Then we can choose g so as to be in general position for every pair (X.,X.) for which
X.!^X. ^
Proof
(including
j = 0 )o
J
Choose a triangulation K of X containing all the X^
as subcomplexes, by Theorem 1, and let K*^ be the n-skeleton of K . By induction on n , use Theorem 18 to homotope
f|K^
into
general position for the pair (K^jK®^"^ ) keeping K^"'' fixed, and extend the homotopy from K^ to K by the homotopy extension theorem.
The induction begins with
n = Xq , by moving
into general position keeping Xq fixed.
fjE^^
At the end of the
induction we have a map g , that is in general position for each adjacent pair of skeletons containing Xq . If
n = x^^ then
X^C k'^ , and so
If
g|x^
is in general position..
X^^ X^. then
-28either
z. =x.
and the gener'^^l position of
general position for the pair
gj-X-
iaplies
; or else
x^
and
(X.,X.)C (K^jK""^) , so condition (iii) is satisfied for J (Z.,X.) because it is satisfied for . J
Remark
In Corollary 2, if we put
^-q = 0
and the family equal
to the family of skeletons of a triangulation K of X , ^e recover, for general position in manifolds, a gener'^.lisation of the primitive general position of K in euclidean space. Corollary 3
Given
f» X
M
and
Y C M , we can homotope f
to general position g such that for each r , dim (gS^(g) A Y) < d^ + y - m „ Proof
Having moved f into general position g "by Corollary 1 ,
we then use Theorem 15large and
"by induction on r , starting with r
S^Jg)=0 , amhient isotope
gS^(g) C M
position with respect to Y keeping The t-shlft of a map
into general
fixed.
The proof of Theorem 18 is like that of
Theorem 15? and uses a generalisation of the t-shift as follows. Given
Xq C X and
f; X
M
general position, then in particular
such that fjx^
fjx^
is in
is non-degenerate,
and so hy Lemma 32 we can first homotope f into a non-degenerate map keeping Xq fixed. Choose triangulations f ; K
L
Therefore assume f non-degenerate. K, Kq of
is simplicial.
let
X, Xq and ,L of M such that K' , L' denote the barycentric
derived complexes modulo the (t-1)-skeletons of
K,L .
-29Then
f,K' -> L'
renmins siinplicial because f is non-degenerate«
Let B TDG a t-siciplex of fK , and let
be the
siniplexes of K that are mapped onto B , which are all t-siaplexes since f is non-degenerate« oor-ie
in other words there is an integer q such that
Xq of
Order the A's so that those in XQ
if and only if
qs;i
(v/ith faj^ = b ),
A^jB
a^jb
be the barycentres
Then for each i
st(a.,K') = a.A.P. ,
st(b,L') = bBQ ,
and if f^ denotes the restriction f^ ; of f , then
f^
bBQ
is the join of the three maps
a^^-^-b ,
O
6
and
P^^Q « The local shift, given below, determines
a new map 0
«
gj_ « a^A^P^ that equals f^^ if frontier
i>q,
A^P^, fixed if
bBQ and is homotopic to f^ keeping the i ^ g „ Therefore, letting i and B
vary, the local maps g^^ combine into a global map
g; X
M
that is homotopic to f by an arbitrarily small homotopy keeping Xq
fixedo
We call the move
Local shift of a map
f g
a t-shift keeping Xq fixed.
The local shift is much the same as
before, except that instead of moving one cone away from the centre we have to move several cones away from each,other« Although each cone is not in general embedded, the movement of
-30each cone can "be realised "by an ambient isotopy as in the local shift of an embedding. Tout of course the moveQent of the union of the cones is only a hoQotopy. As before, choose a honieociorphisia h ; Q boundary of an (ni-t)-siaplex5 throwing (m-t-l )~face T Extend h to
h ; bQ -> A
Define
(V-j 5Vg,. o » a' . .Let on
A
onto the
into the
(which is possible since by hypothesis
c 6
kj^ ;
A
to the map
X < Q ),
by mapping b to the barycentre v of
and joining linearly. simplicialo
^
Subdivide so that
h; (bQ) '
A'
is
v^^ = v , q < i S n , and choose a sequence near v- in general position with respect to be the hoDaeoniorphisni joining the identity » In particular
k^ = 1 , i > q . For
each i ,, define : a.A.P.
°
to be the join of the maps If
i>q
then
f ; A^
g^^ = f^ , and if
—-]
B
i£q
and
h
k^hf ; a^P^^
bQ .
then g^ is ambient
isotopic (and therefore homotopic) to f^^ by an arbitrarily small ambient isotopy keeping fixed the boundary BQ (and therefore the frontier A^P^ ),
This completes the definition of the local
shift. Lemma 33 Proof
In the local shift
= S^(f^) .
The singular sets are unaltered by ambient isotopy.
Corollary
A t-shift preserves non-degeneracy.
31-
Lemaa 34
With the hypothesis of Theoreci 18 let
t-shift keeping Xq fixed.
dia S^(f) £ d^
f
g
Toe a
for all
r<s ,
then the sacie is true for g . Corollary s
A t--shift preserves general position (for choose
sufficiently large).
Proof of Leniaa 34 d = diin S^(g)
.Suppose not°
and
suppose
d > d^ , where
r < s „ Since g agrees with f on the
frontiers of the local shifts, socaething aust go v^rong in the interior of soae local shifts
Therefore
dim S^(g| U^(a.A.P. - A.P.)) = d , and dim S^(g| y (a.P. - P.)) = d - t
.
Therefore if we choose subdivisions (Ua^P^) ' , A"
of Ua^P^^ ,
A' with respect to v>;hich hg is siniplicial, then there is a (d-t)-sii:iplex
D
A', in the interior of A , that is the iaage
under hg of at least r siaplexes.
Select a set of exactly r
siaplexes mapping onto D , and, of these, suppose r^^ lie in n a.P. , 1 < i < q 5 and suppose r^^ lie in a.P. . Therefore ^ q+1 ^ ^ we have q
^ -
9 0
0
^
We shall now choose certain sinplexes i=
each i .
^
C^C F
and
with the properties D. 3 D 1
(*) (Recall
dim D^ < dia ^iS - r^^c .
c = codiaension = a - x „)
D^C. A., for
-32Pirstly if
r^ = 0 , choose
= P and
D^ = A-?
so that (*) is
trivially satisfied. Secondly suppose ^gS base
v^y^O
and
(g|a.P. ) is a sulocone of 1 1
i>1 „ By Leama 33
hg(a.P.)
vvith vertex v. and
1 1
hf S (f|P.) , Therefore there is a simplex
C. . A'
such
that C. C hfS
(f P.)
L. = v.C. ,1
D „
11
Therefore dia C. 1 S dial S r^(f|P.) 1 = dica S^.(f|ai.P,) - t - 1 <
- t - 1 , "by hypothesis since ^ r^< r< s ,
= Q - r^c - t - 1 , where c = codiniension Therefore dim D^ •< Q - r^c - t = diciA-r^c , verifying property (*) . Pinally suppose rp, 0 „ By construction g is the same n , as f on VJa^P^. , and so there is a simplex Cq £ A such that q+1 1 1 Cq C
n hfSr^(f I U
Dq . vCQ
D
p^ ) ,
-33We verify (*) as in the previous caset n
dim
< dim
Cq
=
U
?
n
diQ SpQ(f O
<
) - t -1
- t - 1 ,
Since
rQ:< r < s ,
and so
di[ii Dq < dini A - r^c „ As in the proof of Learaa 30, eoilDed A
In euolidean
space, and denote "by CCj^l "the linear subspace spanned hy C^ , etc.o
As "before there are two possibilities, each leading to a
contradiction. Case (a)
For each j ,
1
.1-1 _ _
A (D-I
and
0 " ^
Case (b)
Not (a) .
13. span I. jj
In case (a) we deduce din A r®-
+
^
=
^^^
0 ~
Sudining for
'D-1 + dim D.
0 L X-
j = 1 ,2 , c . „ ,q , and cancelling, we have diin n
0
r 1 D. I =
^ > dim D. - q dim A ^
1
A
diQ
< =
q. 0 (ci - t) - I r.c , m - t - rc
=
dr - t
<
d- t ,
+
£ (dim D. - dim A) 0 ^
=
by
A
-34contradicting the fact that
D^ ^ C
^ D. C A [D-I » 0
^
0
In case (b), there exists some j , 1 5 0 < 0. ? such that /H [p. 1 and
D.
do not span
.
Therefore
0 r.T^'O 1sp and ^ D. = v.C. „ Also j-1 A [D.] and D
DOtl
D.
A , and
[c / 3span a
o"
proper sutspace, IT say, of [A f « Now the vertices of the 0's ajad D's are all vertices of A' , and our choice of v . in i
D
general position in A
with respect to A' ensures that
v^.^ IT , because the definition of the general position of a V'
pbint Involved sufficient lattice operations to cover this eventuality. Therefore T? A v.C. = C. , and so t] t] Q. D c n ^ . CTTAD. =
J r^ C-CF
contradicting our choice of D in the interior of A
.
This
completes the proof of Leaaa 34» Leaaa 35
With the hypothesis of Lemaa 34 suppose that
dla S^(f) = t > B Gorollary
, Then dim S (g) = t - 1 , S • "' • S
With the hypothesis of Theorem 18, ne can aove f
into general position "by t-shifts keeping Xq . fixed. Proof
By increasing Induction on s , starting trivially with
s = 1 5 and, for each s , by decreasing Induction on t , starting with
t= dlQ S^(f) , we can reduce each singular set s
S^(f) to s
its. correct dlaienslon by Leama 35, at the saae time keeping ®rrGct the singular sets
S^(f) , r e s , by Lemnia 34.
-35Froof - - -of - ILei ,nc-ia-35 . . -Let - - -d ,= -dim — sS (g) . By Lecima 31
S^(f) is s
a t-dlQensional sulocoaplex of the triangulatlon K of X used in the t-shift
f-^g «
By construction the t-shift keeps the
(t-1)-skeleton of K fixed, and so Ss (g) 3 Ss (f) n ((t-1)-skeleton of K ) inplying that Suppose
d > t- 1 d > t - .1 ; then
d>d
, and with one s
clodification the proof is exactly the same as that of Lernaa 34, is to say, substituting s for r « That d-t ° we examine the interior of a local shift, and find D , C A , that is the linage of r, n siaplexes in
\J
a.P. , and
r. sinplexes in a.P. ,
q+1 ^ ^ s = Jr. 0
0 £ r^ < s
1
1 £ i < q , where
11
^
for
0 5 i£q
The inodification that we need to prove is r. < s 1
for
0£i< q ^
in order to he able to verify (*) , and therefore achieve a contradiction in each of the two cases„ establish
The contradictions
d = t- 1«
There remains to prove the modification, and for this we use two pieces of hypothesis that we have not yet used, that dim Sg(f) = t position.
and
^I^^Q is already given to be in general
-36Using LeEiaa 33 and that
S„(f)
is contained in the
O
t-skeleton of K , w/e have for each i
11
s
1 1'
c: (s^^i)
Now trivially d
(t~skeleton of K )
=
(a.P.) A A.
-
^ 1 i^ a. „
s > 2 , "because if
and so we could not have
s= 1
1
then
S (f) = X and s dim S^(f) > d o And a. is
o
S
the only point of Sg(g|
S
I
rmpped by g to v^ o Therefore
= 0 J and so
r^ < s
There reciains the case
for
i = Oc
1< i
n-q.|:s,
' ° ' "'"^n
are at least s siciplexes
then there
capped Toy f into
the t-siaplex B ^ implying
dis S^(f X^) > t > d^ , and S O s contradicting the hypothesis fj^^ in general position. Therefore n n - q <s „ Let Z = IJ a.P. . By definition of the t-shift q+1
g
^
agrees with f on Z , and so S^(g Z) C
Z n (t-skeleton of K ) n = IJ a. « q+1 ^
,, « »
Since
are the only points of Z, ciapped by g to
V , and since there are less than s . of theci, we deduce
Z) = 0 J and so
•0
j^^n® ° The proof of Lecicia 35 is coDiple'teo
-37Lempia 36
Given
both f and
XqC X , Xq< x , a ^
fjXQ
f; S
are in general position.
Let
M , suppose that f-^g
be a
t-shift keeping Xq fixed. (i) W
diQ(S^(f) A Xq) < d^ , for all
r<s
, then the same is
true for g „ (ii) If, further, Oorollary
t= d^
then
dia(S^(g)n X^)< d^ «
A t-shift preserves general position for pairs..
For use the Corollary to Leciaa 34, and LeaDia 36(i) with s large, Proof of Leaaa 36 (i) have
Suppose not„
dia(S^(g)n Xq) = d^ , because
Lemcia 34 Corollary.
Then for soae r , < s , we dia S^(g)
by
As in the proof of Leaaa 34 we exaaine the o
interior of a local shift, and find a simplex D c A, of diinension d^ - t , in the iciage of rQ simplexes of n Z = [J a.P. , and r. siniplexes of a.P. , •> Also q+1
^ ^
D d fXQ o But
1
11
Xq A a^^P^ =
0
, and so
for
Therefore D is in the image of
r^ p^ 0 ^
S^^lg Z) n Xq . But
Z = f Z , and ditnCSr (f) A Xq) < d^Q 0 by hypothesis, and so in the verification of proof of Leriaa 34) we gain one diaension;
dia Dq < dia A - rQC o Therefore in case (a) we have q r -
did n|D.
0
<
r
~ t
(as in the
-38-' d -t contradicting the construction
D ^
A D^ .
C
In case (b) the contradiction is unchanged. (ii)
The proof of Leaaa 36 part (ii) is the same as for
part (i), except for the ciodification of having to show r^ < s ,
for
as in the proof of Lesrna 35and so
r^< s
for
1
0 f: i
Firstly
q «
,
^q^^
Finally
because
D C fZq j
^Q^ ® ' otherwise we
should have s siaplexes of Xq aapped onto B , implying
dir.i Sg(f|XQ) > t = d^ , contradicting the hypothesis
x > ZQ
and the condition
did S^(f Xq) < dg - S ( X - X q )
s
included in the general position of
fjx^ « The proof of
LeQQa 36 is complete, c
Proof of Theoregg l8
We are given
fi X
M
with
fjx^
in
general position, and we have to move f into general position for the pair (X,Xq) keeping Xq fixed.
Leaaa 35 Corollary
shows that f can he moved into general position using t-shifts If
X= Xq
we are then finished, because the general position
of g implies general position for the pair.
If
x >Xq , there
remains to achieve condition (iii) for general position of the pair.
Leciriia 36 shows this can be also using t-shifts, by
induction on s putting
t=d
, and starting trivially with • o
s = 1 . The general position of f meanwhile is preserved by Lemma 34 Corollary.
INSTITUT DES HJ~UTES ETUDES SCIENTIFI~UES
:1..9.ll 1..Fevised-12~21 Seminar
on
Combinctorial
by
E.C. ZEEH.AN.
Topology
The University Coventry.
of Warwick,
VII INSTITUT
DES
HAUTES
SeminE'tr on
ETUDES
Combinatorial
The idea of' 8:,1 enGulfing homotopy
stateme~t
step in passing
into a geometrical
from algebra
map (2)
a homotopy
X
X
aoout
Obviously
is k-connected~
in
is homotopic
is contained X,
it is a key
subspace
M
in the interior
two statements
theorem.
about
that is to say the
to a constant.
in an m-ball
in
M.
Tho first is
and the second is a geometrical
the sGcond implies the first, because
The converse
our first engulfing
for
:E
c
X
is contractiblG.
liI
statement:
be a compact
is inessentJ-&
statement
statement.
the ')rem is to convert a
M, and COIlsider the following
(1 ) inclusion
Topology
to geometry.
For exam::::Jle le t of a manifold
SCIEr~TIFIQImS
is not so obvious~
a ball
and is given by
For the thecrem He shall assume that
that is to say the homotopy
groups 1t.(I\1) l
vanish
i ~ k.
£illd....-~,_~
ffi_
+ k .::'_.1..:.. ..JhelL_X __ is
111.8 S sen tJ~.. L.~tn
it-+_S3_.~cp!,-';.!.~ill9.9-_ in _1L.~11 ~1L~}1y __;tn.!9rior_of"JvI~~
11 if' ~d
onlY-if.
X.
VII
- 2 We shall prove a gelJ.cralisatioY.l in Theorem ~rom which the above result ~ollows at once. proof of Theorem 21 is long9
illustrate
the main idea of an engulfing
19,
called "pipingl:,
which will
theorem.
The proof
48, involves a more complicated. technique
and we ~Qostpoll.e this UIJ.tillater in the chapter
in order not to interrupt
our main flow of thought.
the piping lemma is only concerned with winning
=
for the
four lemmas, the first three of which are straightforward.
The last one, Lemma
x
eince the
s~ecial techniques
we give a shorter proof 07: Theorem
boundary,
requires
involving
However
21 below,
In effect
the last dimension
m - 3. If
L~n'Y
o
in..~!t__ .ill:}d.-:i_.s;._B ._tt~DLe._.~c.ll11~2.illJ? i ~n tis
0
t. 0"0_L....lL1l..t..:Ll
o
Y c B, for if not
First we may assume that
o
isotop
B
onto a regular lleighbourhood of itself in
Theorem 8(3). to
X
The proof is easy to visualise
we push
other bits of was necessary
B B
along with it.
out of the way as we
to have
B
M
so tllat X9 Yare
we can ensure that
y .11..
g09
: triangulate
subcomplexes. collapses
on the number of elementary
Y
expanl!is
which explains why it M. a ~1oighbourhood of
By subdividing
simplicially
simplicial
becausE.:as
that we may have to push
in the interior of
Now for the details in
Notice
lvI, by
to
collapses,
X
if necessary
By induction it suffices to
VII
- 3 consider the case vvhen X ~
Y
is an elementary
(Notice that we do not say anything about the triangulation,
for otherwise
because during the induction
aF
w
the link of
L
F.
f:i."om the face
of
gets pushed around.)
that
across the simplex
X~Y
F
Let
denote the barycentre
F
in
b.
be a simplex of' dimension
M, which
of
F,
is a sphere because
0
c X
F
being a subcomplex
,\
, = ~. and
collapse.
this would violate the induction,
B
Suppose, therefore,
B
simplicial
c M.
Let
1 + dim L, and
1\
choose a homeomorphism
h : L ~~.
Map
F
to tho barycentre
1\
6
1\
of
6, and extend linearly
to a homeomorphism
h : FL ~ b..
0
Since
Y
c
sF
. + we can choose a pOlnv
b
c
B,
in the 1\
interior
of the segment 0
that
abF c B.
Let f
the homeomorphism
.
aF such b.
defined by
1\
mapping hb ~ b., composition h-1
composition
st(F,
M)
therefore
•
keeping fh
b.
fixed,
and joining linearly.
throws on
F
b
onto
to a homeomorphism keeping
fixed, because
Y
g : M ~ M
M - st(F, M) fixed.
does not meet
F.
Join this
to give a homeomorphism
onto itself, ·which l{eeps the boundary extends
The
1\
: ~L ~ ~L
to the identity
to the identity, Y
6. be
->
fixed,
of
and which
ambient
isotopic
The isotopy keeps
step, 1\1), and moves
ab F
onto A.
o
Since
B ~ Y u abF,
o
gB ~ Y u A
=
X.
a more delicate
the isotopy moves
The proof of Lemma version
37
of this result
B
to gB where
is complete.
We shall prove
in Lemma 42 below,
replacing
-4 the manifold
B
by an arbitrary
VII
subspace.
1)1en. fX _':~.1. fY~ Let
Prool'.: f : K ~ L tha t
Y
K, L
is simplicial. J. S
be triangulations
Let
a subcomplex,
K'
a:ad K'
o~
X, Z
be a subdivision
of
such that K
collal!ses simplicially
such to
Y.
Let
=
Ie I
denote
the sequenco
for each '{oJe
fA.
1
across fK
i-1
"'.,
-
fK.
from the face
that
,
.J;.".
1
non-degenerately fB. a face.
o
0
1
Ai u Bi face
of
the simplex
C
X -
Also
Y
collapses, A.
1
and suppose,
from the face
collapse
across
B .• 1
the
1
in general
X - S(f).
to
fB .•
A
1
0 -r11 f\ .L \ .••••
y C
there is no subdi vi sian
i U
Sef).
B.) l'
n
L'
The reason for our claim is
into some simplex
A.
!
,
and
=
n
fE. (notice that we do not claim it is a
linearly
because
K
simplicial
f ~ K' ~ L' is simplicial). mags
f
'::!
is an elsmentary
1
sim.Pli£.-t~1. collapse becau5c such that
K1 '::,\ •••
of elementary
i,
claim that
ball
Ko ''-~
of
Therefore fK. 1
Therefore
=¢
L, and fA.
is a ball
becaus:
fA.1 n fK.1
is the complementary
1
The sequenc;: of eleme';l tar;y-collapses
gives
fX
~;fY.
- 5 The proof' is
x
==
y
Z.
==
result,
v lal if'
>
:It
ill-
3, for th en
-
assume x ~ m - 3.
Therefore
namely
tl'l
VII
the same statement
ChOOB0
We r~~o~ p~ove a weaker
cxcept
that Z is one dimension
higher. Proof of thc.yve"gJcerresulLCg; Let C be the cone on X.
~_2x - m t.31. ..
Sinco X is inessential,
o
X c M to a continuous-map
inclusion simplicial keeping
approximation
fIx fixed.
position
kee~ing
1':0 ~
fix fixed.
Therefore
linear,
l' into general
the singular
set S(f) of
~ 2(x + 1) - m. of C through
S(f); tlmt is to say D
of all rays of C that meet S(1') in some point
is the union
than the vel~tex of the cone. fC, Z --fD.
Them dim D ~ 2x - m + 3.
Since a cone collapses
Y--~·..••Z, and the proof
of the weaker
38.
Since fX
lemma
As in the weaker
=
X, we have
+-.?J~~
(Lemma 48) below.
lemma is long we postpone
J,et
r(~sult is complete.
.£ z ~ 2x..- m Eroof .9..£:....:tl1LSltronge..r_xes1;1J."t For this we need the piping
other
to any suocone we have C -.... :,D,
and since D ~ S(f) we have Y'~Z by Lemma
of the piping
Since the proof
it until later in the chapter.
caso, let C be the con.:;on X, and
o
0
f:C ~ M a (piecewise; linear) Triangulate of X.
extension
X and let Co be the subcone
By Theorem
the
By the relative
theorem we can make l' piecewise
Let D be the subcone
x c
M.
By Theore;m 18 we can homotop
l'will be of dimension
y ==
we can extend
0
18 we can homotop
of the inclusion
X c M.
on the (x - I)-skeleton
l' into general
position
for
VII
- 6.....•
the :pair
Co
-.;r
c Cx+1
X
keeping
Co
U
.JI..
'--'9
:fIX
The triple
f'ixed.
(in tho senS0 of the piping lemma)
is cylinder-like
and so by the :piping lemma we can hornotop 01 c C
and choos8 a subs:pacG
S(f)
n
dim(Co
Let
D
00 U 01 O~,D by
":..\ D U
01 because
Lemma 38~ bcc;::iuSC
fixed,
n
2x - m + 2
~
dim D ~ 2x - m + 2.
01;
-...''"':::\ D and
y == fO, Z == feD
Define
U 01 •
Co
00
f Ix
lweping
C1
< dim 01
be subcone through
f
such that
C
01)
XX9
T11erefol'e
and the result follows
01)
U
D U C:1.:::> S(f).
Co n 01 cD.
Then
Tho proof of L0~~a 39 is complete. c>
We havo
x
is contained
inessential
in a ball in
stai,ting trivially wi th dimensions
less than B;y L.:nnma
in
M9
and have to show that
The proof is by indnction
x == -
1.
Assume
the re suItis
on
x,
true f'or
x.
39 choose
11
Y~ Z c
such that
X c Y
''>J
ZZ ~
where Z
~
.2x - m
~ k, Th8refore
Z
x ~ m - 3.
+ 2,
j.
39
by the l~pothesis
is in-JsscEtial in (Thi s
by Lemma
s one
OJ
Iii.
2x ~ m + k - 2.
But z < x b;l the hypothesis
the place s wher
0
codimensi on
;>:
3 is
o
crucial).
By
Lenmla
Therefore
37
so is
Y.
Z
ic contained
in a ball in
M
by induction.
'rherefore we }-;',9.ve put a ball round
X, and
- 7 -
the proof
19 is complete.
of Theorem Corollary
VII
We deduce
some corollaries.
1.
,then al'}x, subsp~ce, o:C dimel'l§.i.QI,J. ~ l{ is ...Q,Qntainedin a.bali.. The corollary
follows
to
from Theorem
19.
, " (,~veaK
Qorq-llarx ,2•. a »Qm
immediately
0 tC2.123~ Il1- slllle r e...iJ,L ;::5-f~j;.t~.g
J~f_.1 s ..:t-op 0 1o,g,i ca 111L..JL~me omor-phic
Sm • ..We call this the w.:.;alc Poi:i'lcaroConjecture
although
the hypothesis
structure
(we always
assumes
assum0
this)9
homeomorphisffi9 not a polyhedral the proof
Schonflies
proof9
which
upon
does not depend
the stronger
result
that
The stronger
result
for
it in these notes9
In Chapter handlebody
Theorem
9
and which sphere,
upon
9
theory, gives m ;::6.
is also true 9 but we shall not give
from differential
theory,
upon including
r4 :;::o. Let
Then since of
is that
depends
the only lalown proof depends
smoothing 9 and deep results ==
which
is in fact a polyhedral
m == 5
because
proof9
combinatorial
the Schonflies M
The reason
of Mazur-Brown.
using
manifold
gives only a topological
homeomorphism.
Theorem
we shall give Smale's
has a polyhedral
the thesis
that we give here is Stallings'
the topological
e5
that M
because
x
==
m ~ 5
[m/2]
x.~ = m - x .- 1.
and
'P
we have both
M9 and call this complex
M
x, x~ ~ m - 3. also.
Let
X
Choose
a triangulation
be the x-skeleton
VTr
- 8 -
of
x... .,-
M~ and
the dual
largest
subcomplex
meeting
X).
by Corollary
x1j-I-skeleton (which is defined
of the barycentric
Now a homotopy
1 both X~ X*
m-sphere
x.
that
and if they dontt already
X9
X*
IVl= N u N*. ambient ball, ball
Noyv picl{ a regular
isotope A
it onto
say, whose
A*, whose
N.
interior
in the interior
of
the topological
Schonflies
ball.
Therefore
contains
A '.'•.•
=
Theorem
A u C
sewn along their boundaries;
o'~ M.
Then o
..I.
in
carries
E
into another
Similarly
Therefore
H
to Lemma
o
=
A
C
M
is
0
'.-
Therefore
!vi
t.§..Jheunion of
r
balls.
1ustcr.]jLck-Schir~e~man catc~ory
8
ProoL~_
Let
M
be k-connected.
balls
topological
Q.onse.s..ucntl",Y M_is
~ r~
by
is a topological
sphere.
then
a
u A.,.
of two topplogical
in other words
and
(m _i)-sphere
24).
of ~azur-Brown
B9
construct
is a collaled
is the union
illltilthey
of
N.
Then
M,
noi ?;hbourhoods of
complex
A) (by the Corollary
M
them a little
neighbourhood
contains
C = IvI -
Now let
derived
of
if necessary).
of the two ba:.ls to cover
The isotopy
interior
Therefore
B, B*, say.
of' B, B*
then we stretch
in the second barycentric
M~ not
are in the interiors
''''
N, N .....be the simplicial '
Let
do 9 as follows.
i~ ballE
X.,.
(by taking regular neighbourhoods We nmv want th.:;interiors
of
is (m-l)-co~Lected.
are contained
We can also assume balls
first derived
t.obe the
of
- 9 -
Now
~ k <:::> rr/r
[m/r]
VII
< k + 1
<=> m < r(k + 1) <=> m + 1 ~ r (k + 1 ) • Thereforo set G., 1.
{o, i =
the
condi tion
1,
•••
1,
• • • 1
9
[m/r]
~ k is
nl
can be partitioned
r,
each
disjoint
all
1.
if
ClEG
simplexes,
all
t he role
that
play
of the preceding of K
•
1.
each
A~bient
.•
Let
~ k + 1 intogers
Divide
thcver~ices
of
M'
K.1. be
1.
isotope
9
hypotheses
possiblo.
suppose
proof'
ncighbourhood
1.
1.
for
1.
B. by Corollary N..
of
then
By construction,
1.
M
Then M = UA., as 1.
of showing by examples
that best
x = m-2.
3--manifold
OpOj.'l
in the
and 2x ~ m+k-2 in Theorem 19 arC) the
In 19':57 Whitehead contactible
played
Them 1\1= UN .•
Cluestion
of
1.
simplicial
in a ball
r
The K.' s will
1.
A~~ containing
a ball
1.
First
•
1.
m-3
in J ..
1.
and so K. lies
B. onto
x ~
lie
into
M' consisting
subcoffinlex of •
N. be the
:'!c nail' tlH'n to the the
the
the
of a q-sirnplex
compleii1ontary skeletons
corollary.
dim K. ~ k
• denote
thE: barycentre
the
subsets
M'
of whose vertices the
disjoint
t.hat
1'1, and let
by putting
1.
saying
of
th. lJ- becond derived
I" 1," It 1."••L '- '>
i,
complex. J.,
subsets
J.1.
into
derived
to
r
into
contai:aing
Choose a triangulation barycentric
equivalent.
produced.
the
follmving
example
~
M-' (opon moans non-compact
of a
without
boundary
A
The manifold in:;edontial
is
rCT'1arl{abl·,;;in
(since
L~3is
that
contractible)
it
contains but
is
a curve not
S I thE'e t is
contained
in a ball
VII
- 10 -
iu M~.
The mani~old
is constructed
as follows.
Inside a solid torus
T~ in 83 draw a smaller solid torus T2~ linked as shown; T2 draw T3 similarly
D e f·lnc"
7.: ·J.·,"1':';
..
j,
'I:) -',,)
,3 •
a baJ.1 lIi M exam:plu
linked, and so on.
7-
-_
links T~~ then S Ho
then inside
Cf.li t the proof'~ bccau8c
1
is ~ot contained
t..l1G proof of the next
s SiElp18r.
P00naru
(1960) and Mazur (1961) produced examples of a
in
VII
- 11 compact bounded boundary. page
contractible
inesscmtial,
a regular
example
M4has
bu~ not
r2
of B.
L;t
as
spine
contained
were
neighbow~'h')od
interior
Let
of 1illzur1s
it+
3,
see Chapter
10.
Suppose
is
wi th non simply-connected
For a description
In particular is
4-lIlanifold
the
B.
of B, M, under
B by
lies
in the
2
B.
of D
in
There
3, Theorem 8).
(Chapter
this
2
2
reason.
By replacing
neighbourhood 2 D fixed
Then D
following
nQceSSar~T we may assume D
a homeomorphis~n M -+ Ivl1 keeping B1, M2 be tht!, images
for
in a b&ll
be a regular
~1
2 D •
dunce hat
in a ball
contained
if
the
homeomorphism.
Therefore
we have
By the
regular
2 and 3),
neigrbourhood
annulus
jll in
-
B
B:l
Iv1
M2
!Vl1
tho
top
arrow
1\:1(M) ~ O.
is
0:_'
83 x I
'"
'"
M
cOI!lnutative •
Therefore
the manif'olds
x I
triangle
induced
by inclusions
(M:\. )
an isomorphism,
R3marl;:~ one
'"
0
0 M
1\:1
the
(Theorem 8, Corollaries
we have 0
Therefore
theorem
D It is
2
is
is
not
and the bottom contained
significant
open,
an,d the
that other
group
zero,
contradictin!
in a ball. L1. the is
two examples
bounded.
It
is
above
- 12 -
conjectured
that
no similar
VII
example exists
~or ~ose~
manifolds.
More precisely: Con,ject\l}~e..:. Corollary ~--
Observe ~lat
this
the Poinc8I'c Conjecture m
=:
is
3,4 the conj3cture still
unsolved.
an (m-2)-connected
__
for m ~ 5, because
In the missing
conjecture
true,
if the Poincare
dimensions In
in a ball.
th0n tll.e proof =:
dimensions
to the Poincare" Conjecture, Conjecture
is true,
m-manifold~ m ~ 3, is a sphere,
is contained
missing
for k .., =: . __m-2.~
--=::;:-.-
is true
for m ~ 5.
is equivalent
For,
true
••;:a~",,-,;'
conj0cturu
is true
subpolyhcdron is
1 is
~
then
and so any proper
Conversely
of Corollary
which
if'
the above
2 v\I"Orl\:s for
the
3,4, because there are complementary sl{eletons
of cod.imer~ s i on ~ 2. Bing has shown that
in dimension 3 a tiGre delicate
result
7..
will
suffice:
he has proY.:;.dthat
every simple closed
Wi;;;
curve lies
if 11
next give an exam:glo to show that
X ::;:Sn, embedded in Mb;y f'irst
locally,
and then connecting
in which
then M3=: S3.
in a bal19
2x ~ m+k-2 is nocessf:'.ry in TheoreD 19. let
is closed manifold
the hypothesis 1
Let M =: S
linking
x
Sm, n ~ 2, and
two little
n-spheres
them by a pipe running around the
\
st.
VII
- 13 -
Notic0
that
2n+1~ x ::: n~ k ::: 0, and so the
III :::
hypothesis
:rails
by
one dimension
I, Ia
2x Next observe acrOSf:l
tho
n to a poin.t by pulling
that
we can hOlilOtope 8
other
and back
ball In
in a ball,
9
(by Theorem the universal
countable
set
theref'orG
link.
spher'cs,
i'or
since
otherwi se we could
n~m-3) and span it
of disjoiJt
disl:s,
There w'hich suggests it
is
its
be contained
if
"vo cannot
sn
lifts
the pair
homotopy gr'oups 0qui'vDlcnt
to 'iC
i
(M, C) is 1C
i
(£:1,0)
saying
(F) .il
,
tbat
would lift
This
to a
could
to a countable
in the
"-;IT!bed X in a ball,
set
of
contradiction
i'or
that
i ~ k.
thB inclusion
II
\VG
lS;3t example, migl1t try
to
of 1\1.
5'~be:pace 0 to be a k-core is This
to
say the
condi tion
C c M induces
I'olati
ve
is
isomorphisms
i < .':., an.a. 8n epimorphism ~_(O) -c.
Ii' M isk··coflJlectud, 01' ar~:T collapsible
disk
obstruction
t:-connected;
vanish
it
an (n+1)-disk.
two l1oighbours.
r.1ore ~p'(:cisGl~l de:f:'in8 a closed of I'd if
in thi s
with
some BOl"'tof 1-dimE.:1"lSional II core
in
unkIlot
in a ball.
a 1-diD.GllSional
that
8n cannot
none of whose bourldaries
But by construction
8n cannot
otherhand
On the
R x 82n of' 81 x fJ2n the
COYer'
one end
81 .
the
inclssi:mtial.
any one oi' ·'Iihich links
shows that
engulf'
around
S::.1 is
Therei'ore be can ta ined
+ k - 2.
set
in IvI is
C:.
k-corc.
thclJ.
a point,
or a ball,
- 14 ~~ampl~J.
VII
The k-skeleton
of a triangulation
of M is
The k-sk8leton
of a triangulation
of a
a k-core.
3.
~~e
k-core is another k-core. Exa~~~ __4.
A regular neighbourhood
~~qlAQ~~
If D'~ C then C is a k-core
of a k-core is another
k-core. if ~~d only if
D is a k-core. Exam~le
6. If p ~ q then
Defini~la~~2f
SPx point
cng£1fin£.
Let X, C be compact subspaces ~_X
is a (q-1)-core of
of M.
We say that we can
b;L..ll.\ll>hin/L.0llt Ta• .f.9.e1c£=fr:.qm t.h~_Q1'..!L.9., if
thel~e Gxis ts
D,
such that XcD~C
din (D-C) ~ x.+1. More briefly
vie dGscribe
or ::.m£1l.lf' ..;X:Jn ~. applications
this by saying ~np;ulf'J.,or ~:n,g\l.lf X ::from ..Q.,
The fesler is D-C, and it is important
that it be of dinension
special cases of the same dincnoion chapter we shall engulf singularitics may introduce
new singularities,
for
only one more than X (and in as X).
For exawple in the next
of maps, and the feeler itself
but these will be of lower dimension
th.an the ones we started with, and so can be absorbed by successive engulfing.
Rewriting
Theorem
19 from this point of view, the core C
would be a point, and X would be engulfed
in a collapsible
set.
- 15 The proof
that this statement
by the following
VTI
is equivalent
to Theorem
lem~a.
LeD}11::'llt0. Let
C-,- X
9.e~pact
subspac~J? of M.
"Q..e_"Wgulfed:'r.£l11 C if. 8d~n..kY _if..JL_ilLcOR~ned
Proof...:..If X can be engulfed regular
neighbourhood
of Oy because
N of D, which
N ~ D "'",C.
simplicially
elemental"Y collapses
is also a regular
Order
dimension, of dimension
Performing
11.
collapses
all those
D, say.
Then
we have only removed
simplexes
of
the rest of the elementary
collapses
gives
N01l=.90lTIPact_ collapsing
and~~ci
sion.
We shall always aSSUDe X compact,. but it is sometimes to have the core 0 non-compact, 22 below.
So far collapsing
and we extend
the definition
as for example
in the proof
has only been defined to non-compact
where
{ D-O -~
for compact
spaces,
spaces as follows.
Define
D-O n C
the right hand side is compact
non-compact
the definition
useful
of Theorem
-D_-C compacty and D '-"~.~O if
N
so that N
simplicial
Perform
';;:: x+2, leaving
in
neighbourhood
if necessary
the elementary
by Lemma
dim (D-O) ~ x+1, and D ~ X because dimension,;;::x+l.
in _~ re.R~
in Dy then it is contained
and subdivide
to C.
in order of decreasing
Then X can
Oonver'sely given X c N --:)0, triangulate
so that X, 0 are subcomplexes, collapses
19 is given
collapsing.
If C, Dare
is nCVlr; if they are compact
then the
- 16 -
de~inition
agrees
hand
we can triangulate
side1
elementary
with conpact
simplicial
collapsing1 and perform
collapses
face of any elementary
VII given the right
1
the same sequence
o~
since C does not meet the free
on D1
collapse.
because
imm.]diate consequence
Jill
of the
defini tion is the C:i;SJ.J2Jon property A·~
because
the condition
use this property Given D~C,
n
A
B
<=>
1'01"both
in either
u
1.'....
B '..... :;.,B
direction
when we say triangulate of D-C such that
triangulation
~A:B n
B.
Whenever
we shall say
9J
~xcisio~.
sides is X':B
D-C
D-C n C? and choose a particular
the collapse collapses
sequence
we mean
simplicially
of elcoentary
we
choose
a
to simplicial
collapses. The definition remains
of engulfing
from a non-compact
the same, with the new interpretation
given
core C
to the symbol ~
~.2msn:K· Stallings He envisaged swallowed
introduced
a dif'1'erent point
an open set of M moving1
up X.
Rewriting
amoeba
of the ball containing
the connectlon point
between
like, until
19 from this point
Theorem
open set would be a small open m-cel11 the interior
of vi ew of engulfing.
our definition
X.
and we could
of view, the
isotope
The following of engulfing
it had
lemma
this onto illustrates
and Stallings'
of viow. kemmCLlI:1 ~
L';Lt If[ b~",§:",l~aniJ'_Ql_Sl wi tll.ou t""£0llnCll!P1l-'l-..
.9 onIQ.Q.S~ t .8.}l.b s.J?§.~.()~~ ~L e t~.Q_J)8....§l.,"£ 1 0
Sl,§§ ~-l:2..s.pa co ~1l2..t
n~...£e ssar ily c:ompact L..
- 17 -
prgo? regular
I? C is compact
n0ighbourhood
and umbient
° in U,
o~
neighbourhood
Mo of D"":C tn
C n Mo, Do == D n Mo·
of Co, Do in
SOElC
thE::interior
of
to M by keeping
and another
Thon
of) Theorem
the rest of M fixed.
fine, then U ~ N
case when X u U = M. engulfings?
-p
hU
C
because
is th& tit
is contained
For example
the amoeba
apyroach
eacl",;. neYI engulfing
in
was over X.
consider
the
is no good for
may mess up what
The advant.age of' the feeler approach successive
colla:t;)sibilitycondition
q-collalJsibili ty below).
FJ~:HC
neighbourhoods
If the triaDg~lation
price that we havo to pay for this advantage a certain
Let
and so U will be isotoped
tha t the core C stays f'ixe':~ while
satisfy
to a regular
isoto}? NC into ND keeping 8(3). Extend the isotopy
U in IJe:nmaL~1.
Therefore
has a.lready beE.m engul?od.
° fixed.
Let NC' I-jD be second derived
and so we can anbisnt
MO?
X, by Lenmla 40,
containing
which will be compact.
tr ia~J.gula tion of 110'
In general
successive
is easy, for choose one
confine attention
]\/1,
Co u Mo fixed, by (the proof
sufficiently
the proof
isotop one onto the other kee~ing
If C is non-compact,
°0 ==
VII
feelers
are addecL
is The
is that the core must (see the definition
of
ii f\lrther aclv8Dtagc of the' f'eGler approach
can handle bounoJ3.ryppoblcE1s which pre sen t certain
- 18 -
difficulties. consider
Before
the special
the general
case9
case of a collapsible
core.
19~ we shall be able to deduce
of Theorom
from the general proof
handling
VII
Theorem
21, but again
however,
we
As in the case
the following
it is worth
Theorem
giving
20
a short
separately. Th_~~.m
_manif.9ld...l_L~m-3~ ~O.~ ~eJ: l~~mb.2_8.k7.9.QP.nE,).c~2fl. ~~
L
both J-11
Le.1-J2J?e Sl;_9Q:J-lgJ2.§Aple~ s~bsl!ac...sL_sll£l X >.§_ COElPEl.9"t..§..1£1?.m2~
the int oct.o_r-_Q.f....~,l._~~If~LLt.J_~~t]1.iUl-1LEL.9_~p._e~.Kll)J'J."t11 a_£Q.1J.D.;gsi ble .1:rq£ll1Ja.c!L1?jl.?-~lli~~=inj~ .. erior of_Ji.L_tha.1..j.s t
Proof· so that
Triangulate
C is simplicially
simplicial
collapses
claim that
it is possible
the complex There
C u Xy
is no trouble
C u X, and subdivide
collapsible.
of C in order to perform
collapsing during
Order
the elementary
of decreasing
dimension.
all those
of dimension>
it to Xo say, dim Xo
collapses
of dimension>
cannot g...,t in the way.
There
with those
x+1, for COIlsider a collapse
of dimension
from the face EX. principal
looks
It is possible
if necessary
as though
=
'iiJe x on
x, as follows.
x+1, because
there might
X
be trouble AX+1
across
that F c X, but since F is
in X, F is still a free face of ii, and so the collapse
is
valid. o
Now Xo is contained Therefore
in a ball
in M by Theorem
19 Corollary
by LGlJma 37, C u X is also contained. in a ball,
1. o
B say, in M.
o
We may assume
C u X c B by taking
D.
regular
neighbourhood
if necessary.
- 19 Now a ball interior,
is a regular by Theorem
neighbourhood the proof feeler
of Theorem
neighbourhood
8 Corollary
B of C.
By Lemma
of any collapsible
1.
40 vIe can engulf X.
by one in special
This completes of the
cases.
If there exists X
Let
set in its
X lies in the regular
Therefore
We now show how the dimension
20.
can be improved
VII
x
.
such that W ...•... "J X,
and X n C = W n C, then we say W can be furled. to X relative __
or, marc briefly, where
Cis
W can be f~rled.
•••
t~~
••
~_
~
'4
_.
__
••••
to C
_' .. ~
~,
The term comes from sailing,
a ship, and the 2-dimensional
the i-dimensional
._
x < w,
sails
Yif
can be furled
to
masts X.
o
111.,1.1 ~.~--1.f... w call1?..L._fill:l~.J~.](_
~ h. theJ~lL..£P...a1l:JL_ 'N. <;. j)
=...Q
o
1n._l'L....~~1h.ql?-_th8;"t_lliLlP=C2!i ..J'-~ Notice
that there is no restriction
and that the feeler has the same dimension corollary
as W.
To provo
we need a lemma, vfliichis a more delicate
We take the opportunity version,
on the d.imension of W,
while proving
sharpel'"the.n.is needed
h0:C'C,
Lot X"': .••Y in tho manifold
version
this lenmla, to prove \';hich
'.iill
M. INrito
the of Lemma
37.
a sharpened
be l-~s(:f'ul later fo:.'"
VII
- 20 0
if
X-y c
M
X~Y ~, X~ ~-.Y
if
X-Y c
1~11
if
X "-~~(X n M) u y~y.
The ambient manifold understood. and""~
0
X~Y
0
M is not included
in the notation
We call ~--; an interior _c,.Q.llajLs£" ~
but is always
a bou,n.dar¥_co~,;L.ill?~
an ~!li.§..~tp:t~~c21]'...§.p~ . .At the end of this chapter
define j.,nili[§..rd,g., co~llo,psing '-~ admissible.
Notice transitivity
shall
was introduced
by Irwin~ and inwards
both f'or t:ngulf'ing spaces that meet the boundary.
of 0, ~ is obvious,
int8rch~ngod,
\1'8
is in a sense o:9posite to
tho.t all three relations
tl~t two elementary
simplicial
because
o since it lies in M.
are transitiv.:.;. The
and that of a depends u~on the fact
collapsGs
. 0
_ R
K~ ~
K2~
the froe face of the second rGDains
free in K~,
. -X._~ gJ.ven ~ Y ~ .~ Z, tricmgula to and
Therefore
push all the interior collapses
which
Adnj.ssible collapGing
collo.:psingby Hirsch,
collapses
to the front,
leaving
all the boundary
z.
at tho end, X-~
If N is a derived then N
.
neighbourhood
of X in M
-..g,--;. X. If a ball B in M meets
B is admissibly
collapsiblo, If X ~
X in M is a ball ~e8ting Exanmle _
admissibly
. ."..
••
;.,j,ijI.
4.
collapsible.
in M in a f'ace, then
B~O. 0, then a derived
neighbourhood
of
M in a face.
A ball properly
embedded
in a manifold
is not
VIr
- 21 -
.;LLX':'-a,~ Y
c.
f.~~ed such
that
LW.l!I,L
then
Z i
a,
X u Z* -'-:4
s
ambiE:}~ ... t=i130t..opic
to _Z..:.: •..}cG~,EJ~g
I
~~:'
Remarlcs. -1.
The spaces
. 2.
are
not
nocessarily
compact •
X-Z"A.•. C X-Y •
3.
dim(X-Z~)
4.
The lemma is
~ dim(X-Y), true
if
by 2. a, is
replaced
by 0 or
~, again
by 2. El?oof. Triangulate if
necessary,
triangulate __ ....• r ,or .1'..-1.
A from
tho
face
x=Y in
M, and by subdividing
collapses
of Le::lr'la 37, by induction
proof
sir:rplicial
when X ~ Y is
the
of
..••.0 -.........\ ...•.
As in the elementary
a neighbourhood
collapses,
an elementary F,
say.
it
sufficos
sim:Qlicial
There
are
to
collapse,
on t he number
consider across
two CD.ses according
the the as
of
case simplex
to whe ther
VII
- 22 -
x
Y or
'~:;1.
X
---f3~ Y.
case A~ F c M. was to exclude
st
In the first case A, F
(The purpose
M, and in the second
of the admissibility
the third possibility
.
st
A
.. M~ F
in the hypothesis
eM) .
Firs.t caScLL-.:b-J!..,--i_1!,. As in Lemma 37
'IiVG
to be a subcomple:;~ of the triangula tien, because isotoped
around during
Thorefore
Z,..• '."
the induction~
we may expect A
n
until
do not assume Z
Z is going to be
it reachos
the position
Z to be an arbi trary subpolyhcdron
of i>.. 1\
F
a
1\
Lot F be the baryccntl'o of P~ and let A' be a subcUvision
of A
1\
containing
aF and A
n
Z as subcoIDplexcs.
Choose a point b in the
A
interior
of the segment
aP, and sufficiently
for there to be no vertices
of
i~'
in
cloSG to the . abF other than those
We claim th3.t
--oi o
abF
abF n Z.
point a in aF.
VII
- 23 To prove this, recall that Z ~ Y ~ aF, and so the idea is to collapse
abF onto aF, but leave sticking up those bits lying in Z. let P be a simplex of AI meeting bF and not contained
More precisely, in Z.
=
Let Pi
=
P n abF, P2
P n bF.
Then Pi is a conveA linear
cell with face P2, and so we can collapse Pi from P2• o
collapse
is interior,
because
0
Moreover
this
0
Pi U P2 cAe
for all such P, in order of decreasing reQuired
0
M.
Perform
dimension,
cullapsos
and this gives the
By excision
collapse.
•
0
abF U Z "'-::,\ Z.
i
Since F
M, the link of F is a sphere, and so we can define the
homeomorphism abF onto A.
in the proof of Lemma 37, throwing
g:M ~ M described Let
Z~;:
=
gZ;
then Z is ambient
isotopic
to Z* keeping 0
The inage under g of the colll;;lpse abF u Z~
y fixed.
Z
is a collapse
0
A U Z:::l:
But A U Z>;-. = X u Z .... -., because
Z, .,...•
X
=
A u Y and Z* :J Y. o
Therefore:; we have the
required
collapse
X u Z~;:~
Z*.
S iq£ olld._c .... a_s_e_:_l:~ F c 1\1. In this case,
but
case
the definition
L = lk(F,M) of dimension
is
no longer
the
construction
of g is
different
a sphere
r
1 + dim L, and let
of abF is because
but a ball.
as in the
first
the link
Let 6 be a simplex
be a top dimensional
face
of 6,
1\
with barycentre 1\
F
1\
-+
r,
r.
Let h be a homeomorphism given
0
L
-+ tJ. -
r,
by map~ing 0
and. joining
linearly'.
Then hb E
r,
because
A c M.
1\
There is
a homeomorphism of
6. detel"T'lined by r,mp9ing hb
-+
r,
keeping
o
tJ. -
r
fixed,
and joining
linearly.
As before
this
h08eonorphism
- 24 determines
a homeomorphism
of
~,!=,
isotopic to the identity
abF onto A and keeping Y fixed. as the first case.
VII
The rest of the proof is the same
The collapses
this time are all boundary
The proof of Le~~a 42 is complete. reason for avoiding the third case ll.. the required
throwing
g
Notice .. i M, F c 111is that
collapses.
that the otherwise
isotopy would have had to push stuff off the boundary
of M, which is impossible.
We have to show that if W can be furled, then it can be engulfed with a feeler of the samc dimension.
Recall that furling and X n C
means there exists Xx, x < w, such tfilltW~X
=
W n C.
u O.
This implies that W u C~X
Since x :::; k, we can engulf X c E-·":..!C, dim(E - 0) ~ x + 1 :::; w, by Theorem 20. ",
WuO
such that
Apply Lemma 42 to the situation
0
..... "'>'XuOcE
(the collapse being interior because all subspaces are interior to M), we can ambient isotop E to E* keeping X u C fixed, such that ('Nu C) u E~>~ E*. E*~'O
because
W c D~O.
Let D = (w u 0) u E*.
Then W c D',"",Eojd
the pair (E~;l' 0) is homeomorphic
to (E, 0).
Finally we have to check the dimension
~*)
:::; aim[(W =:
u 0) - (X u C)]
dime-IV - X)
:::;w.
Therefore
of the feeler:
by the Remark &fter Lerr~a 42,
dim(D -
and
VII
- 25 dim(E~ - C)
=
dim(E - C)
~ x + 1
:::;; w. Therefore
dim(D - C) ~ w, and the :proof of the Corollary We have now completed
that
\~c
shall
the proofs
need in the ensuing
lemma (L0mma48). prove the generali
For tho r0st sation
that
of the engl11fing theol'ems
chapters,
ot' this allows
apart
chapter
from the pi~ing shall
W8
go on to
for more complicclted cor0s,
permi ts both C and X to meet the boundary of generalisation
is complete.
To
'fl.
stnte
and
the
we need two definitions.
pe f i.~lJ.j..2!';;..,~'lt.~<1=.2.9J-1£J?1l.t1JJ~1.itY... We
descr'ibe
tho collCtpsibili
ty condition
was mcnt,ioned in tho Remark aftc-:r I;emma1.-1-1. of ;JI, not :c.eces sarily
compact.
Let C bG a closed
th9.t subspace
Define C to be .a•.~cq]J-...? ... Jl.,flLP1..e~ __ ~p....11 if .•.••• C
is a subspDCG ':j such thE).t C ""',~ ~~and dim(
theru
on the core,
Q
n
0
~ q.
iK)
If dim C ~ <1 th8n C is q-collapsible. .~~. co 11apSlG~e 1"\:n-;1 closed
If C is neighbourhooc1 of C, of C
n
:'11, is also iDl
BUDspace
subsrace
o
of ~ is
of :'1: is
O-collapsible
Q.-collapsible
compact and q-collapsiblc9 that
• for all
q.
then any regular
meets flI in a reguln1'" neighbou:r-hood in M
q-collapsible.
arc properly
embedded in a 3-ball
Let C, X be subspaccG
of M •
i~ the inclusion map X c M is homotopic
is not O-collapsible.
---._
We call X .•..... C-inessential ~.-~~
in
~9
keeping
in M
....•~:.;:""-~
X n C fixed,
VII
- 26 to a map X
C.
-4
£lJCam1?.J.:!L~If C is a point in M.
the dei'inition reduces
to X inessential
'rherefore the cO~1cept is a generalisation. If C is a k-core
for if X is triangulated no obstruction boundary
hemisphere
so that X n C is a subcomplex~
to deforming
fixed,
and dim X ~ k then X is C-inessential;
into C each simplex
in order of increasing
then there is
of X - Cl
keeping
its
dime:;'1sion.
If C is the northern hemisphere and X the southern n of S ~ then X is ll9~ C-inessential • .~theOI'~TIL?J ..:.
r....Q.t C~coll
ElllsibJ..lLl~o~Qf
j;h.£.
9 ..£..Jll::.2..:-1&j~_XX be ..£qmp~~:t ancl C-:iIl~~~li~ntJ~L...._illl-.L.~'tll?J?02e
(1)
.
d:Lm (~:LLJ. ..JIL..:S...J<. ~'ld
(2)
..
p__nnJYL==_J..~.~Q.L.D.J4• Notice
that the converse
C-inessential,
because
is trivial~ tile collapse
if we can engulf
X then X is
D-~C gives a deformation
r'etrs.ction of: D onto C, vifhichhomo tops X into C II l~eeping X n C fixed.
- 27 -
VII
Before pl 0ving Theorem 21 we give some examples and corollaries. 1
Let 1.1 be k-cormected, X
o
le t C oe a point
in M~and le t
o
be inessential
in M. Then w'e C2~1e~1.gulfX in a collapsible
A reg"ular neighbourhood
of' D is
8.
bc,ll,
set D.
arn so we deduce Theorem 19. o
Let Mbe x-connected. If C is a collapsible set in M, x 0 then C is an x-core. If X c M, there is no obstruction to deforming
X into C, keeping X n 0 fixed, \7e can engulf ~9JIrqJSL-3~
X~ alld
Theorem 19.
X is
X
illustrate
Yle have l,jI ::: Six
Then 0 is a i-collapsible can be engulfed;
contained
be easily
example~ vv.hichwas described
['.round the S1.
in M by be i11g self-linked
us that
Therefore
so deduce TheoI'em 20.
Consider' Irwin's
Example 3 after
81 x 82n•
and so X is C-inessential.
in a regular
abo\,-e in
82n $ and X ::: S11embedded
Ohoose C = 81xpoint
(2n-1)-core.
in
'l:heorc;m21 tells
by I,emma39 this
is equivalent to saying ne ighbourhood of 81• Of cou:..'se this can
seen by eleme:"1tary methods - but the purpose was to how Theorem 21 can be applied
in situations
where Theorem
19 fails. ~§i~J2l~
We give an example to show that
the hypothesis
(l+X~ m+k-2 in Theorem 21 is the be st possible. Let M = E2n-1 ::: En x En-1, n > 3,0 ::: 81 x En-1, X::: sn-1 x 09 where 8n-1 cont.ains
81 in its
interior .........
---......
.
1'-..--,---,: ,.--'--' i
I
C
!i---~
I
I -''''---
J
I
I
~"-.t-~ I -- -.-------I
!
i
!I
...•.....
~j
\
- 28 Then C is an n-collapsible
i-core o~ M (it is not a 2-core
z).
=
~2(M,C) ~ ~1(C)
because
VII
=
Putting x = n - 1, q
n, m = 2n + 1, k
=
1 we see that
q + x ~ m + k - 2 by one, but all the other hypotheses X is trivially
C-inessential
because
are satisfied.
it can be shrunk to a point of C.
we could engulf' X.
SUPT)OS0
Let A be a regular neighbourhood is an open set containing h moving •
X lies lnnbnll by Theorem
1
n in E.
0
Then A x E
Let N
i:a IT by TheOl~em 19.
=
h(A x En-i).
Spun
-r
Ji.
Since ~ (N) n
••
,nth a dlBlc D
to C
a fundamental
class in H~(C
l;
E
Hi (0) be the image of thj.s class under the inclusion
o
II
~ D· c C.
disle
..
l;n thJ..sball
N keeping X = oDn fixed (which is possible since X n Then C II D is i-dimensional, oriented by orientations
of 0, X, and therefore possesses
1"
0,
II
does not TIleot0).
C ,.lI.v
Tl
=
9. By Theorom 15, isotope nn in N into general position
with respect
.p O.L
n-1
so by Lemma 41 there is a homeomorphism
0, and
this open set over X.
of' S
Then l; is independent
n E,2n-1. ' We verify
in En x
o.
But ~
=
of D
that l;
J
n
because
II
Dn).
Let
homomorphism
it is the linking class
0 by spanning X with the unique
0 from the commutative
diagram of inclusion
homomorphisms
~
···_'::"~H1(N) = The contradiction
z.
shows that X cannot be engulfed.
E~.. PW..9"~_
We give an example to show that the hypothosis 2x ~ m + k - 3 is the best possible in the case that X c M. Let M = 81 x B2n+1, n ~ 2,
and in the boundary
M
=
S1 x S2n let X
=
Let C be a point, which is a O-collapsible
Sn be as in Irwin's example. O-core.
Then all
VII
- 29 the hypothGSGS 2x
i
arc satisfied,
m+k-3 by one dimension.
We cannot engulf X, for if we were able
to, then X would be contained ·a disk in M.
except that
in ball by Lemma 39, and so would span
This disk would lift to a couiltable set of disjoint
1 , - cover l:\. "B2n ne un1versal x +
't' d'1S k S In
0f
The boundaries
bv v
we know that adjacent bOlli1dary spheres are linked.
The cont:padiction shows that X cannot be engulfed, hypothesis
of these
l' 1 d'1n R un~ln~e ~ x s2n, but
d' 1Sk s wou 1d therefore be homologically construction
',rM.
8.l1.d that the
2x ~ m+k-3 is best possible. I cl0 not know whether
that X c M.
x ~ m-4 is best possible
It would be the best possible
if the following
in the case conjecture
were true.
90I.lJ..~~9>:t1IT~ .. L~,U1~~J2~~ ~a__c:t;l}l,.t~",",m§:n.~ fg 12-~ ..?~.Sk
4 let.- =""'-'"""81~_.~,,~-.., be essential _"""~_ ,__ ,,,,,,,,,.~"._,.,.;;:-.:.-=- in •.__ ""'="' •... _.,_.••-. ...~
M
Then 81 .••...does bOlmd a disk in M4 . ....• "'._.~_ •.••••... ::w_.~....."._ __ ~ •. -'" _.~.not ..'~ ...-.. ;
~
•.•••
~.-"<--
Observe that S1 does bouncl a P..iDgy.~_~ disle because A good candidate
for this conjecture
is Mazur's
.__ ~,. __"'._.~~..".,~
tvi4 is contractible.
manifold
" M4 (see
, S1 x pOln . t c (S1 x B3)' , eh ap t er 3 , page 10) , an d lor t' 11e curve cnoose ..P
0
drawn so as to avoid the attached
2-ha:1.o.le.This curve bounds
with one self intersection,
seems impossible
Before proving we state and prove Corollary
Theorem
which
21, which will require
a disk
to remove. several lemmas,
two corollaries.
1 to Theorem
~"""'''*'"'~-'-~'~'~'~'.-~.'''''''''''.'"""",4.~~.-._~-~~,--
21. .•,....,.,_
~Jt t._Q.h :x ..b EL a 8,. Jn ~.Jl''".<::~?l'£l;'L.?1.• _~.Lf_~~V~cpn .2~~L~("i9.Il]".ts~ il?&~tlJ,r 1 ~,:l.J2 _~, then we can engulf' W with a feeler of ."'_'*...-....~· the same dimension. .-._o......, •.•• _~••.-;.·.•..•.••••.••..••.... _.....
~.~~
•••.•.••--.'~"
....
, ...•••.,'._"'~~""'-'
.""'-'---.,.-Z"'_'-'a...- •••••••,.:
;:.:a~._"".• ,,,,.-..,..._ ..~.-,,,-_~"---.••••.c.=·
The proof is the same as that of the Corollary
*_',.:.:.._o_.~
to Theorem
209 with the
VII
- 30 proviso that it is nacessary to verify admissibility C.£.r:.2J~l,q£L_?__~2~...T1J.PB1~~1!L,,2.:1.DrY£.1nJ
at each stage.
.
J.e t.J!.~ J:_L~~Llc..-.2..,9-I?-l1.~L9.!.esl_"_m.ap: i t.o1.q,P... Jc_ £..ID=.3.La.lli1"1 ej~ Q~~
h-connected ~'_~-<"
.•.". ~,
submanifold of 1ft
...-oa.~''''''_'''''''~_''''~'''''' __ '._'~'_'"''''''''''
__
"'O-!
__
m-4.
11 ~
~_''~''~'~'~~''''''''''''
Let C ,·-9:,.;0 in M __
•••••
"'-.......-.,...
••
_--'_
••
¥''''-'''"-'"~''''-'''','''''''
"' __~"""'&~a.
o
~,!lll
and .'"
0
£LD_.JL~('..~.:...1t§ll ..... 1?..~.9.9nrQ~.U_j.. A."Jv~.,_JS. ~ lSt,.~QL1~L+ Y-.,t )Ln_.Lc;...;,8. wi th
9-llItlllJLlU
_~j}.:-._.TJl;.~ll ..w ~ ~~~~l).Ji1lU' ..X._c:.1L~~_~__ pu 911." t l}.al
..
£li11!.lD-:-c) "i.xz1 al14.Jl..D_1L;::.jl~ ....1L2.LjJ_Ni_~Con.s8Quel'Lill X is q,Qntaip.!2,.cl o
in a ball that meets M in a face in ~.
_
.•.. --"
..••• .;,,~,---,-'
•.•••.. ,,~-"'_••. --...::..-~~.~
.--.~-~ .., ~-_.__.. -.......:..••.=. -.••..•. ,~ ••..•..•••••.••
o
Apply Theorem 21 to X n M, C n M in Q and engulf •
X n M
0
E--~C n M, where dim(E-C) ~ dim(X n M) +1 and E
c
o
E u C --y
R
c
Q.
Then
Now apply Theorem 21 to X, E u C in M to engulf
E ~O.
X c D~--..iB u C, where
u 0))
dim(D-(E
~ x+1 aDd D n h
Because of' the last remar1:::, the colla:9se D~:8
=
.
(X u E u C) n M ::::E.
u C is interior.
Therefore D .,0..;:} C. Therefore D
a.,
o~ and so a derived neighbourhood of D is a
----j.,
o
ball meeting M in a face in Q. Vie
now pI'oceed to the Drool' of Theorem
2"1.
The last statement
of the the 81 s is talcen care of b;)rthe following lemma •
.
ctJ2gS.§:;~J?._so.~]}lttJ2 ...J1.Jii
.
=._lX
lL..9J..,it}l·
The iden is to isoto!) D inwards a little, keeping X u C fixed.
X u O. Let Y ::::
We can assume M compact by confining attention to ~,
a regular neighbourhood
....
,-
of D-Y.
Let c : M x I ~ M, c(x,O)
be a collar given by Lemma 24 Corollary. triangulation
of M
x I
=
x,
X E
Choose a cylindrical
(that is one such that the projection onto M
M
- 31 is simplicial)
such that
C-
VII
1y is a suhcomplex.
of M x I in itself as follows.
Define an isotopy
For each vertex v E M n Y keep
v x I fixed.
For each vertex v E M - Y isoto}? v x 0 along v x I and stop before hitting (v x 1) u C-1y; extend to an isotopy of
v x I in itself keeping v x 1 and tho intersection
with c-1y fixed.
Now extend the isotopy cylinderwise
A x I, A E M
in order of increasing Wl.t'n
d c -1y...,.. J.lxe.
to each prism
d.imension, keeping A x 1 and the intersection
The image under c extends to an isotopy of M
keeping Y fixed and moving M - Y into the interior.
The restriction
to D gives what we want.
We introduce useful
a notion duo to Moe Hirsch, which will be
in the proof of Theorem
21.
Let s denote the simplicial
collapse j7"
•••
The symbol s includes collapsGs.
~L..§.
"-:.'J.\.
'~.•••••
-:--
T
1 ">-\K
n11 the given ordering
Let W be a subcomplex
as follows, by induction in~uctiv0~trail+
of' K.
on n.
=.u.
of elementary
simplicial
We define the E..~t...1of W If n
=
0, define trail
W has been defined for the collapse
s
W = W.
t:
v
Ko ~
and
sunpose K n-·1~K .1:'_
K:t. '::, ••••.
-..,;. Kn-1 •
n is across
the simplex A from the face F •
Define trajl . s W ==
,trailt W,
F ~ trail
.<
~A u trailt W,
t
W
F e trailt W.
VII
- 32 ~~en there
is no confusion we shall
Geometrically
the trail
the deformation retraction K~. L.
drop the suffix
is the track
K 4 L associated
left
s.
by Wduring
with the collapse
We leave the reader
to verify
the elementary properties:
(i)
trail
K = K~
trail
L.
(ii)
trail
(Wi
(iii)
trail
(trail
Therefore
"trail!!
u
W2)
w)
is a closure
L =
= trail
Wi
= trail
~\!•
operator
subcomplexes of K, and the trails fibering
u trail
W2•
on the set of' all
form a sort
of combinatorial
of the collapse. Remark
Notice that
the trail
depends upon the triangulation
the order of the elementary collapser. collapses K',::,L
are re-ordered
then the trails
are different.
For example a~~issibility is an admissible
triangulate
so that
in the boundary.
simplicial
collapse
a simplicial
Therefore
collapse
the trail
is not
invariant.
However trails
X-yY
to give a different
are used to define
then the trails
a piecewise linear
If the same elemen.tary
turn out to be the same, but if different
elementary collapses K~L
and
can be related
is a piecewise collapse
the trail
linear
invariant.
invariants. If
(in a manifold now) then we can
of anything in the boundary remains
Therefore admissible
to ilboundary preserving"
to piecewise linear
collapsing
in terms of' trails.
is equivalent
For Lemma53 below
'. -,.)
we shall introduce inwards
-
73
VII
another piecewise
collapsing~
which
linear
is equivalent
invariant
to "interior
called preserving"
in terms o~ trails. We;
now prove tvvo important
Lemma
propertie s of trails.
dim (tra,11...1V) _:(, dim (L ('Lt~il Let s denote K
=
Ko ~.'
K:1.---.; •••.....•.... ">,
starting
K
=
n
L.
and dim (K1'1_1n T) T then trail
results
hold for s.
Therefore
dim A ~
Let t be the collapse
trailt'fJ.
By induction
on n~ Ko--""
K. 1 11-
dim T ~ w + 1~
.l.
S
If F E T then F E K 11-1 n T, and so dim F :(, w. 1, and so dim (trails
Also L n trail s W:= L n (A
u
w) :=
u
T) ~ w
+
1.
~
~ w.
For each elementary
collapse~
either both A~ F are in the trail~ or neither ordcr~ all those elementary in the trail, giving K
dim (A
T)
A u (Kn_'}.l. n m)
which is of' dimension
Proof
is by induction
Suppose K '--"K n is across A from F. n-1 TT W == T~ and T n ,no := m n K and so both n -,..., ~ n
w +
C
..:LJ.
vv.
:(,
l:-
then
Jil._~VI.
The proof
=
Vi
c K9
'N
the collupse
with n = O.
trivially
of' length n - 1~ and let T
If' F
W
It.K-::,.1is a.simpli,£ial coll.m~d
)+~
"'",,>\
collapses
L u trail W.
across A from F~ say~ are.
Perform~
in
for which neither A, Fare These elementary
collap ses
VII
- 34 are valid, because A principal
if A principal
in Ki u trail W;
and F ~ trail W, then perform,
F
Proof
\l"v2
=
trail W, then
also if F a free face of A in Ki,
is a free face of A in A.1 u trail W. collapses,
to Lel.lli'TIa 45. If K--~ L is a simplicial
Let s denote the collapse K--~L.
the induced collapse L u trail trailt
1
in order, the rest of the elementary
Corollary
i
in K., and A
s
trails W2 because W2
C
giving
collapse.
Let t denote
L given by the lemma.
Wi -~
Wi.
Now
Then
Now apply the lemma to t
to obtain L u trails 1N1.--"-"; L u trailt W2• s for t in tho right hand side gives what we want.
Substituting
Corresponding inessentiality to generaliso X
-t
C.
to the genoralisation
to C-inossentiality,
in the theorems from
it is nucessary
from the cone on X to tho mapping
cylinder of
It was for a similar purpose that Whitehead
the mapping
cylinder in 1939.
introduced
We need a relative version
mapping
cylinder here because X n
pointed
out in Chapter 2, the mapping
rather than a piecewise
in the proofs
C is kept fixed.
of the
As it was
cylinder is a simplicial
linear construction:
it is a tool rather
than an end product. Let K, L be complexes meeting
in the subcomplex K n L,
- 35 and let f:K ~ 1 be a simplicial The relative_~apping
cYlJnger
(Since our complexes one of sufficiently
high dimension
¢.
of K uLand
=
1.
~K u L defined
inductively
B E
=
each other).
are
If A E K n L
.
a(~A u A u fA)
AS.
dimGnsion~
where
Define
~K = Ui~A; The relative
choose
so that the new vertices
in order of increasing
..
space9
a say.
If A ,-K - L~ define
~A
= Ui~B;
of f is a complex
always lie in some Euclidean
independent
define ~A =
~A
map such that fiR n L
For each simplex A E K - L choose a new vertex~
as follows.
linearly
VII
ma~:9ing cylinder
K u L as a subcomplex.
A E
Kl.
is ~K u L.
In particular
it contains
VII
- 36 b
c
d
Let K be the 2-simplex~ diagram
abc, and L be the 1-sirnplcx, ad.
shows the relative
~~K ~ L given by ~a
= ~ =
mapping a, ~c
=
The
cyli~deI' o~ the simplicial d.
map
VII
- 37 I&mma_~~
.CP
(NewmfUl)
For this proof
in the complex ~K.
view we replace formulae write
a
~A
=
The symbol ~A will
the statements
0, fA
=
Therefore
o~ = ~o chains.
inductively
If B is a ball,
and so by the ambiguity a general
~A
To empbasise
=
¢,
~A
We replace
=
for a
the chain point
flA is degenerate
2
of
by th9
u by +, and
a(~oA + A + fA).
We ded~Ge
+ 1 + f
on simplexes,
and extending
additively
then oB is the chain of the complex of our notation,
.
oB
=
B.
to
• e,
(Of COl~se for
chain 0, 00 is defined but 0 is not).
The proof is by a double following
stand ambiguously
0, respectively.
for boundary.
by verifying
K - L t,hsm_vA is an
only we shall use chains modulo
and an (n + i)-chain.
subcomplex
E
A.
+ 1)-ball 'y!i th face p~
If An
induction.
Let 1, 2 denote
the
statements'
1(n):
if An E K - L then ~A is an (n + i)-ball.
2(n):
if AB E K, A E K - L, dim AB
=
n, 0 ~ dim A < n,
then p,(AoB) is an n-ball. Notice
that the lemma follows
non-zero
coefficient
and A is a vertex for dimensions
in
o~A.
from 1(n) because
A occurs with
Both 1, 2 are trivial when n = 0,
(A ~ fA because
AiL).
We shall assume
1, 2
< n, and prove first 2(n) and then 1(n).
The proo~ o~ 2(n) is by induction
on dim B.
of 2(n) when dim B = q.
denote
the statement
begins
with 2(n, 0) trivially
Let 2(n, q)
The q-induction
implied by 1(n - 1).
We shall prove
VII
- 38 2(n, q - 1) => 2(n, q), 0 < q < n. Given An-q-1Bq E K, write B
=
xOq-1•
Then
lJ.AoB= !lAo(xC) = !lAC + j.lAxoC
= X where X, Yare
+ Y, say
n-balls by 2(n, 0), 2(n, q - i),
respectively.
Then X + Y is an n-ball provided that X, Y meet in a common face. Now X n Y
=
oX n Y, because X
=
= (fJ.(oA)C+ fJ-AoC+ AC
=
aoX~ a i Y + fAC) n fJ-AxoC
!lAoC + (fAC n fAxoC)
coY.
=
Therefore X n Y
oX n oY.
There are four cases.
= o.
(i)
fAC
(ii)
fAxC ~ o.
(iii)
fAG
~ 0,
fx
fA.
Therefore fAxoC
= o.
(iv)
fAC
~ 0,
f'x E fO.
Therefore fAxoG
=
Therefore 1-'ACn fAxoC E
In each of the first three cases X n Y (n - 1) ball by 2(n - 1).
=
=
fAoC.
fAC.
fJ-AoG, which is an
In the last case X n Y
=
fJ.AoC+ fAG,
which is the union of two (n - 1) balls meeting in the common face fAoe (because fAG ~ 0), and consequently X n Y is a ball. Therefore X, Y meet in a common face, so that X + Y is an n-ball~ and 2(n, q) is proved. l'T r,e
write A
=
now
h ave t 0
prove
XBn-1, x E K - L.
2 ()n
') G1-vel" An E K - L., => 1 \.n •....
Again there arc four cases.
VII
- 39 (i)
B E K.
(ii)
B
i
K9 fA
(iii.) B I=. K (iv)
9
J O.
fA = 0
B ~ K9 fA
=
9
fB
fB f O.
=
O.
In the first case we prove, by a separate induction on n9 that ~A
=
(xyB),9 where y = fX9 and the dash means take a first
derived complex modulo xB + yB.
For if this is true for
dimensions < n, then ~=~xB ==
a(~xoB + xB + yB), since ~oB = 0
~ a«xyoB)' ==
+ xB + yB), by induction
a(o(xyB))'
~ (xyB)'.
In case (ii) ~B is an n-ball by 1(n - 1). ~B + fA is an n-bal19 because ~B n fA face.
Also ~xoB
+
(~B
+
==
Therefore
fB, which is a common
fA) is an n-ball, because ~xoB is an
n-ball by 2(n7 n-1) and ~xoB n (~B + fA)
==
~oB + fXoB
~ ~oB + fB, by a homeomorphism, ==
o~B + B,
which is (n - 1) ball, because it is the complementary face to B of the n-ball ~B9 by 1(n - 1). We have shown that
~oA + fA
==
~xoB + (~B + fA)
VII
- 40 is an n-ball.
=
But o(l-LoA+ fA)
is an n-sphere9
and
joining
aA.
Therefore
to the point
l-LaA+ A + fA
a gives
~
nn
(n + 1 )-ball. Case
(iii) is simpler
than case
l-LaA= I-LB+ I-LxaB9which
Thererore are n-balls
meeting
is an n-ball
in the common
since fXoB
= (n - i)-ball and ~
Case (iv) is yet simpler
because
fA
=
O.
I-LB9I-LxoB
face
~B n I-LxoB= !-LoB+ fB9
Then !-LoA+ A is an n-sphere9
(ii) because
=
fB9
as above. an (n + i)-ball9
because
as before.
this time
!-LBn p,xoB = !-LoBy
= = The proof
o!-LB+ By since fB (n - i)-ball
as before.
Q9ro:l;lar"y_ to Lemma Lt6..:.,
l-hILu
Proof
~
Collapse
across
A E K - L9 in order of decreasing
hkILuL
of' f:K
L can be obtained
x
= x
x
=
Eroot'
=
09
X
from A9 for each
x x 19 fx
=
x
X
t,2
We construct
cylinder9
simplex
dimension. cylinder
from K u K x I u L by identif'yin.z X E
K
X
E•.K
x
E
a continuous
cp : K u K x I u L
onto the mapping
L~~.:..
1...o'Dological:j..y_ th_e rela..!Jve mapping
-+
fx
09
and L8mma 46 is complete.
of 1(n)9
J;&l@1a4L
=
K n Lx t E I. map
.-.-.--, ~ u L
that realises
the identifications.
We
VII
- 41 emphasise
that the proor
and not piecewise
(or this lemma only) is topological
linear.
Dorine
~IA
and ror A E K - L construct order or increasing
step~ lot Sn n h :8 t
=
~or the moment
.
L..
~ (~)
By induction
the identi~ication~
.
~
"
"-"
hi
~
//
is a homeomorphism.
the pseudo-isotopy
enables
/
/
1jJ
8n/
~, 1jJ
the identi~ication.
exists.
so as to realise
and so this map can be ractored
Sn
We shall show
n ~ 8 such that ho = 1~ ht
that this pseudo-isotopy
~lsn has already been constructed
where
•
(A x I).
ror 0 ~ t < 1~ and h1 realises
is a homeomorphism
in
as rollows •
that thero is a pseudo-isotopy
Assume
u L to be the inclusion,
x I : A x I ~ ~A by induction
dimension~
For the inductive
~IK
Using Lemma
46
~ to be extended
or A x I onto a collar or~.
By filling
that ~A is a ball~ to a map of a collar
in the complementary
balls~ we obtain a map of Ax
I onto ~A, such that the interiors
are mapped
This is the required
homeomorphically.
map
~IA
x I,
a-
because
un.der the identification
no point
of (A x I)
is identified
wi th any other point. We now construct spanning
the verticos
each vertex
the pseudo-isotopy.
of A n L (possibly
B
Let B be the simplex
= ~).
of B~ and so
fB, = B
=
A n L (~ Aj beeauso A ~
t).
Then fb
=
b for
VII
- 42 We first construct
the pseudo-isotopy ht:B x I ~ B
on B x I by defining
[t, 1]
X
by mapping the segment x x I linea~ly onto
to be given
x x [t, 1], each x E B.
ht keeps B x 1 fixed, ho
=
1,
for 0 ~ t < 1, and h~ is the required
ht is a homeomorphism identification
Therefore
B x I ~ B x 1.
Next we construct
the pseudo-isotopy
on A x 1, as follows. Lift A ~ fA; that is to say choose a simplicial embedding
=
g:fA ~ A such that fg and t E I, define at with vertices
=
1, and B c gfA. (1 - t)a + t(gfa).
Given a vertex a E A, Let At be the simplex
~at; a E A~, and define tIle simplicial map ht:A x 1 ~ At x 1.
Therefore
ht keeps gfAx
1 fixed, ho
=
1, ht is a
homeomorphism
for 0 ~ t < 1, and hi is the required
A x 1 ~ gfAx
1.
identification
Moreover ht is compatible with pseudo-isotopy
already defined on B x I, because both keep fixed the intersection B x I n A x 1
=
B x 1.
We can show by elementary
geometry,
that
given 8 > 0, there exists 0(8) > 0, such that given 0 ~ s < t < 1 such that t - s < 0, then tIle isotopy from hs to ht of B x I u A x 1 can be extended
to an 8-isotopy of Sn, keeping fixed
of h s (B x I u A xi). Choose strictly monotonic sequences 8. ~ 0 and t. ~ 1
outside any chosen neighbourhood
2
such that t.
2+
1 - t.
l
< 6(8.). l
Choose a sequence of neighbourhoods
V. such that nV. = B x I u A x 1. 2
2
l
Suppose inductively
have chosen the isotopy ht of Sn for 0 ~ t ~ tie
that we
Now extend the
- 43 -
VII
isotopy ht of B x I u A x 1 for ti ~ t ~ t. 1 to an e.-isotopy ~+ ~ of Sn keeping fixed outside h~ (V.). Therefore [ht.l is a lJ. ~ 1
Cauchy sequence
u
of homeomorphisms
limit map hi exists? making
~
of S ? and consequently
[ht; 0 ~ t ~
11
the
a pseudo-isotopy.
Any point of Sn outside B x I u A x 1 has a neighbourhood
outside
Vi? for some i? which is kept fixed for ti ~ t ~ 1. Therefore tho point is not identified with any other point under hi. By construction B x I u A
X
hi makes the required
1? therefore
the pseudo-isotopy,
on Su.
•
(V x I) u (V
the construction
of
of cylinderlike
Let V be a manifold?
=
Therefore
on
and the proof of Lemma 47? are complete.
p~fin~~ion
W
identification
X
and V x I the cylinder
on V.
Let
i)? the walls and base of the cylinder
(perversely we regard V x 0 as the top and V x 1 as the base of the cylinder). We call a triad xx? J~ if there exists a manifold
C
JX+1 of compact spaces QllJpderlike
V and a map ~:V x I ~ J such that
~(V x 0) c X
and ~ maps (V x I)-W homeomorphically V x 0 1" --.I
•.• --
.- .•-
i
w 1.
'-'.,
!
!
I
I
.".'-
V x I
I
~ 1
~
-.--.-)
onto J - Jo•
VII
- 44 Example
1.
Let XA- be a complex, subcone
on the (x
cylinderlikc.
A.1
~ by mapping
to a homeomorphism
for each i.
Then X, Jo c J is
For let V be the disjoint
Define
extending
of X.
i)-skeleton
fAil in 1-1 correspondence
x-simplexes of X.
J x+1 the cone on X, and Jox the
-:Jc have
union
with the x-simplexes
0 isomorphically
x
of a set of
onto B., and 1
of' Ai x I onto the subcone
alread.y used
thi s example
fBiJ
on Bi,
in the proo1~ of
39 above.
Lemma
of g to the (x - i)-skeleton
Let go be the restriction
let J~ be the subma:pping cylinder cylinderlilw.
For- choose
x-sirn.plexes {A.l
1
of X-C.
Define
homeomOFphically
~ by mapping onto I1.Bi'
to example
Then X, Jo c J is
to be tho disjoint
in 1-1 correspondence
In the special 2 reduces
V
of go'
Bi
of X, and
union
of a set of
\frlththe x-simplexes
fB.}
1
the pair A. x I, A. x 0 1
(by Lemma
1
46) for each i.
case that C is a point,
not in X, example
1.
1emm[L~..lThc_.P1J2.ip.,gJ2mmaJ be a man5.f.9J2:.2.. an4_l£J.
L~tM:
:~..9_J~.,.£..Jx+1
be..£Y.J.i.12§c£like,
o ~2
J£"..iL-J 0 an.£.S1:Lch that ..£( J
f.:L=J
-t
]1.,. homgtopi c
t}L f
-=...J0)
eM. _
Th~2} thqre
exi ~i.s.§... IQ,aJ2,
k..£.9 .. EJng X j,LJo-±:ixed..Jancl.. a sut? .§J?.fLco_J:l. . c;: J".2.
- 45 -
VIr
such th~t o
11(J 7_3.02
.§lf1) S!ll!L31
c Li. c
~ 2x - I!l.. + 2
dim J..J0 ..D.~J1)
3 ~
~ 2x -
m---i"-1
3 0...!..L.31 -.:...30
The meat of the lemfJa is the combination condition J~Jo
of the collapsing
u 31 together with the dimension
dim J~ ~ 2x - m + 2.
In order to achieve this the homotopy
f ~ f1 has to be global~
rather than local liko tho homotopies
of simplicial
approximation
and general position. Before proving the lemma we introduce notation.
The proof
will then follow~ and involve three sublemmas.
Let V be compact.
9X.±',ll.td.r:.ts&
of V x I
if the suo cylinder through each simplex is a sub complex •
Given any triangulation9 Theorem
We call a triangulation
we can find a cylindrical
1, by merely making tho projection
Given a cylindrical
triangulation
subdivision by
~:V x I ~ V simplicial.
we can ohoos0 a cylindrical
derived complex by Lemma 5. In a cylindrical
triangulation
simplex~ call A ~~~Etal call A ~~tical
if ~A = ~A.
if ~IA:A
~
there are two types of ~A is a homeomorphism,
In an arbitrary
triangulation
and of V x I
VIr
- 46 it is possible vcrtica19
to havG simrlexes
but in a cylindrical
thercof9
that are n0ither horizontal
triangulation~
every simplex is either horizontal
Q;z1.jlld..Q£VYJ...~J.§.~j.Ylg
nor
or any subdivision or vertical.
•
Let K be a cylindrical triangulation of V x I. For each a simplex A E ~K, the subcylinder A x I consists of horizontal a-simplexes
and verti~al
(a + 1)-simplexes, alternately. interiors
arranged
Removing
the
of the top horizontal
and the top vertical an elementary Proceeding a co 11a p seA
simplex is
simplicial
collaps~.
in this way we obtain
.
x I '~,; (A x I)
U
(A x -1).
Do this for all simplexes A E ~, dimension,
and
W0
in some order of decreasing
have a simplicial
collapse V x I ~V
Let P be a sub complex of a cylindrical V x I. of P.
Call P sQl.L
x 1.
We
triangulation
of
every simplex beneath
a simplex
P is solid if P = trail P under a cylinderwise
collapse. A subcyJ.indel'is solid. Examnle....,.~ 2. -~ .
The intersection
and union of solids are solid.
If V is a manifold then W is solid.
and W the walls and base of V x 19
- 47 -
VII
The result follows immediately p
=
from Corollary
trail p~ Q = trail Q under a cylind~rwise
1 because
collaps~.
~Nithout loss of' generali ty we can assume J.l compact~ for
otherwiso replace M by a regular neighbourhood z
=
2x - m + 2.
hyDothesis~
of fJ in M.
Let
Let ~:V x I ~ J be the map given by the cylinderlike
and let ~
=
V x I
"
f~. .----.
_CJ2.
.. __ > M
/ ~, J /f ~I
Let Z = ~-1 S~1"·. I
)
Th~n Z has the properties
( 1)
dim Z :;:; z.
(2)
dim [Z n (V x I)" ]
These properties position~
~ z - 1.
follow from three facts: firstly f is in general
implying dim S(f) :;:; z~ and dim (S(f) n (X u Jo))
~I(v x r)
seconilly~ is non-degclierate because and thirdly ~-1(X u Jo)
=
(V x
~ z - 1;
o
is a homeomorphism;
r)·.
If we could now find a subspace Q ~ Z such that dim(Q n W) < dim Q ~ z V x I ---../N u
I~ ---.,
'[if
then the proof would. be finished by defining f1 In particular
the collapses J---;Jo
u J1~JO
=
f and J1
=
~Q.
would follow from
VII
- 48 Lemma 38 because
W ~ S(~).
However in general
VVhatwe have to do is to homotop f to 2:1.
== ~-1S(~1)'
so that
there
~1~
does exist
a
no such Q exists.
and replace
Z by
con"taining
,';:1,
Z1 and with
the above properties. Digr'cssion. =_._~-~,. obstruction
We digress
to finding
intui ti ve idea behind Then
1CZ x
I is
for
a moment to explain
a Q containing the proof.
the subcylinder
Z~ and to describe
Let 1C:V x I through
the the
V be the :9rojection.
-+
Z and wu can collapse
cylini:lervJ'i se
It
is no good }JHtt.Lng Q, ==
high.
f.,nd the trouble
cylinderwise horizontal the is
tc try
j,s that
if
V'/C
this
start
is one dimension
collapsing
stuff
in these
underneath
them.
Bimplexl;s in order
of a map (wh~ch is essentially we alter
Z minus the punch-holes.
x
f
too
I
thG dim...;nsion by one 1 then the
How the only way to "punch holesOi
Roughly spea~ing
1CZ
of Z ~or'm an obst1"'uction to collapsing
(z + 1 )-dLnel1sional
underneath,
I because
X
and reduce
z-simolexes
to punch holes
1CZ
what Z is) to
f:1.
a homotopy froEl f to f1 keeping
to release
(V x I
r
V\'O
to alter
shall
fixed~
X u Jo fixed,
the idea the stuff
in the singular
and Z to Z1 so that
More prcc:isely
homotopy ~rJm
is
'I'hercfore
away
the map.
Z1 equals
describe
which will because
set
a determine
~ I (V x I)
o
is a homeOIflorphism. The way to punch holes
in a simplex AZ E Z is as follows.
VII
- L~9 Since A is a top--climellsiom-11 siL1plex of a singular from where interior
two sheets of ~(V x r) cross one another.
point
sheet containing sheot.
of A, and a neighbourhood A.
Then pipe
The "free encl."moans
dividing
set, it arises
N into a central
1'1"
Choose an
N 01' this point
in the
over the free end of the othel'
~(V x
0).
The
d.isk N' surrounded
pipingll
i1
is done by
by an annu:::'us Nil, and
re:ple.cing ~N by
U
-Plhere~:1.NH is a long pipe running
along a path in the seconcl sheet
to the fr0c eno.9 and
docs not meet the rest or ~(V x I).
had to have
w)
in the interior
The following
show how the riping want.
pictures
illustrate
enables us to perform
Of co~~se the pictures
that we
of M was to make room for
tho pj.pc and the c~rl?over the e-11d. Define V x I.
The reason
~i
:;:
the idea when z :;:1, and the collapse
are inaccurate
tllat we
in that x ~
ill -
3.
VII
- 50 -
!J$FORE fl"ee end
---------t 21
1
path:
\....~l . ,
,'.., N .... , 1-0;-'
-
·1
I
..-;.----. ."
'\ Z
v
I
I
/'
x I
obstructed
by Z
.....,;
.. of
,
pt.
... 7.•••••. t.~
I
~.::.'_:
----7
'
".~~~""'''''':.
~,/':
.L:-
~
I/
J.-...-
1--
.----"'/'.'~
f
i --
iY
I
,
\ .1
!-..--
~._
'
'---
._.~/
.
VII
- 51 -
A tecru~ical di~ficulty it is impossible with respect
in the proo~ is that in general
to ~ind a cylindrical
to which
the map ~:V x I ~ M is simplicial.
reason is that we cannot make both ~, ~ simplicial, by Examplo
1 after Theorem
1.
sometimes want 9 sioplicial, necessary
o~ V x I
triangulation
A.S
The
as is shown
the pI'oof progresses
we shall
and other times ~, and so it will be
to switch back and forth.
Q.()A..tJ...nuing th.£..J2£.2~f, of Lem~L48. Let K, L trian~~late successive
subdivisions
denote the induced subdivision
of K.
the pair V x I, Z.
K1, K2?
subdivisions
•••
of K, and L1?
of L.
(for a suitGble
L2,
•••
will
First let K1 be a cylindrical
Next let K2 be a subdivision
~:K2 ~ M is simplicial
We shall construct
of K1 such that
triangulation
of
M).
Then
L2 has the properties:
(1) dim L2
~
(2)
The z-simplexes
(3)
9 identifies
identifies
interiors
of Z above.
of L2 are interior
the z-simplexGs
of L2
to K2• in pairs~ and
of those simplexes with no other points.
The properties
Theorem
z.
Property
(1), (2) (3)
follow
(1)? (2)
from the properties
comes from the general position
and
17. Su 121crnm.§LJ..:..
WL®1l.-qp.oos_~~~,JC
.L41
80
thi?t~La .E?.!L"ti sfi_e~§.. t.h.e __:further prQJ2.~:
If A.J s a ll<2!i,&ontaJ._2!,-..EimJ21.ex oLL2~
.
~b._n
q~.~~.L7Sl::..
- 52 Although Ki
Eroof. general satisfy
(4) because
is cylindrical~
two horizontal
lie in the same subcylinder~ under~.
VII does not in
z-simplexes
of Li may
and therefore have the same image
If they do, then we can move one of them, A say,
sideways out of the way as follows. of Ki
Li
Choose a point VA E st(A, K')
modulo the z-skeleton.
such that ~vA
i
7U~~ which is possible bocause dim A
since x ~ m - 3.
=
z < x
=
dim V~
(This is one of the places where codimcnsion
~ 3 is essential.) the linear join
Let K' be a first derived
Since both VA and A lie in a simplex of Kt,
v.r!:"
the interior of
is well defined.
By property
(2) A lies in
and so there is a homeomor:phism
throwing A onto vAA, and keeping the boundary
fixed.
~r
of
·st(A~K')
Extend
1/r
to
a homeomorphism 1/r;V x I ~ V x I
fixed outside st(A, K'). ~vA
i
(z-skeleton
Since 1jr keeps Z-=-A f'i.xed,and since
of ~K')I we have
~~A n ~*(Z - A)
c ~1/rA.
Now choose a point v.1-1.. for each horizontal that the imagGs {~vA} are distinct. E.L'e
z-simplex A ELi' such
Since the stars {st(A, K')}
disjoint ~ we can define the homeor.iOrphism
all the A's simultaneously.
~ = 1/r keeps
~1/r-1:v x I ~ J, for the cylindGrlike
(V x I)" fixed. ~ =
so as to shift
Let
which is a valid alternative because
*
Let
g-1S(f)
=
1/rZ.
hypothesis,
- 53 Let
K =
L=
WKi1
V x I1~.
WL1.
Moreover
Then
R1
VII
t is a triangulation
any subdivision
if we replace
K in our original
will automatically
(4).
satisfy
satisfies
(L~)1
property
of t also satisfies
anc1 consequently ~1
L
by construction
of the pair
construction
(4).
by ~1
This completes
R
Therefore then L2
the proof
of
Subler.llna1. 9,p1.±§jrugj:,i?11_of 0n.~ -oipe. So far we have constructed
v
X
11 Z satisfying
barycentric of K3.
a triangulation
~()roperties (1 J 21 31 4).
first derived
Let K5 be a cylindrical
(3).
Let K3 be the
Let K4 be a cylindrical
of K2•
second derived
Let "'~1A~:: be two z-simplexes property
K21 L2 of
of
subdivision
K4.
of L2 identified
-by 'P1 by
I.• abeJ. thc~n so that one of the following
cases
occurs:
(i)
both vertical
(ii)
both horizontal
(iii)
A horizontal
One of these cases always
and A •.... vertical • occurs because
either ilOrizontal or vortical1 cylindrical
because
have 2UY (z + i)-dimensional get rid of. Lot
Ther0for~
J~be
be tho vertical
of K2 is
K2 is a subdivision
K1 (this was why we bothered
thore is no need to do any piping1
overy simplox
with Ki).
because
neither
stuff underneath
of the
In case (i) A nor A* will
them that we want to
we can assumo A is horizontal.
the barycontro interval
above
of .A1 and let P (p stands for path)
A
joining
A
to
~A x
O.
By
- 54 -
VII
,..
construction
A is a vertex of K3 and P is a subcomplex
of K4~ and
so the second derived neighbourhood DX+i == N(P~ K
5)
is an (x + i)-ball meeting V x 0 in a face~ DX say.
Let
DZ _ A ()DX+i, " in A5 (A being the which is a z-ball, being the closed star of A 5 subdivision of A induced by K ).
5
7
I
<
(
p ..._....~ .....
x+i
D
'-....
(.
<
Bx+1 ....·..::;/\
\
< ~
1
•• __
.,.
.• _
- 55 Let
*
DZ' =
which
.B.~l;
DZ n cp-1 cp
(3) abovcs
is a z-ball by property
a subcomplex
o~
KS'
Both P and
because
in K "
disjoint
(4)"
embedcHng,
Their
second derived
=
..1 DX+1 cp cp
=
D
x+1
Let K6 be a subdivision
are
neighbourhoods
TherQ~ore
of KS' and
are
cplDx+1 is an
MS
a triangulation
Consequently
D~ becomes
of a
of K ,
Let K , M7 be barycentric second deriveds of 7 Then cp~K7 -? 1,17 is simplicial because cpis non-degenerate,
6
Bm :::N(cpDx+1, EX+1 = N(Dx+1, x+1
Eo Notice
4, and
o~ K
KS"
z u D~.
M such that cp:K6 -~116 is sim91icial.
K6s M6, Let
DZ•
on
and
(5)
subcomplex
simplicial
are subcomplcxes
DX+1 n Z
Thcre~ore
5
cp is not in general
~(Z-:~A) x I
disjoint by property
but is not in general
=
of balls,
7
K
)
7
z
N(D*, K7)·
that these aI'e three balls,
neighbourhoods
M )
because
they are second derived
Also BX+1 meets V x 0 in a face, BX say
'lJecause J,~x+1 dJ.'d): B~,':':+1 \ v ," lies in the interior of V x I (because D~ did); and Bm lies in the interior of M (because cpDx+1 did), From· ():)) we deduce (6 )
qJ -1 (Bm\)
:::
x+1
B
X 1 u B '" +
cp-1(:8m)= (:SX+1 -
BX)
·x+1
u B ','..,..
VII
- 56 -
By Theorem
6, V
I is homeomorphic
x
to closure
(V
I _ DX+1),
x
and so we can choose an embedding
l1:V x I tha tis
(v
X
T
tho id.cntity _
nX+1).
arc both proper ambient
01)
-7
V x I
tside :z?:+1 i m:.d maDS V x I onto closure
mh ,~e t wo ;[laps
and agree on the boundary.
isotopic by Theorel~ 9, because
o:::dstsa homeomor'phism k:Bffi
-7
Brn,
Therefore
th0Y are
x ~ m - 3, and so there
keeping
the boundary
fixerl, such
that -nX cp I B x ::: kcph I Bx :.0
Extend k by the identity
-)
m B ..•
to a homoomorDhism
of M.
Defino
cp~ to be
the composition V x I
h
" V x I
__ .-52_._) M
_-E__
> M
-_._--,-"---_._----,-~ CPi
N o t·lce th U't
.~ OU t SlQO
1 c 1 "o.x+ D U B:::. -1-.
* ,
..p '"' an d th 0 on 1 y d l~~eronce X 1 . Bffi• ux+1 In b e twoen cp, 91 l·~ ~ to -~ltnr ~ tl1.e emb~dd'ng '"' l S OI~ B + , .LJ;;: 9::: 91
Theref'ore 'Pi is homotopic
to cpo
Also 'P1 ::: 'P on (V x It
, because
X tho o~ly place where they might not agree is B , and hore they
agree by choice
or.k.
Therefore
We have completed that the neighbourhood £:+1. .'..•
91 ~ cp keeping
the construction
of one pipe.
Notice
N I'eferred to in the digl"'essionabove is
irhe pipe VIas thrmvn up indirectly
rather than by drilling
(V x Ij fixed.
directlJT
by the homeomorphism
k,
along the path cpP (which might be
VII
- 57 -
= ~ -
. x pre tt y rugge d II Q
could then be locally
Recall
knotted
along p).
x
Since homcomorphically
=
onto ~B9
CPi
=
is com.muta tive.
Since l' = fj. 'NO
--1
'Pi
S ( f 1. )
>M
/'
/1~1 JI
/
outside
have f 1. (J - So)
i;B, and f( J - J 0)
o
c M, and
o
c Al as requil'cd.
Let
•
T,et.B -_ BX+1 U BX... ,+1. .... t';B.
into ~B, we can
-1
~
t;
B~ and sinco t'; maps B
'Pit; lli~ambiguously. Ther0fore
'0,
=
'P outside
anQ maps no other points
V x I ..------"
Z:l
.'1
the CO~;ll;lUtati vo diagpam
v
define f1
X 1 .oDx+1 ~ ~,,~n the:. _~ v~mb{~dc~l'J:~~ ~ ~ .~~ cpID + ~
7 ~,because
Therefore
Since h keeps Z - B fixed,
~.Lnen f ,1:1.~ agree
~ 'd e QUlJSl
- 58 -
VII
Now h~-1S(f1
GB) =
I~B), by
h~-1S(k~h~-1
1
= S(k~lhB)?
because
definitions
of ~1' f1,
h~-1 l~B:~B ~ hB is a homeomorphism,
=
S(~lhB), because k is a homeomorphism, z Oz z oz) - (D - D ) u (B* - L* "
= Let
f
°z U D~J °z Z - (D
each i we construct
3
. as rcqu1recl.
(Ai' l\;i)J be the set of pairs of z-simplcxes
of.'ty:pe (ii) or (iil), where
K ,
9
.. , K7 of K2"
i
runs over some indexing
a pipe as abovG, using We now verify
set.
of L2 ]'01'
the same subdivisions
that the constructions
are mutually
disjoint. Firstly
the paths
tP. 1J
are mutually
from UIA.. by pr'operty (L~) above" ·"1
disjoint,
Also these arc subcomplexes
and so their second dcr>ived neighbourhoods
[~D~+11 arc mutually
disjoint
their second derived neighbourhoods Therefore 1
subcomplexcs
(5) above, the
of'M6•
{B~J are mutually 1
we can define h, k so as to construct
B·~J. simultaneously.
As before def'1ne
~1
of K4
fD~+'l J are mutually
o.isjoin t? and_ o.is j oint fl"um UA;;~i" Therc fore, using images
and. disjoint
Therefore
disjoint.
a pipe inside each
= k~h? f:1. =
CP1
r;-1 •
VII
- 59 pro.2f..
Since h is an embedding
suffices by Lemma 38
it
to show that h(V x I) --"'';!h('i,r u T u Z1) --->h7J.
(4) each
By property
path P. is disjoint 1.
froD 0 u T1 and therefore
. d s nx.+1 1 BX the second derived nC1.ghbourhoo .+1 are a 1so d'" ISJoln t f rom 1.
W u T.
h keeps W u T fixed1 because
Therefore
the ~B~+1~.
1.
Therefore
it suffices
h only moves inside
to show
..I.
h(V x I)~W
u T U hZ1~~.
Now h(V x I) is the sub complex of the cylindrical
triangulation
K5 obtaine d by rer:lovingthe open simpli cial ne ighbourhoods
fp.}.1.
h(V x r) is solid because
Therefore
meet UPi1 neither
sub complex of .ill =
K5
beneath
A1 and let
being the intersection Therefore
VI!
it.
A E L21 let A' denote the
z-simplex
UA", the union for all such
solids.
if a simplex docs not
does any simplex beneath
For each horizontal
f
.Ii.
It
=
ii.' n h(V x I).
Then A'
L.•
or the
Let
is solid, ard
so is L"
of solids, :cmd so is 13being the union of u T u E is solid.
We have
h(V x I) ~ VI! u T u E ~ V x 1 and so h(VxI)--,lWUTUE cylinderwi se by Corollary
We do this by examining according
to whether • 1.
2 to Lemma
each
Ail
Lt5.
Next we have to show
separately.
There are two cases,
A is the first or socond member
is the first member
of a pair.
Now A'
of a pair . is a
- 60 (z + i)-ball,
because
pro:9oI'ty' (2),
and so A' triangulatos
beneath
A is
VII to the prism
~
I by
x
(z + i)-cell
the convox linear
st(A., A') meets
The subball
.i..•
interior
A 'in
the commonface nZ,
and so tho complement AI1 is also
a (z + i)-ball
collapse
across
is
'.'
=
collapses
Thorefore
we can
,\ II J.1. .•.• ,
'.'
Z
froLl D.... or
In this
What is left
case after
all
is ~ u T U hZ1 by Suble~TIa 2. T
1:/ U
D
'~A
o
/
1'l..." •..•.. -
horizonta.l
hZ1--.. W u T by collapsing
U
°z
A -
nZ."..•.' -.,..
• • ~ ~.~...• ..•-.
z-sim:plo::es A or A~;. in L2•
W u T-"j W cylinderwise.
a;:·
A').
the second member or a pair.
Next colla:psc
all
A')
with face F = lk(A,
h~ and so we can collapso
these
for
st(A,
-
from F.
All
A .. A~
:;: J.l.'
Finally
colla:pse
We have shown
rcquir·ed.
Defino 31 the thr00
To prove
:::
u i;(T-:-W).
8(f1)
We must verify
conditions:
(i)
dim J1
(ii)
dim (Jo
(iii)
J ---.30 u 31 ·~Jo.
(i)
observe
~
z
n 31)
that
~
z - 1
dim S(f1)
~
z and dim T < z.
that
31 satisfiJf'
VII
- 61 To prove; (ii);
n 8(f1»
Jo n J1 = (Jo
u (30
n F;(rr"-
= (30
n S(f»
u (1;W n r;(~f-
= (30
n S(f»
u r;(W n
dim
(30 n S(f»
dim
(W n
T~
w» w»
T- 11).
< z, by general
position
of f.
W) < dim T ~ z.
dim (30 n 31) < z.
Thercforo (iii);
To prove
JQ
U
J
=
i;(V
31
=
F;W
=
r;(W
=
r;'Fv.
30 Therefore
the collapse 37 because
by Lemmn
W ~
3"""" 3 0
S(r;).
I)
'x
u r;(ir- - w) u r;Zl
u
u
T
,J:1. ~
u Z:1.)
30
follows
The proof
from
Sublemma
of the piping.leIT~a
3 is
com:plete.
(c.f. Lemma 39) 1~:t~C_J?.~" .Q•. £lp~s*~g." ..l?-u..b.~.:g.Q...2..~. _'?J>~}:~l:L1.;lch t1~.:t.~iLtIILj.Q. ....D_. ~J_~~ I&..:t)c..J)§_C?..9~.P1LC~C_-:.U1~s ~n t.t~J-.~.,mLZ'"-..CLl!_c*S...~...._3_h..§.n. _th~_§iL.Qr·e, .. coona c t §~9.
G!dJ'd .•L 2~~..2..• s~L(~!tJlill.i X u C c y u C "-CS, Z u C ---=-"".--~~,.~~--,,-------...-.,.--...-----
Notice fol:ows
trivially
....
(X u C) n M
=
from
that
the interiorness
the other
(Y u C) n M
=
of the collapse
result~because
(Z u C) n M
=
C n M.
VII
- 62 The lemma is trivial if max (q, x) > m - 3, because
=
choose X
y
z = max (~9
=
Z.
Therefore
x) + x - m + 2.
assume q, x ~ m - 3.
then
Let
There are two cases according
as to
whether x < q or x ~ q.
Therefore
z = q + x - m + 2.
Vie can assume X = it then trivially the inclusion h:X x I
->
C,
because
if
it follows for X. X c M is homotopic
M keeping
not necessarily
'lye
loss of gener'ality
:prove the resul t for X .: c,
The C-inessentiality
Both f, h are continuous-maps,
linear, but before we make h piecewise
linear we first want to factor it through a mapping Without
loss of generality
of heX x I).
cylinder.
we can assume M to be com}?act.
For if not, re:Dlace M by a compact submanifold neighbourhood
means that
to a map f:X ~ C by a homotopy
;~ n C fixed.
piecewise
Without
(Construct
1\1:;:
M:.;;
containing
by covering
a
heX x I)
with a finite number of balls and taking a regular neighbourhood of their union).
Replace
C by C n
M:;.;'
If the result holds for Nt..•.
then using the same Y, Z it also holds for M, by excision. Therefore
assume that M is com}?act, and consequently
pair M, C is triangulable.
By the relative
simplicial
theorem, we can homotop f:X ~ C to a piecewise X n C fixed. J =
Triangulate
identifications
of the topological
cylinder.
approximation
linear map keeping
X, C so that f is simplicial,
~X u C be the relative mapping
the
and let
Since h realises
relative mapping
the
cylinder, we
VII
- 63 can ~actor h through J by Le~na 479
x
x I
h
"~ where glX
) lVI
/
/g
J/
u C is the identity.
Again using
the relative
make g:J ~ M piecewise
simplicial
linear keeping
approximation
theorem,
X u C fixed.
J
=
Let Y
g(~)9
which
is one of the spaces to be ~ound.
I,e t Y 0 = Av
Then Y ::) X an d d· 1m Y ~ x + 1 • Yo c
Y.
In particular
~he identity, position
glYo
Yo
fixed.
by the hypothesis homotopy
oJ>"
18 we can homotop
U
-Pv) .I..A
Y - Yo c M.
X n I\'I c C.
of g keeping
Therefore
The homotopy
X u C ~ixed, because
extends
it is
gl~X into general
0
There~ore
Then
•
because
At the same time we can ensure
o
g(~X - Yo) c M.
= g (v
is in general position,
and so by Theorem
keeping
U f-;r.J>..
Y n M
=
that
YonM
trivially
~X n C = ~X.
c C9
to a
Since
VII
- 64 dim (~X) ~ x + 1, the general dim
position
S(glp,x)
~ 2(x :>;;
"beeause x < q. y
=
g(~X)
general
IIlay
implies + '1) - m
z - 1,
What we want is dim B(g) intersect
position
:>;;
z - 1, but as yet
C in too high e. dimension,
move is neces8ory. The ..! dim Co :>;; CJ. by hypo the eis.
Let Co = closure o
Since Y - Yo c M, by TheorG~ 15 we can ambient Yo f'ixed,
Qnd so another
l.Ultil Y - Yo is in generD.l posi tion
isotop
Y
keeping
with respect
to Co.
The:::>efore
= ~~ -.. ',.. Tr.:.ei sotop~r of' Y keeping Yo fixed x u C 1'ixed" that
does not alter
determines s(glp'x)
a homotopy of g keeping
hat
does reduce
S(g),
because g ( ~X)
n g (.J - p.X) ::: Y (, (C - f'X) ==
Therefore,
Ii
C
-- (y
f'X - X)
.-
1..-'-
Yo)
-
(y
Yo)
'~T
Ii
Ii
C,
since
J ==
Ii CO?
s~_nce Y
-
~X u (J - p'x), we see tlmt
- S(g Ip.x) u S(g IJ
p.X)
U
clusure
g-1 (g(p.X)
XnCcfX
C
8lnce g is non-degoner8te~
Writing S(g)
(Y - fX)
n g(.:J - p.X»?
0
Yo c M.
VII
- 65 of which
the
second
made of dimension
term
is
empty?
~ z - 1.
and ~he other
two terms
wa have
Therefore
dim B(g) ~ z - 1. Novytriangulate lJY the
J ~T
the
collapse
J'-;>C
to Lemma L~6)?
Corollary
u C by l,emma 45.
of the
and let
Therefore
T
mapping ::c
trail
cylinder B(g).
(given
Then
(T u 0) by Lemma 389 be cause
gJ ~g
T ::) S(g). Let
Z - gT n Y, which
is
the
other
space
that
we had to rind.
'l'hen dim ~ ~ dim r7
m .L
\ ~ 1 + dim Sf\.g},
~
=
Now Y u 0
gJ, [J.nd Z u 0
=
z.
(0"m <.;>-l.
n Y)
== (gT u C)
= =
by Lemma 44
u V,..,
n (y u 0)
g( '1' u 0) n gJ geT
u 0).
Therefore XUCcYUG~ZUC9
which
completes
the
Therefore th(; :piping simplicial, cylindel'
z
lemma. ~nd. let
proof'
=
of' case
2x -
ill
~LX
of f.
'Nitl1out
loss
dim 0 ~ :;;:; for
otherwise
Jet
This
ascume M9
As before ,..1 -
+ 2.
(3)
U
C be the of genernli C(x)
time
we shall
C compact, relative
make f:X
sim:plicial
ty Vie
c~')nnte the
need
C8.n
to use -+
mapping
aSS11Ine
x-skeleton
C
of C.
- 66 -
VII
Then fX c c(x)~ Qnd if we prove the lemma for c(x):
X
c(x)
u
c
Y u c(x)~ z u C(x)~
excision9 because C - C(x) c 1\.1 by the
then it follows for'C by o
hypothesis dim (C n M) ~
x.
Y. ~
Hence dim J ~ x + 1.
Therefore as surne dim C ~ x.
=
Xo be the (x - 1)-skeleton of X9 and 30 cylinder' of
f
IJet
~Xo u C the submapping
I Xo. As before construct from the homotopy h a
piecewise linear m3p g:3 ~ M such that g!X u C is the identity. In particu18r glX u C is in general position9
and 3 ~ X u 30
~
X u C,
and so by 'l'he(,rcif\ 18 Corollary 2 we can homotop g into general position for t~e pair J, X u Jo keeping X u C fixed. o
At the same 0
time we can ensure that g(J - X - C) c M.
Therefore g(J - C) c M,
o
because X - C c H by the hypothesis X n M c C. This tilne let Y = g3. y n M c C beco.use Y
-
C
:::
Then
- gC
g3
c g(tJ
x x X , 30 c 3)(+'1 is cylinderlH:::;, ~ g(3 Therefore
b;)'
the pj.ping 10m.Hie;
Y ~ X,
-
(LCl"fll.1t:t
-
dim Y
,,;;
x + 1, and
0
0) c M. 0
Now the triple x ~ m - 3.
30)
c M and.
Li.8)
we can hom.oto};) g l-::eeping
o
X u 30 fixed and g(3 - 30) c M~ and choose J1
~
S(g) such that
Trinll.gulatethe mapping cylinder collapse J 0 •••.•• '":1 C given by Lemm<: :+6 8or~(.)11nI~Y ~ and let T = trclil (30 n d.im
1":"1
J•
~
"
.
~ z..
dim
3:1.) •
(30 n cJ 1 )
Then
by Lemma
44
VII
- 67 Also
u C by Lemma l-\-5.
Jo~T
Jo n J1 c T.
because
Therefore
Z
dim J1
T9
~
=
(T u 31. U C),
g(T 'u J1); then
z.
Jo
u J1.-;T
u C u 31
Therefore
Lemr:c.a38, g.J ~g
by
Now let dim
Therefore
We have
=
gJ
31 ::> S (g)
because
dim Z ~ z becQuse
=
Y
.
both
Y U C9 and geT u 31 u C)
=
Z u C,
and so
x u which
com};'letes
Definition:
the
y u
C c
froof
C"'-J Z u C,
of Lemma 48.
a 16gul:"'lr nei ghbourhood
~dmt9sibl-~
if
the
N~X
collapse
9L7~j·I'Jt~~_.l~e t :t:J2.e
If
adrnissible.
ar&
ElD.,,,;'l9l11issibler~£Lu.h1!'11.eJ1illbourhoQ.d
X·~ Y the
resul
~ is
Ther>e:foro ~ssume Y -2:,.x.
N -2;. X ~Y.
called
jJ1E: fl'_Q.n..t.Ler_,=9f~~\Lj.LL1!.~_~Jf ~.£.N - ~.2
z.:$Y orL$];.J_1Jl.E;:ll. N is .:;,Q....§.£..
~oj:.
is
N of X in IvI is
trivial
There
aro
because
then
two C'3.ses: the
o
absolute
case
when X c E and t:i:1e reL ..tive
case
when X meets
H.
o
In the
al)sclute
therefore
the
CDBC:
bath
re:;sul t follows
The rela ti ve case of the
proof, (i)
to
Lemma
-]4).
N, Y must
leaving
the
A derived
1 [3
also
li8
from Theorem similar
~etails
7
is
.•_
6 CorollaI'Y
q:nd we indica
to the
neighbourhood
in 1:1, nnd so F -
t3
~',
j'T
0
4. the
ste~9s
reader. admissible
(c.f.
Corollary
VII
- 68 (ii) ambient
two admissible
111.lY
isotopic
(c.~.
Theorem
If' N:l..is
(c.f'.
Theorem
adn.issible
that
8 Coroll~ry
Now~ given regular
the
I'egular
then
there
~(x~
=
0)
is
J.JGllliUa 50 is
regular
neighbourhood
a homeomorJ:1hism
.
0,
X E F and
tlle
lemma 9 let
~(F x
I) -
.
MnCff-l{),
2).
s:~tuation
in
neighbourhood
c;}i":::"indel'vviseby
neighbourhood
are
8 (2».
of Y in N - F.
..p an a drnisslb'] ..e reg-u'1 8.1' neighbourhood. ~ oj.
Henco N~:N:t
neighbourhoods
8.1'lother admissible
or X in M~ and N:1.c N - F, N:l..such
regular
-v'
J).
In
-.,
IVl
(iii).
TherefOl~e
of Y because
N -S;N1 -~Y.
N1 be an Then
oecm.lse N is
N:l.. ",T'
.1.'1
is
also
---="C(, Y ~. a. v
an admissible
The proof
of
complet.e.
We now prove In :Qarticul[~l~
thlS
Theorem eovers
2-j
in
tbJ:; case
the
8pecial o
1."-:11 or:. X c M.
case
that
X n M c C.
- 69 f~!l~~
Let
VII
C 12.e~a __£1:"colJ.~siQJe
~_:-~~~c;>£~e~9f--,~-,~~n
~~ __1;>8 . ~.Q.L1ill.~ . ~.J..~C_-:igesSel1J;ja~~:~LD.,Ji_c C.
Le,1
o
c = dime C n ~,~).
Lot tl1~ .I._·t
o
(d,
('Iv•••••.••. •
0
r""
.r1-1c> -.~
We consider
dJ'• m1 (0.n nI 1.7) -" q• V_",;, .ll. ""
separately
the
three
The Q-collapsi"bili c > q in
Of course
(1)
cases:
ty
general.
(2)
c ~ q
means
c > q > x
(3) c > q ~ x. (c.f.
x == -
Thl~ ~9roof' is
')Y induction
Assume th0
::>e8u1t true
1.
y~ Z such
choose
the
proof
of Theorem
on x, for
starting
trivially
dimensions
< x.
with
By Lemma L.l-9
o
tlLi.t
X c Y U C-----:lZ U C, Z n 1·,Ic C9 and
z == dim Z ~ maz (q, rrhere:fore
19).
x +
z :r; k by th.e hy:,;othesis
TheI'efoJ'(:3 Z is C-inessential.
x) + x -
~il3.X
(q,
x)
b;y th2
Also z
ill
+ 2.
~ rn + It - 2. hypot:1csis o
X
~
with to
m -
3.
ThereforE
b;! inductjon
d:.m (}~ - C) ~ z + 1 ~ the
A,
Z c E ~C;.
we can enguli'
and
L n
iVl
C n H.
:::
Apply
Lemrria Lj.2
situQtion o
y. u C'~Z
ancl a' nbient .
(y u C) u
H, to
isotop
E
u C c E,
l\:eeping
Z
U
C fixed
so that
0
Since
E';"---'IE;;;.
D.mblent isotopy
preserves
interior'
<)
we h:cive 3:---..,;C.
coll::psibility o
we hnve
D = Y u C u E,.,.~
Thcre:~-'ore putting
-.'
0
D --:I E,.. --.:-''::;
j
2nd
80
)~c
o
ri--~C.
Also dim (D - C) ~ x + 1 because D - c -
(y -\
E.) "j
u (E.
- 70 dim Y ~ x + 19
VII
=
and dim (E.;; - C)
dim E - C ~ x.
Qas_q_l?J.; .. _._CL~L..9..2~"~ o
Let C-~Q be given by q-collaJ?sibili Q so that
Xn C
=
X n Q; for
some triangulation
by Lemma44, and C~~,
fact
it
o
C--~ Q,.
u T by IJemmaL~5.
as Q in the definition
we cell 8u:p:poseX n C = X n Q.
C ~ Q ensures
Q-inessential
also.
Q
u T is as
of q-collapsibility.
Therefore
retraction
(X n C) under
Therefore
becnusa X n C = X n (Q u T).
X u C ---=-X u q by excisioit9
can choose
Then dim T ~ x + 1 ~ q
is better o
'Ne
T = trail
not let
of' the collapse o
good a candidate
if
ty.
In
Therefore
and X n M c "~' The deformation
that
if
Therefore
X is C-inessential
by case
then X is o
(1) we can engulf
X c E ~Q.
Apply Lemma42 to the situ&tion o
Xu
and e.mbient isotop
:ill to Eo"
'j'
o
(x u C) u E,., --....." E.,.• •'
o
E '~~.
Let F
.J.~
Therefore
C"-,.."X
F :J C
QcE
IJ
keeping
=
X u (~fixed,
X u C u E~. 0
:J
so that o
0
Then F-;.K,.; ~Q9
because
0
':J9 F ~.~
ane. C -'''; Q,. o
If we could deduce th:J.t F --.-) C we should be finished'9 this
cannot be deduced,
Chapter 3.
Therefore
as is
shown by the example at the end of
,"V8 have to get I'm.met
a regulRr
neighboul"hood of F;
but it
attention
to a compact subset
in order
C19 F19
1'.1;1
be a regular
that
neighbourhood
~h denote th0 intersections
the diff'icul
is necessary
shou.ld exist. Let
but
first
tbB r8gular
-_.-~----
ty by talcing to restrict neighbourhood
of F - q in M.
Let
of M1 with Cjl F9 Q respectively.
VII
- 71 o
Then Fi --~
l~hand Cl
in Mi.
C:h in VIi because
of Hi.
meet the frontier of Fl
0
~
Then Cl
Let:n be an admissible
Adding C - Cl C
of N because
also have N an admissible
regular neighbourhood
c F
regular neighbourhood
does not meet the frontier
-r.! ~l'
x
none of the collapses
Fi
of Cl
by Lemma 50.
to both sides, N u C ~C C c N u C.
U
in Ml,
the collapse N u C--"'/C.
Let D
=
by excision.
trail X under
44.
..
N-->Ci•
Now
some triangulation
of
by Lemma 45, and
Then X c D~C
dim (D - C) ~ x + 1 by Lemma
Therefore
By Le~lla
X u C fixed so that D n 1'.1= (X u C) n M
=
C
43
isotop D keeping
n M.
This completes
the proof of case (2). C~ s~j)~~~->
..9.. ~ "_x
o
Let C ~~~
be given by the q-collapsibility.
Triangulate
X u C· .:.~Q, so that X is a subcomplex,
and subdivide
that C ~Q
Let TX be the trail of the
collapses
(x - 1)-skeleton collapses
of C~
of dimension
X u C because
although
of some elementary and therefore
where Xi
Q.
~ x + 1.
T u Q by elementary
It is valid to perform
an x-simplex
it is nevertheless
principal
a free face when X is added.
(C - (T u Q»)
u T.
simplicial these on
of X may occur as the free face in X,
Therefore
We now want to engulf Xl from
Q. Observe
so
o
Then C ~
collapse,
remains
= (X -
simplicially.
if necessary
that dim Xl ~ x because
dim T ~ x.
Also
VII
- 72 -
x~ n
M c (X u C) u M
=
=
C n M
so is X u C, and therefore r'etr'action C ~ Q, ensures
Q n M c Q.
Since X is C-inessential,
so also is X~.
The deformation
tha t X~ is Q-inessential.
case (1) we can engulf X~ c E ~
o
Therefore
by
Apply Lemma 42 to the s1 tuation
Q.
o
X u C -;:I and proceed
as in case (2).
The problem
Q c E1
The proof
of proving
.that the deformation type collapses,
X:1. U
of Lemma
the general
case of Theorem
of X into C may involve
and so the engulf
interior-to-boundary
involve
type expansions.
at the end of the proof
introduce
a device
of Moe Hirsch
some boundary-to-interior
But when we try to expand
stuff out of the way.
situation
21 is
the inverse process
interior-to-bolil1.clarywe hi t an obstruction, room to push other
51 is complete.
because
there is no
We drew attention
of Lemma 42.
Therefore
to this we
to cope with the difficulty.
InwaL.C!&. 9gl:l;.apsing.
~~JEi_tjoA: simplicial
Let K ~ L, J be complexes. collapse
every subcomplex Example
K~L
x
O.
J
if J n trail W
=
J
n W for
W c J. Let K be a cylindrical
Then any cylinderwise
X
is ~vau.fy~
We say that an (ordered)
colla:;?seK ~base
triangulation
of X x I.
X x 1 is away from the top
VII
- 73 Let s denote induced
collapse
simplicial induced
K'~L
collapses
a subset;
K--~L,
o~ t are a subset
set of simplexes, therefore
while
V.
Definition. ..~ --
Let X~Y
to V
by adding
Therefore
t V c J n trails V Therefore t is also.
s is away from J.
by adding
trailt V is obtained
V c trails
trailt
The elementary
of those of s, with the
J n V c J n trail
because
and t the
u trail W given by Lemma 45.
If V c K, trails V is obtained
ordering.
an ordered
the collapse
=
in the manifold
J n V,
M.
We call the
'
collap se l.llwa.r.d~9 and wri te x....:f..;y if, given any tria.ngula ti on of
X~~,
there exists
.......,~.-=-..•..- ~;~-'~ X - Y
'A
definition
-
':{
a subdivision
n Y away from
;~':II.~ .A -
that y is invariant
x '~X
n
and a simplicial Y n
TIt'(
.'.u
under Y
It follows
collapse at once from the
excision:
<=> X u Y ~
Y
;t0c...§mp 1e • Let M be compact, of' the collar. there exists
X a collar
Then X --1;; Y, because
a cylindrical
subdivision
on M, and Y the inside boundary given any triangulation and a cylinderwise
of X
collapse
away from M. L-iLI}ll!l~L~J __(JLi 1's.£hl L~..t,Q..J?..9_~0....9..::.9~9l:tan sibJ-a k..::Sl9t::'L ot:.JvL~.9JJJ2.Il.o_sa y~~ Z.z.. aJ:~~
.c 21f1E.a,c b_.:J.
:J_,j~_.Q. ,9_::::S.;.z...-1L~q..&-,g;....J.l=L_Q.::.i.ll§ s~en tJ1?-l 9 anQ..~-Ll. ..rt_s.....9• L~ t
- 74 -
VII
/
Z\J
M
I
y
The proof is by induction with z
= -
19 for then choose D = Y u C.
true for dimensions Since z ~ Y9
the hypothesis o
£9
starting trivially
Z9
Therefore
assume the lemma
less than z. of Lemma 51
is satisfied for Z9
dim (~ - C) ~ z + 1.
and so we can engulf Z c E~C9 ambient isotoD
on
By Theorem 15
keeping C u Z fixed9 lli~tilE - (C u Z) is in
general position with respect to Y. Let W
=
closure (Y n [E - (C u Z)]).
Then
dim W ~ y + (z + 1) - m. Now N c '(Yu C) - -(Z u
cr.
so that W is a subcomplex. subdivision T = trail W9
and a simplicial
Therc:fore triangulate By the definition collapse Y u C~Z
.
then T n M = W n M.
(Y u Cr-:
(Z u C)
of y there exists a u C such that if
VII
- 75 -
~e claim that Y~ T~ E satisfy the hypotheses for Y~ Z~ 0 in the le~na.
Once this claim has been established~ we can appeal
to induction~ because t == dim T ~ 1 + dim W~ by Lemma ~ y +
44
z + 2 - m
< z~ by the hypothesis y ~ m - 3.
Therefore by induction engulf Y c D~E. o
E ~O.
Thererore Y c D~O
Therefore Y c D~
because D~E~O.
0 because
Finally
dim (D - 0) ~ max (y~ z + 1), because dim (D - E) ~ max (y, t + 1), by induction~ and dim (E - 0) ~ z + 1~ by choice of E. There remains to establish the claim about Y, T~ E satisfying the hypotheses. o
First E is q-collapsible because E --~O. k-core becauso 7C.(E~ 0) == O~ all i. J.
is compact.
Next E is a
Next T is compact~ because W
Next Y ~ T because Y ~ W and so~ taking trails under
the collapse Y u 0 -"1 Z
U
0,
Y == trail Y ~ trail W == T. Next Y u E '~T
u E by excision~ because Y u 0 -y"'T u Z u 0 by
Lemma 52, and (Y u 0) u (T u E)
=
Y u E
(Y u 0) n (T u E) == (Y n (T u E)] u 0 ==
T u (Y n E) u 0
=
T u [tV u Z u(YnO)]uO
==
T u Z u O.
VII
- 76 Next T is E-inessential
=
t
because
E is a k-core~
y + z + 2 - m
~ k, by the hypothesis Finally
the dimensional
hypotheses
y + z ~ m + k - 2.
are satisfied because
t for z~ and t < z.
change is to substitute
and
Therefore
the only
the proof
of Lemma 53 is complete.
3e=1a.t:i..y_~__~olJar:s~~ Let M be compact~ V mod C in M as follows.
C c M, V c M.
We construct
a collar on
Let c:M x I ~ M, c(y~ 0) = y~ Y E M be
a collar on M.
Choose a cylindrical
a triangulation
of M such that V, C arc full subcomplexes
simplicial.
triangulation
(This can be d.one as follows:
triple M, V, C~ noxt take a first derived, c simplicial~
then subdivide
extend the subdivision
V - C to
vertices
the
next subd.ivide to malte and finally
8 > 0 such that M x (0, 8]
Let f:V -~ [0, .s]be the simplicial
by mapping 8.
Choose
and c
first triangulate
to make M x I cylindrical,
to M).
contains no vertices. determined
of M x I, and
of V n C to 0 and vertices
map of
Define
Vi
=
V2·-
tc(v, t); V E V~ 0 ~ t ~ fvJ tc(v, fv); v
E
vj.
We call Vi a ~oll.§..r o.n":'[J!l...2.¢l. .Q.J.lL1!, and we call V2 the ip.§J"g.e
bOBn~£Y
of the collar.
VII
- 77 -
c
V
Suppose, ~urther,
dim (X n t.')
. ,Iii) .,I.'U.
that we are given XX c M such that
< X~ and that we chone the tri~ngulation
o~ M so as
have X a sub complex.
1en~.
o
ii
i.L-V 1. ::;::->:::!_y _ilL
1Yl__ ili!r.~.J. X
('1"..
~11·
0 V 2 )~jC-:..
P~q-.2f.. i)
o
If v E V - C~ then v E A, some si~plex A
.
i
C.
The
fibre v x [0, fv] c int [st(A x 0, M x I)] because the triangulation choice of
8.
is cylindrical,
Thererore
and because
of our
the image c(v x [0, fv])does
VII
- 78 not meet C. V2
n
C = V
=
V2 n M ii)
V:1.
--r....,.
Therefore
n C because
=
1'-10
V
n
V
=
C
V1. ()
n C c
\[2
l{ext Finally
C V1..
V 2 because take a cylindrical
that V1. is a subcomplexs
C.
V n C is full in V.
because
C9
V ()
subdivision
such
nnd collapse cylinderwise
away from V. iii)
Let p:M x I ~ I denote the projection. meets
V1.
-
V
then pA is 1-dimensiona15
is a convex linear cell of dimension
If A a E M and A n
V2
a - 1 separating
one of which is A n V1..
A into two componentss
CQ11crpse across A n V1. from A n V2 for all such simplexes A not in Xs in order of decreasing o
and we have the collapse V1. --, iv)
If
v
.r!. E
.lI.,
then either
.M) V
V u
eX
dimensions
n V1.).
~!!s vvhence
A Ii
2
Ii
< Xs by hypothesis~
C
dim (i~t
Ii
else
n V2 contains an interior points whence
"
.L.~
V' ~ dim eX 2 )
< dim
dim (A n V2)
A
~
x.
Therefore
or
dim eX n V2)
We m.'8 given X c M: to engulf, where ;~ is C-inessential, C is a q-collapsiblc
k-cores and
q~m-3 .i~
:.f;
m -
4
q + x ~ n + k - 2 2x ~ m + k -
3.
< x.
- 79 -
VII
Let Y be a col12r on X mod C, with inside bOill1dary Z.
We now
want to apply Le~~a 53, and so let us check the l~potheses. Y u C
~z
u C by excision,
Lemma 54 (i) and (iv) • furnishes because
a homotopy
because
from Z to X keeping
hypotheses
by
.
Iv~?
Finally
43.
Finally
the
y :::Z + 1, z :::x. Therefore
Next
z + 1) ::: x
we can choose D so that D n
ThG proof of Theorem
H~~
because
Lemma 53 we can engulf Y c D~C.
dim (D - C) ~ max (y,
Lemma
X n C :::Z n C fixed, and
by LemlTl1J. 54 ( i ).
are satisfied
XcD~CbecauseXcY.
the collar
by hypothesis .
Next Z n C :::Z n M c
Therefore
Z and Y n C :::Z n C by
Next Z is C-·j.nessenti2.lbecause
X is C-inessential
dimensional
Y ~
First
..Q.. . c ltl..L X £.91~ ct. t
"+1
A
!£1_eIl~r~r~~i si. l!iT
21 Case
CL Q -:-i ne
cU}.•
+
n M by
is complet3 .
9..§_e.~]..i,t~J~.2~ ..a.n.CI._,.§.j1!L.1J;..
x
~~
W .•1L
9 ::;Y__V-_.9%-:S;'~~-.Jd,..,.Q
-k~ti~l_.Q=-~
/S--£"
(2)
.M :::(Y n1 . C)
.. §.§!F~!-...LaJ--1-. §l!-cll. that
.
n_ID_<: ~
VII
- 80 •
---
---
The important part of the le~~a is to get Z n M c C and the v-collapse. Without
P£.99f. compact5 in M5
for otherwise
and perform
loss of gen2rality
we may aS2ume M
replace M by a regular neighbourhood
all the constructions Xo ==
--~~-
eX
therein.
of X
Let
n l~) - C
Co == Xo n C. We assume Xo Triangulate
J ~~
otherwise
the lemma is trivial: X
M such that Xo and M n Care
first derived.
=
subcomplexes
Let Vo be the closed simplicial
=
W == Y
Z.
and take a
neighbourhood
of
Xo - Co; in other words Vo is the union of all closed simplexes of M meeting Xo - Co' points
(Notice that Va may not be a manifold
of CO~ and is thorefore not a regular neighbourhood
general) .
'Ne deduce
in
at
VII
- 81 (1)
Vo·--'" Xo
(2)
Va
n
by Lemma 1 L.j. Corollary.
9
C = C09 because
of the first derived.
Let V~ be a collar on Va mod C in M9 X a subcomplex appli cnble) •
l,et X~ be the subcollar on Xo of the collars.
n X -.: Vj.
V~
(4)
dim (X n V2
(5)
V~ u X u C~X
(6)
V~ ~
on Fo'
~
X
=
X2
denote
We claim:
V2
-
1
u C.
F:t n X Vj.
V2 -....,.Xj.
frontier of Va in M9 and let F~ be the
n X
= =
Fo
=
Fo
Therefore by construction
=
Co
n Xo
009 because Va is a neighbourhood
F is a collar mod X as well as mod C.
by IJemms 54 (i).
n X=- V:t
u (X n Vj.)
of
n C.
= = = =
Therefore
X n (frontier of Vj. in M) X
n (F:1. u V2)
Co u (X n V2)
X n V2•
follmvs from Lemma 54 (iv).
V:1.-~Vo
and let V 29
Then Fa
Therefore
)
n
X
9
0
U
Xj.
To prove (3)9 let Fa
(4)
=
(3)
that we have
(so that LemIJ1s54 is
during the construction
the inside boundaries
subcollar
with the proviso
To prove
by Lemma 54 (iii).
(5)
observe that
Next Vo u (X n V~)~X
n V:1.
VII
- 82 by excision
from
Therefore
(1),
Next
Vo,
V2
(6) ~X2
n (X
U
and so Vi U X U C ········~x U C
C) = (Vi
n x\I.J
U
=
~r) n ),.
U Co
V:t--Y.JXi
observe
that
by
because
(1),
(Vi
the
Also V2 n M = Vo n d
Xo.
=
Co
=
X2
U
V2
pair'
There f ore
V ::3.-.....-I X2
Xi
X2•
Let
by
(2)
again
by
trio.ngulation
is
Le~~a
( \1
(2)
away from
homeomorphic
Vo•
to
54 (i).
o •
= trail of' the
W
Y Z
Xi
Therefore
The proofs Ti
X2
Lemna 54 (i)
n M,
by
cylinderwise
by
.)
n C) by Lemy)1a54
(Vo
V2,
o
n V2 =
X n Vo = Xo.
because Vi
To prove
=
Vo n (X n Vi)
n Vi by composi tion,
Vi---.,lX
by excision,
because
(3,
of
(X n
Vi)'
4,
5, 6)
T2
=:
X:t
u (.ri
U
(X
- Vi)
Xl
U T2
U
(X
- Vi)
X2
U T2
U
(X
- Vi)·
by
are
trail
collapse
composite
= = =
U V2---~X1
(6).
excision
because
complete.
(X n V2) Let
under
some
.-
83 -
VII
•
M M
/' x:t.
.......... ~ Y
·x
c:
".I
I
~
X:a
I
T:a
'~
z
w
We must now show that W, Y, Z satisfy the properties
"
in
the Lenlina. First dim ~N ~ x + 1 and dim Y, Z ~ x because dhl X:t ~ 1 + dim Xo, ~
because
x, by the hypothesis
dim Xs ~ x - 1 , because
X2
Xi is a collar
dim (X
. () M)
on Xo,
< x.
to Xo· is ho:t:i.eomorphic
dLn T1 ~ 1 + dim (X n Vi) , by Lemma 44 ~ 1
dim T2
~
+ x
1 + dim (X n V2)
~ x,
by
(4).
Next Vi u X U C is (X u C)-inessential C-inessential
because
also C-inessential
x
by
X is C-inessential.
because
(5),
and therefore
Therefore
they are subspaces
of
V:t
is
W, Y, Z are U X U C.
Next
c Vi! because
o
Next we have to show W u C-~Y
u C.
First observe
that
VII
- 84 U T2 by Lemm~ 45 Corollary; moreover
X~ U T~~X1
is interior for the following reasons. under the first part of
.
If
this collapse
rri = trail (X ()Vi)
(6), then
Ti n M = (X n Vi) n M, bJi7the propertJ.
T "(,
= X n Va
=
Ti
Also T~ (6),
Xi U V2,
C
because it comes frow the second part of
.
flnd so (T:l.- Ti) n lV1 C (x~ U v2)
Thorefore (X~ u T~) n M Xi
-,r
'/:"'0·
o
U
T~--~ X~
U
T2•
=
Xo
=
=
by Lemma 54 (i) .
.Xo,
(Xi U T2) n M.
We can add C u
=
Co,
C
T2•
lr i n X"~ V;
n M
by
X""":v;..-
Therefore
to both sides because
Leoma 54 (i)
n
X- -
v7
C
V~
=
X n V2 by (3)
o
Therefore iN u C "---; Y u C by exci sion. X2 by Lemma 54 (ii).
Next X~ ~ C u T2
U
X - -v7 to both sides because Xi n C = Co
x~
Ther'ei'ore
We can add
n
X· - V"";
C
X2,
C
Xi
by Lemma 54 (i)
n
V2
=
X2,
by
( 3)
•
Y u C --Y.,. Z u C by oxci si on.
Next X, W, Y all ~eet M in the same set because X n Vi' Xi U T1' Xi U T2 nll meet M in the same set, namely Xo.
Finally
- 85 Z
n
VII
M c 0 because (X2
u T2) n M c V2 n
lVI
by Lemma 54 (i)
~ Vo n C c
C,
and
(X - V1) n M c (X - Xo) n
i
c C, by definition The proof
of Lemma 55 is
of Xo.
complet6. ~,.
9£_ TEE3.2Fem li~Q~~,§~~~
Proof We are X is
given
C-inossenti~l
.
where dim (X n M) < x.
X c M to engulf,
where C is
a q-collnpsible
k-core,
and
3
q, x ~ rn -
q + x, 2x ~ m + k - 2. B;y
Lemma 55 choose
Since
Z is
C-inessential
by Lenuna 53. and ambient
yX, ZX such
vyX+1,
and Z n M c C we can engulf
A}?ply Lerllfl1a42 to the isotope
('IVU C) u E,;,--.E,,:. X c D-~ C because
X c \IV U C -~.}y u C ~:£.":" Z U C.
that
Let D == W u D -~,E·~C. D
-
W U C ~y
(Y u C) u M fixed, (W u C) u
==
E;;;.
u C c E, so that
'l'hen
Also dim (D - C) ~ x + 1, because Ti'., C "1,1 ~". v
dim (D E~> '6"
E;;
o
situation
E to E~;:, keeping
Y c D ~C
-
-
Y,
E.,. ) ~ dim "
C '" E
-
V{
==
x + 1,
C,
dim (R., - C) == dim (E - C) '" ~
lilax
==
x
(y,
+ 1•
z + 1)
- 86 Finally
VIr
we can choose D such that D n M :: (X u c) n M by Lemma 43. This completes
Theorem
the proof
of our main engulfing
theorem,
21.
We conclude a theorem of En.
this chapter with an application
of John Stallings
More precisely,
which
given
implies
the uniqueness
two piecewise
linear
(::pOlystructures)
on En (there arc obviously
they are piecewise
linearly
pieceWise
linear
manifold
homeomorphic, structures,
n :: 2 is classical,
trivial, of Moise.
We shall prove
and n
Conjecture
proof
structures
infinitely
provided
and n
I 4.
of structure
many)
then
that they are The case n :: 1 is
3 is the Hauptvermutung
Theorem
the case n ~ 5 •
.Q.~_stj.9.p.~~ ~~2.Y_s.E.J The proof below
=
of engulfing,
fails
t
_t£:H~_!2J' n_=
41-
for the same reason
that the Poincare
fails.
This is the Hauptverwutung
for manifolds.
The obvious
case
to look at is:
~~2..1:L.i:
~~.jJ1e_. d_'2..\!_ble s.us.]ensi91L.2f_ a PotI}ca~_~...Jillllere ~2.lL()}_9,g.~gall::Y_.Acmlq2llorJ2l1i.q .to_S 5?
By a Poincare is a homology
3-sphere,
. sus~9Emslon 0 f'
1\T3.
m
sphere we mean a closed
3-manifold
but not simply-connected.
.. 8'1.,.-,J[3 1S t he same as t1,. 110 J01n "'lla
•
M3, which
The double This car..not be a
VII
- 87 polyhedral
sphere, because
suspension
ring 81 is M3, not S3. M 9.~n
Call a manifold boundary,
the link of a i-simplex
if it is non-compact
and its structure has a cOlmtable base.
equivnlent complex,
to saying there is a triangulation
in which the links of vertices The key idea of Stallings
Let M be 2-connected.
is the following
We call M 1-connect~d
i8 2-connected. structure
s8quence~
The property
This is
of M by an infinite
definition.
a~ i~fin~t[
if given
Q c M (Q is not necessarily
such that M - Q is i-connected.
by the exact homotopy
without
are (m - i)-spheres.
compact P c M there is a larger compact a subpolyhedron)
on the
This is equivalent,
to saying that the pair (M, M - Q)
is topological~
independent
of any
on M. If m ~ 3, Em is 1-col~~ected at infinity. Whitehead's
gamplCL1...~ after Theorem
19 Corollary
example M3 (given in Example
3) is a contractible
1
open 3-manifold
not
A
i-connected
at infinity.
In fact, if S' is the curve not contained in a ball~ and Q is compact ~ 81, then the fWldamental group of M - Q is not finitely
generated. The interior
after ~~itGhead's
example)
1-coru~ected at infinity.
example M4 (given
is a contractible open 4-manifold not In fact, if D2 is the spine, and Q is
compact ~ D2~ then the fundamental
~i(i~)as a subgroup.
of Hazur's
group of M - Q must contain
The dimension
4 is not significant
in
VII
- 88 Mazur's
examples,
because
Curtis has given similar
examples
for
~ 5.
dimensions
It is no coincidence illustrate
that we use the same examples
non-engulfability
fact the idea behind connectedness
and non-connectedness
the proof
at infinity
of Stallings
implies
a certain
theorem
to
at infinity;
in
is that
engulfability.
T.hg...ol'-.?m .1~_._~~1lJ11~)
L~j:,MID be~_~9.9ll.i£ftCLti ble _o~n manif.'old, i:_9.Q.nIlecYed __at :m
iD%J11i.tx. If ..E12~ Proof. --.-".' -=-=--
tp.~n M
= ~m
Let P be a compact
of the proof is to show that Pis engulf P directly Therefore
because
of M.
The main step
con tal.ned in a ball.
it is not in general
we have to start indirectly
M l1away from P".
subspace
We cannot
of codimension
by engulfing
~ 3.
a 2-skeleton
of
So choose a tl""iangulation of M by an infinite
complex. By hypothesis,
choose compact
i-connected.
It is important
subpolyhedron
in general
i-connectedness for the moment.
Q ~ P such that M - Q is
to observe
that Q is not a
(in order tl1at the definition
at infinity
be a topological
of
invariant).
Forget P
Let
x
= union of all 2-simplexes
meeting
Y
=
not meeting
union
of all 2-simplexes
Q Q.
VII
- 89 -
In the diagro.n the 2-skeleton
is represb~ted by
2-conn0cted h0li1oto:picin M; keeping assume f is piecewise approximation
X
1'1
Y fix0d~
linear,
ambient
tv a map t':X
-7
thE;crem in M - Q~ keeping
X n Y
linGor manifold).
in ~ 5 dim0nsions~
vYe can
(Since M - Q
By Theorem
X n Y fixed;
that fX - (X n Y) is in general posi tion wi th respect Since both are 2-dimensional
X c M is
simglicial
fixed.
isotop the inagc fX in ii/I •• ;~ keeping
i-skeleton.
M - Q.
oy using the relative
is open in M it is also a piecewise 15
th0 inclusion
P.
so
to X n (:/I - I.;J).
they are disjoint.
VII
- 90 Therefore
fX n X
C is connected
=
X n Y.
because
Let C
Then C c M - Q.
fX u Y.
it is the image ~~der
of X u Y, whicll is connected,
being
~1(M, C)
~o(C) = 0 and
Therefore
=
f u 1:X u Y ~ fX u Y
the 2-skeleton
= O.
Therefore
of cOllilectedM. C is a 2-collapsible
i-core. Now X is C-inessential the inclusion ?~tting
X c M is homotopic
k = 1,
q
are satisfied.
=
x
=
to f:X ~ C keeping
Then U :J 11 -
Q
:J C.
h:M ~ M isotopic
such that hU :J X.
Therefore
hP does not meet the 2-skeleton
notice
of Z u Y.
hP is contained
neighbourhood compactness,
X u Y.
a regular
:J X U Y.
Choose
hP in a ball.
19 Corollary
1
it does not
a second derived neighbo~rhood
of M such of X u Y.
second derived Again using
in the second derived
of Z, because
Therefore
our first
th8t since hP is compact
subspac-: Z of the dual
By Theorem
(1)
This mLl1ceshP "effectively"
(m - 3)-skeleton.
hP is contained
neighbourhood
21 part
by Lemma 41
We have achiev0d
in the complementary
of the dual
of some compact
of Theorem
hU :J X U C
that hP does not meet the second derived Therefore
X n Y fixed.
Therefore
of codim 3, and so we c~n now start engulfing
meet a neighbourhood
X n Y, and
to the id~'l"l. ti ty keeping
P off the 2-skelcton.
More precisely,
=
we can engulf X fro~ C.
there is a homeomorphism
objecti ve of pushing
X n C
2, m ~ 5 the hypotheses
Ther8fore
Let U = l':~ - P.
C fixed,
in M because
neighbourhood
(m - 3)-skeleton. Z is compact,
Z is contained
the N
N is now
and so N--;Z.
in a ball.
- 91 Therefore
N is contained
P c h-1B9 because
VII
in a ball,
hP c NeB.
B saY9 by Lemma
We have completed
the proof, which was to show that any compact
37.
Therefore
the main
subspace
step of
is contained
in a ball. We now use this result of balls
{B.] as follows.
Choose
1.
is connected
to cover M by an ascending
the triangulation
a triangulation
is countable,
A. u B. 1. 1
contained
For i > 1, define
to be a regulQr
Then
1.-
neighbourhood
{B.] is an ascending
of its successory
The proof
of Theorem
of a ball containing
sequence
1.
in the interior
UB.1. = UA.1. = M.
Since M
and so order the
.... B.1. inductively
of M.
sequence
of ballsy
each
such that
22 is completed
by:
~e2llin~2h If. Mm .J, s th~_ un:Lon in thEt,~nterior Proof.
Let Em
each in the interior f1:B1
~ ~1'
combinatorial
-_
then --_ Mm ~ Em. ~--
of its --. successor, ......•.••....•
= U~., the 1.
m....a 52
sequence
of m-simplexes9
Choose a homeomorphism
extend fi_1 to fi:Bi ~ ~i by the
theorem
(Theorem
---~ ~ sm-1 I that Bi - Bi-1 ~ ~i - i-1 x. The two corollaries to Theorem Q.Q£ollar;y:1.
..~.~
ascending
of its successor.
and inductively annulus
....
8 Corollary
3) which
says
22 are also due to Stallings.
.. Th _.ep1.ecowJ. so l' lnear _~t ruc t ure. of ~IIm !\
is uni..@e up to h0.11!£2.rn.2£l?hi sm.
- 92 -
VII
m q Le-.i.1V1,Q: be contrScctible _2lli'3n ma:g,ifolds. m q If m + q if: 5 the11.lL~_("l E + • Copolla"rX 2.
=
This result examples
is interesting
above. It suffices
at infinity,
to show that M x Q is 1-connected
and this is done using
if m ~ 2 then
compact
in view of the non-trivial
Mill
algebraic
H~(M)
has one end because
topology.
= Hn_1(M) =
First,
o.
Therefore
if both m, q ~ 2 we can find arbitrarily
large
subspaces
A c M, B c Q such that M - A, Q - B are connected.
Therefore
(M is 1-connected
x Q) -
(A
B)
x
= M
by Van Kampen's
with amalgamation
both
x (Q -
Theorem,
sides are killed
On the other hand if m > q so choose
B to be an arbitrarily
is homotopy
equivalent
again is 1-connected While Theorem
=
u
(M - A)
becQus0
by Van Kampen's
x Q
in the free product
by the amalgamation
1, then Q
=
large interval.
to two copies
discussing
B)
.•
the real line9
and
(M x Q) - (Ax B)
Then
of M sewn along M - B, which Theorem.
Em, we mention
the analogous
result
to
10 for spheres. ~mma
5..L..
!:mL.S?£.ierl1f.rt.:L2.n __pres~£'yjtlg l.l
i§2.t2l?i£.to ._ the ideIl~1;l t,y~ Proof. f to g, where
Given a homeomorphism g keeps a bell Bm fixed,
f, first
ambient
as in the proof
isotop
of Theorem
10.
VII
- 93 Nmv
embed Em - 13 in n simplex
barycentre~.
/J.monto the complement glE
The restriction
ID
h:/J.~ /J. by mapping
pi_ecewise linear except h is isotopic
extends
to a continuous-
"
,...
homeomorphism
B
-
/J.~ /J.,that keeps
" at /J.. By l•. lexanderts
to 1 by an isotopy
of the
/J.fixed
and is
(Lemma 16)
Theorem
H:/J.x I ~ /J.x I that keeps
!J.and
,...
~ fixed,
and is piecewis0
linear
except
on !J.x I.
Therefore
the
,...
restriction . t opy lSO
of H to !J. -
0f ~m ~
. movlng
Therefore
which
deformation
the subgroup
retracts
of Em.
However
because
the Lie group topology
induced
from P.
The higher
linear
of linear
onto the orthogonal
both P, L have two components,
the two orientations
linear ambient
the group of piecewise
of Em, and let L denote
honcomorphisms, group.
a piecewise
1
Let P denote
g,emarlt..:.. homeomorphisms
go.t
/J.deter~ines
corresponding
this is a deceptive
to
remark,
on L is not the same as the topology
homotopy
groups
of P arc not known,
but they ure known to differ from those of L.
1
INSTITUT DES HAUTES ETUDES SCIENTIFIQUES
1963
Seminar
(Revised 1966)
on
Combinatorial
by
E.C. ZEEMAN.
Cha.Qter8
Topology
EMBEDDING AND ~~~OTTING
The University o~ Warwick, Coventry.
INSTITUT DES P~UTES ETUDES
1963
Seminar
Chapter
(Revised
SCIENTIFIQUES
19661
on
Combinatorial
by
E.C. ZEEMAN
Topology
8
In this chapter we wish to classi~y of one manifold
in another.
into equivalence The natural
IIClassi~y" means
relation
In Chapter
o~ algebra9
notion
is a horrible
a nice embedding
the virtue
o~ homotopy
something
is done by
computable, interesting.
Geometrically idea, because
gets all mangled
the during
up.
But
theory is that the homotopy
o~ maps are o~ten ~inite
--- ..
theory.
a homotopy
~requently
Listing
and the way to pass into algebraic
is via homotopy
o~ homotopy
quality
5 we saw that ambient
isotopy was the same as isotopy.
topology
to use is ambient
this has the same geometric
as the embeddings.
means
sort
classes and then list the classes.
equivalence
isotopY9 bec~use
embeddings
or ~initely
generated,
classes
and
and so out o? the mess we get There~ore
our classification
- 2 -
technique will be to map the ambient of embeddings
(geometry)
maps (algebra). the algebra
isotopy classes
into the homotopy
classes of
If this map is an isomorphism
classifies
the geometry;
then
if not then we
have a knot theory to play with. Let M be a closed manifold manifold
without boundary
M ~P2t£
Otherwise
A knotted
isotopic
,\'=---4
We say
M c Q are
homotopi c.
we say M knots in Q.
E~a~~l~_iil
circle.
(open or closed).
in Q if any two embeddings ambient
and Q a
The classical curve is homotopic,
Similarly
kind of knotting
by Corollary
example is S1 knots in but not isotopic,
s3.
to a
Sm knots in Sm+2, m ~ 1, and this
is characteristic
2 of Theorem
of codimension
2.
9 in Chapter 4.
It is the latter example that we want to generalise to arbitrary
manifolds,
and in Corollary
1 below we
giVE: ou.L'f'icicl"J.t condi tions for M to unknot proving
an unknotting
an embedding between
theorem it is natural
in
Q.
While
to prove
theorem in the same context, the relation
the two being explained
as follows.
Let
- 3 -
lso(M
C
Q)
=
[My Q] = [M c Q] =
ambient
isotopy classes of embeddings
homotopy
classes of maps (algebra)
homotopy
classes of embeddings
(geometry)
(hybrid).
There are natural maps lso(M c Q)
l-L "._.~---~ [M surjective
23 and 24 below, which give sufficient l-L
to be bijective.
lsoOd Remark. ~.. "='~'
.•. ."
Q].
u
C
Q)
conditions
In other words conditions
for there to be a classification
<--="
",
The main results of this Chapter are
is bijective.
for A and
A Q] injective ----~[M
in Q is the same as saying that l-L
To say that M unknots
Theorems
c
isomorphism
Q].
~~ [l\!I9
We think of a "knot" maJ2.wtst?,9 as an isotopy
class of embeddings. to think of a "knotll class of subsets.
Other authors,
s~t\IViseas an isotopy
Clearly
finer than the setwise9 more knots. are stronger.
notably Fox, prefer
Therefore However
the mapwise
(or homeomorphism)
definition
because potentially
is
it gives
our mapwi se unt:notting theorems our preference
rather than a setwise approach
for a mapwise
is dictated by our aim
to classify knots in terms of homotopy. St§ttt?l1L~toJ~-=m_a.in th.e()rELms. (with or without boundary). 11 is compact.
Let
Ir
9
Qq be manifolds
We shall always
We shall state the theorems
suppose that
in relative
- 4 -
~orm;
the absolute
M =~.
Throughout
~orm can be deduced this chapter
=
d
The letter
let
2m - q.
d stands for double-point
this would be the dimension an a~bitrary
dimension
because
o~ the double points were
map ill~ Q put in general
Ern» E? d..c1 iDE. The ore1!l23.
by putting
( Irw in)
position.
Le t .~ :M :-+ Q.. t?JL a" II§.J2~
1~1 ts a1l.~e.Q.cli..nJL_oL 1~ in~~~TheJ}J~.J,? .s.Ecll~JJ1a ~
lCT~~tPK. ~lLtx~d7 h,9I11otopic. to iL pr:9;Qel?_.e.rnl2.e_ciCiing
J2X'ovi-
{m ~ .•£1.-.3
{J M 'T~~_4.::.5?..?~1J:11~e.s,~~ d
1..5? is .1-d+ll:-cC2p.ne ~t.e
.~---~
Remark,
piecewise
As usual we always linear,
to the contrary. Because
to assume
f'IM is a piecewise the existence
Theorem
simplicial
linear
In other words
In the following
23 is an exception.
approximation,
embedding; linear
theorem
we can still deduce embedding
==)piecewise
everything
it is only
map such that
this is the strongest
hypothesis
to be
draw attention
that f is a continuous
of a piecewise
continuous
everything
we explicitly
However
of' relative
necessary
to~.
unless
assume
homotopic
way round:
linear
is piecewise
thesis. linear.
- 5 -
lL~:qE~t.tinKTheorem
k~_tf'?_Jl:~.1 ..:::... Qp_~"~t}'V_q~
24.
__~u~h_ t~?.!,LIiL~~~ W_<m.er SlTJ?!~.£..?JJ1~
~f ~.E.._~1'~
homot()]i c lfeeJ2ing r:1J'ixed--.1;~e1L.the.x. are ambJenJ....J; ~Q.!2.Q..t.£
Q
keeping
f'ixed, provided { !1L__~Jl==-.2
{&1-),S . (g.+1l-_<;Lonn~£~~q.
i 'i is
(dt2.l-co®ecJ~~d.
\,
In the absolute
case the two theorems
can be combined
to give. Cor_ollarY 1.
LE?t_1L"t>~_c;lo~~9--!llLg .~t!,ithouDo~E~qa~l..:..
'rl.ten.M.unli.n.Sll§ .. 2n Q.L.Qnd Jso(M~~...Q.)~
.b1,
QL
provid_~d
r ill...;·~•..JL ..::•.2 " M. is,l d+1J ... _~,OYl.l2-~~cteq
LQ j)L.Lq._T_~.=qOIln ~~c t 82, . The proof's of' the theorems ingredients balls,
of' the last f'our chapters,
covering
and we give
isotopy,
general
some remarks
some problems, dimensional illustrate
namely
position
of' the unknotting
and engulf'ing,
the proof's at the end of' this chapter.
bef'ore we give them, we deduce make
are a mixture
about
some more
restrictions in Theorem
corollaries,
f'urther developments,
and give counterexamples
suggest
to show that the
are the best possible.
25 how the theorems
But
We also
can be used to
- 6 -
classify
certain
of spheres follow
links
of spheres
in solid tori.
First
in spheres,
the corollaries:
irnnediately from the statement
and are obtained corollary
we put Q equal
C9rol,l2-~rx~2.
to Euclidean
they
of the theorems,
M or Q.
by specialising
and knots
In the first
space. m
Any__closedk-co:q'pect~~§.JJLnifold
M_"-.2...
k ~.m -. 3-2_c.?~~. be ._e~m.E=edded~nE2m-~? .§Ed unknots. in one p_igher .diJ!lf?ns ion .~ In particular
any homeomorphism
by an ambient
isotopy
S1 can be embedded
can be realised
of E2m-k+1,
Sphere
in the same way that
in E2, and any homeomorphism
by an ambient
The next corollary to a sphere,
M ~ M can be realised
isotopy
is obtained
and is a higher
of s1
of E3 (but not of E2).
by putting
dimensional
M equal
analogue
of the
Theorem.
9_0.ro,~la~. m ::;;' ...9...,-=.-
If Qq is l2]1:9.±jl.::.cOn!L~2~Y~_.\~_~Fe
:2. _tJ:1\3n~1fx..."el=e~lJlent-9..L7i:m(~can b_El._r_EU?I'es~nt_~~q o
Pl.: ..all.n~::sJ2.l1ere e~b~(~l.c~te_d iA. Q .nI(j:~ is_one connecte~Uhen
?em C~.L£;J..as §lif.ie. s tlle.ar£l."9iELl)...1.i sotC)]?y ..J?lass~.s
m gf, S S:~l2.rovill.ed 0
By Theorem
.IT.!.
.?:..~J..:..
24 the ambient
same as the homotopy no base point
.htgher
trouble.
classes,
isotopy
classes
are tl~
and since m > 1 there
If m ~ 1 then the isotopy
is
classes
-7are classified words
by the free homotopy
the conjugacy
classes
Co£olJ-ary 4.
The next
~._~_.
A special
case
If Qq is m;::",connegte
,:ttJ-ell..EPYJ!!..oel!lbeddin~Sm
analogue
of ~1(Q).
in other
3 is~
of Corollary
to a disk9
classes9
corollary
c,9llreambi~nt
is obtained
and is in some ways
is_o~t.2.,]J~
by putting
a higher
M equal
dimensional
of Dehn's
Lemma and the Loop Theorem. . m-1 . m 1 L.eJ1., S_ - S Q and suppo se S LS
---.-
Corolla:t:X 5. .•
iIl(;sse,!,ltJ& i~L,,_~"=Jf. ,g 1-l?~U~1!l~.g •.t1J-~gnnected
9
_wh~r_~
m ~_9." ~ 3, t11&JtSm-1 caI}_Re.s.:illlnnY.9-~.lL.IL~oper1Y. m ~1J1bed.9:ed~9-~~k D c ..;(h..-1.f Q.. is op_~_tJ-igh~~_c0I:J.ll~c~~2:. ttLen_~m(9}
cl.§.s§..ifi.e_s~the.~§:nib,l...e-ILLJ_~c>t9J;lL. cl..§sses of
.§..~£l1~§J~f?Jc_fi2_k.e elL~~
The correspondence and elements
rl fUC~' between
of ~ (Q) m
isotopy
classes
is not natural
but is obtained
by choosing
not exceptional
because
of disks
as in Corollary
29
a base disk9 D~ saY9 and associating with any other disk Dm the difference element of ~m(Q) given by Dm u D~. This time the case m = 1 is
on
we can choose
a fixed base point
Sa. We now make
some remarks
about
the two main theorems.
- 8 -
Remark_l:
Hud~on's
improvements.
John Hudson has improved both theorems by weakening
the hypotheses:
manifolds
to be connected he re~uires
to be connected. k-connected
instead of re~uiring
only the maps
We say that the map f:M ~ Q is
if the pair (F, M) is k-connected
F is the mapping
the
cylinder
This is e~uivalent
of f.
to saying that f induces isomorphisms for i < k, and an epimorphism
where
~(M)
?c. (M) ~'](. l
l
(Q)
~ ?Ck(Q). As before
let
Hudson's
=
d
=
2m - ~
t
=
3m - 2~ = triple-point
improvement
double-point
dimension dimension.
in the Embedding
Theorem
23 is to
replace M d-connected
:
} by
Q (d+1)-connected and in the Unknotting
Theorem
{
f (d+2) -connected ~ by
Q (d+2)-connectedJ In both cases the connectivity Q
because
of homotopy
(t+1) -connected.
24 to replace
M (d+1)-connected1
for
L'lL
(d+1)-connected
{
M (t+3)-connected. of M implies the same
t ~ d - 3 and so f induces isomorphisms
groups in the range concerned.
Hudson's proofs are too long to give here, and so
- 9 -
we content stated.
ourselves
with proving
His main idea is combine
given here with those developed smooth case.
Another basic
Two embeddings is a proper
by Haefliger
cQncoJ:dant if there
F:M x I ~ Q x I that agrees with f there is no requirement
that F should be level-preserving In codimension
than isotopy:
for the
idea is to use concordance.
at the top and g at the bottom;
weaker
as
the techniques
f~ g:M ~ Q are called
embedding
is in isotopy.
the theorems
in between~
2 concordance
for example
as there is strictly
the reef knot
-:::J --y J
is concordant because
to a circle by a locally
it bounds
a locally
flat disk in the 4-ball~ but
the reef blot is not isotopic flat isotopy.
However
to a circle by a locally
in codimension
shown that two embeddings concordant
flat concordance~
are (~~9isotopic~
~
3
Hudson has
-
and so the unknotting
10 -
theorem becomes
of the embedding theorem.
a corollary
However we shall prove the
two separately. Remark_
:So
Co¢iiIIlens ioIl 1.
Our results are essentially ~ 3.
in codimension is fundamentally
The situation
different~
group.
2
occurs and
In codimension
is again different, because
Sn knots in En+1 because
results
in codimension
because knotting
is detected by the fundamental the situation
unknotting
1
of orientation.
two embeddings with opposite
orientation are homotopic but not isotopic, and therefore Iso(Sn c En+1) contains at least two elements. We do not know whether the piecewise unsolved
there are more than two elements because
linear Schonflios
for n ~ 3.
equivalent Co&e_cture:
In fact the Schonflies
is still Conjecture
is
to: Isol~~.!i:'~ta§~_t1lJ2.-eJeme.~tf:>..
Again the situation different,
Conjecture
in codimension
zero is quite
and there are many more unsolved problems.
If M is a closed manifold namely the quotient
then Iso(M
C
M) is a group,
of the group of all homeomorphisms
of VI by the component
of the identity.
It is called the
- 11 -
~J1le.9~topx group of 111"
Only three example s of homeotopy
groups are known. n The homeot opy group of 8 is Z2 by The orem 10
Ex_argple_ iJ)
.
of Chapter
4.
determined
by the degree ~ 1.
The isotopy
Iso(8 J3lxampJ" e 1) i~. of 81 X
2.
0 Q
x)
=
Th e f" lrs t two f ac t ors correspon d
(69 PeX)9 where Recently
Pe is rotation of 82 through Browder
from a theorem
group of a 2-manifold group of ~1(M) modulo
example
of Baer that to the
inner automorphisms. unknots
In
in itself9 but
shows that this is not true in general.
~:~m~l~llvl.Browder
3 has shown that 8 x 85 knots
although
known.
He gives a homeomorphismh
the homeotopy
of 83 x 85 onto itself
that is homotopic
but not isotopic
Choose an element
U E
~3(80(6],
in
group of 83 x 85 is not yet
itself9
representative
3.
is isomorphic
each of these three cases the manifold the following
has proved
1 n for 8 x 8 9 n ~
the same result It follows
automorphism
group
2 of 81 and 8 9 and the third factor 2 1 1 x 8 ~ 8 x 82 by the homeomorphism h:8
6 about the poles.
the homeotopy
words
reversals
Z2 is generated
(unpublished)
0 ther
n n [8 c 8 ] ~ Z2"
=
Z 2 x Z 2 x Z 2'
to orientation
angle
n c 8 )
In
is
Gluck has shown that the homeotopy
lS
given by h(69
n
class of a homeomorphism
to 1.
We sketch the proof.
~ Z9 choose a smooth
f E u9 and use f to twist the fibres of the
- 11a -
product
bundle
83
x
~ibre-homeomorphism
85 ~ 83•
i~
F5
degree 1~ then
~3(F5)~ ~8(85)~
h is not topologically is not piecewise non-concordance
to h.
to 1.
(x~ 0)
I~ h were concordant
P1(Th)~
that a E Z classifies
and so if a
o~ Th is non-zero. S
3
X
J
hi
To prove the
=
torus, obtained (hx~ 1) for all
s5
x
s1.
the Pontrjagin
0 then the rational
But the rational
But lot
class
Pontrjagin
Pontrjagin
~ 1 SJ X 8 is zero, and is a topological
so we have a contradiction.
then shows that
to 1 then Th would be
. 11y h omeomorp h"lC t 0 T 1 -- 83 x t opo 1oglca transpires
linear
to 1~ and therefore
the mapping
from S3 x 85 x I by identifying x E S3 x S5.
Browder
isotopic
let Th denote
To place outsclves
choose a piecewise
concordant
linearly
o~
Z24~ and so a is killed
category
hi concordant
to 1~
o~ 85 to itself
~3(80(6)) ~ ~3(F5).
in the piecewise-linear
We claim
o~ 24 then h is ~ibre-homotopic
denot8s the space of maps
by the homomorphism
homeomorphism
is a smooth
h o~ 83 x 85 onto itsel~.
that i~ a is a multiple because
The result
class
class of
invariant~
and
- 12 Examj2le lv). In smooth theory can knot in itsel~. homeotopy
group o~
6
group of 8 preserving by using
For example
S6
In piecewise
by the Poincare
group D
228 corresponds
the homeomorphisms
there are no exotic
the piecewise
o~ S
linear
spheres
Conjecture9
• 28
The orientation
to exotic 7-spheres9
6 to glue two 7-balls
theorY9
on the other hand 9
(at any rate in dimension which we shall prove
~ 5)
in the
chapter.
Remark
k.
Higher
h0l!lQ..ts2J1Y. grollPlL_'J\i OJ c
Our so called classi~ication Q has only touched generally
the sur~ace
Ql.
of embeddings
o~ the problem.
of M in More
we can study the space (M c Q) o~ all embeddings
o~ M in Q9 regarded (RS
linear
is 229 but the smooth homeotopy
is the dihedral
subgroup
together.
next
it is well known that a mani~old
in Chapter
particular 'J\.(Mc Q). l
something
either as a piecewise
2) or as a semi-simplicial
linear space complex.
we can study the higher homotopy
groups
SO far in Theorem 24 we have only said about the zero homotopy
~O(M c Q)
=
group
lso(M c Q).
For example we might generalise
Theorem
9 the unknotting
o~ spheres by: Conjecture.
In
(£~ c 'J\i
S~.2 =
09
l2.rovidedi + m :::;; ~.
- 13 -
Remark,..,.j.. An 9PsJ;ruction th..§..ory. In the critical dimension, not sufficiently developed
connected
when the map is just
for unknotting,
Hudson has
an obstruction
for homotopic maps to be isotopic, a quotient of in/the first non-vanishing homology
with the obstruction
group of the map (with certain
coefficients).
would like to develop a more general and fit it into an exact sequence, the terms 1t.(M l
25
to Theorem
C
Q) ~ 1t.(M,
Q],
l
One
obstruction
perhaps
i ~ O.
theory,
including
In Corollary
below we give a non-trivial
example
2
that
looks as though it ought to fit into an exact sequence. We now discuss dimensional
counterexamples
restrictions
to show that the
in the two main theorems are
the best possible.
In each of the six cases we relax
a single hypothesis
by one dimension,
theorem
then becomes
f
EmbCddi~-g ~odimenSion
theorem
false.
2
i
:4 \--------~5T-,-
Unknottin~
Codimension
theorem
----------------
and show that the
2
2~ only
\~
only
~--"-"-l
I d-connected - .----~--!-.---------------
I (d+1)-connected
1----
I M only
I I 1
\-~-
. Q only
d-connected
(d+1)-connected.
..---, .._--_._.._-----------!...- .._--_._-~----_..... _._-~,---!
- 14 -
This is the only one o? the six cases where the counterexample than proved. 4-nani?old
is conjectured
Let D2 be a disk9 and ~+ a contractible
with non-simply ?:D2 ~
connected boundary.
embeds D
onto an essential
in Q4.
Such a map exists because
4 in Q.
By the Conjecture
79
Let
Q4 ·2
be a map such that?
in Chapter
rather
curve
the curve is inessential
6 a?ter Theorem 21
in Example
the curve does not bound a non-singular
disk in Q4 and so ? cannot be homotopic kee~ing
the boundary
the connectivity
?ixed.
to an embedding that n2 and Q4 satis?y
Notice
conditions because
they are both
contractible. p01.11rtereXam..12~e~ 2.. Let M
=
Let m be a power of' 2 and m
pm9 real projective
space9 and let Q
Then d = 1 and P just f'ails to be i-connected. Q is 2-coruLected and the codimension caru10t be embedded homotopic
is
;<:
=
;<:
4.
E2m-1. Meanwhile
3. Since pm
. 2m-1 m 2m-1 ln E , no map P ~ E can be
to an embedding.
9o~p~~~exa~~e"~.
(Irwin)
Let m
;<:
3, and let ?:Sm ~ S2m
be a map with exactly one double point9 where the two sheets o~ fSm cross transversally. We shall show that such a map exists in a moment.
I?
ill
were allowed to
- 15 -
equal 1 then the figure 8 would give a correct picture. Let Q2m be a regular neighbourhood
of f'Sm in S2m.
claim that f':Sm ~ Q2rn cannot be homotopic
=
Notice that d
a and Sm is a-connected,
to be i-connected.
In f'act ~1(Q)
=
We
to an embedding. but Q just f'ails
Z, generated by a
loop starting from the double point along one sheet and back along the other.
Notice also that the codimension
is ~ 3.
m m Wrl·t e Sm = D a u Sm-1 x I U D l'
J?ro9.:L.Jhat f exists:
m
m
Embed the two disks transversally as DO x 0 and 0 x D1 in a m m 2m 2m little ball DO x D1 c S . Let B be the complementary ball, and extend the embedding
± ~
disks Sm-1 x
of the boundaries
i32m to a map Sm-1
x
I ~ B2m
of the
Now use
Theorem 23 to homotop this map into a proper embedding, keeping
the boundary
f'ixed. The result gives what we want.
FRoof _tha~ fi~mbedding. f'was homotopic
Suppose on the contrary that
to an embedding
g:Sffi~ Q2m.
Let p2m
denote the universal
cover of Q, which consists of a m b e d together coun tab Ie n~~ber 0f· coples OL~ Sm x D , p 1urn
in sequence.
We can lif't f',g to a countable nmnber of ,.,ffi
maps fi, gi:o embedding, once.
By construction each f'i is an and, for each i,f.Sm cuts f'. ism transversally ~
Meanwhile
P, i E Z.
1
1+
g.Sffiis disjoint from g. ism because 1
1+
g
- 16 -
was an embedding.
Here we have a contradtction,
because
the intersection of the f's is homological, and must algebraically be/the same as the g's because f. !:>! g., each i. 1
More precisely,
1
m of H (8 )
if ~ is a generator
III
and '
is Poincare Duality (where Hc stands for compact cohomology) 2m then in H (P, p) we have the contradiction c
o =
u
Dg.~ 1
Counfer.§xample
Dg.1+1~
.l±..
=
Df.~ 1 u Df.1+ 1~
f o.
8m knots_ i1l.sm~2..:.-lIL ;:::. 1. 1
m-1
.
8 x S knots 1n 8 Q_o_ujltel:exlLmJ2.Le-2,. (Hudson) 1 Notice that d = 0 and 8 x Sm-1 just fails to be ...."
~~~~
2m
m ;:::
.•.•••.:.o.__ ~'~
3.
(d+1)-connected. Given an embedding we shall define a knotting that it is an invariant
knotting
numbers
Let T
=
~.n()t
isotopy
two embeddings
class of
fO' f1 with
0, 1 respectively.
f(S1 x Sm-1), the embedded
a E 81, let Sa ;[..emmp 5 8..
number kef) E Z2' and prove
of the ambient
f. We shall then describe
f:81xSm-1
=
torus.
Given
f(a x Sm-1), an etlbedded (m-1)-sphere. 11 A·ln 82m• spanning 8a ~here_,js.an m":;:pa
1fl~:tin .£J:~at1l.
-+
82m
- 17 -
proqf.
Since m ~
3,
Sa is unknotted
can be spanned by an m-ball, ambient
isotop D, keeping
D say.
D =
S
in s2m, and so
By Theorem
15
o
fixed, until D is in
a
o
general position
with respect
T in a finite number
of points,
one by one as follows. Choose YES,
a
to T.
D meets
which we can remove
Let x be one of these points.
choose an arc acT
arc ~ in D joining xy.
Therefore
joining xy, and an
We can choose the arcs so as o
to avoid the other points S
a
only in y.
of T n D and so as to meet
Let ~2 be a 2-disk
in S2m spanning
a u ~, and not meeting T U D again (this is possible
- 18 -
by general position
since m ~
3).
Triangulate
and take a second derived neighbourhood S2m, which
is a ball
B
2m
of
since ~2 is collapsible.
everything
A2 In . Consider
the set
2m
x= Now X consists common face B intersection
2m
B
n
together
a
we have removed moved back.
n (T
in B
point x.
2m
with one self
U D)
()m-2
2m in B •
X
-sphere
- 2m
B
faces glued
n
Sa'
the torus meanwhile,
point x;
but We have
and so we must now move
U
71)'
are the m-balls
Y
=
2m B n T, B2m n D.
and the (m-i)-ball
-2m
B
n D.
Since~,
with the same boundary,
Corollary fixed.
Similarly
YT U YD
where Y , Y are the cones on the (m-i)-sphere T D
B2m
it
We can write
where 7~, ~
2m
Let
If we replace X by Y, behold
the intersection
X = ~
in B
along the
Let
of the three complementary
along the common
Y be a cone on
glued together
S , and embedded
at the interior
consists
D).
U
of three m-balls
X = B2m which
n (T
1 we can ambient
B2m
n T
YT are two m-balls
and since m ~ 3, by Theorem
isotop YT onto XT keeping
This moves the torus back
into position.
9
- 19 -
Meanwhile
the isotony
replacing
D by
carries Y
D to Y' D say.
(D - XD)
U Y
has the e~fect o~ reducing
Then
n
the intersections
of
o
T n D by one.
After a finite number
A as required.
This completes
and we return
to the construction
o~ steps we obtain
the proof
of Lemma 589
of the knotting
nunmer
k(f). Choose
1 three points a9 b~ c E S .
choose three m-balls respectivelY9 choose
number
and not meeting
the balls
another~
A9 B9 C spanning
in general
By the lemma
Sa~ Sb~ Sc
the torus again.
position
relative
and so each pair cuts transverBally
of points.
Let AB denote
of A and B9 modulo
2.
=
k(f)
the number
We can to one
in a ~inite of intersections
Define AB + BC + CA.
We have to sho~ that k is independent
of the choices
made.
First we show k is independent
of A.
denote
the interval
denote
the immersed m-sphere SBC
=
of S1 not containing
B U f([bC]
Let [bc]
a9 and let SBC
x Sm-1) U C.
' I l'lnk'lng number mod 2 o~ T hen the h omo I oglea ~ Sm-1 and a
m , S2m, , b SBC 1n 1S glven y L(Sa~ SBC)
=
AB + AC9
- 20 -
which
is independent
because
of A.
isotopy of S
a
L(Sa' SBC).
B, C, b, c. an ambient
carries with it the whole
different
k is independent
because
numbers.
of an embedded
A, B, C disjoint
Construct follows.
m s1 x D •
fO' f with 1
one given by the
Then we can draw
as in the picture below.
the embedding
isotopy
the embedding
fo:S1 x Sm-1 ~ S2m to be the obvious boundary
Clearly k is
of A, B, C.
embeddings
Define
of
any ambient
construction
we have to produce
knotting
[be]), then the
k is well-defined.
isotopy invariant,
Finally
of a1
does not alter the linking
Similarly
Therefore
Therefore
Also k is independent
if we move a (without meeting
resulting number
x Sm-1) •
A does not meet f ([] bc
because
1
f ~S 1
Therefore
x Sm-1 ~ S2m as
- 21 -
- 22 -
First
link two (m-i)-spheres
with linking number
1.
into each hemisphere.
in the equator
Then collar each o~ these Finally
the two collars by a cylinder
connect
only moment
the tops o~
I x Sm-1 in the tropic
o~ Cancerp and connect the bottoms by similar cylinder
S2m-1
in the tropic
o~ the two collars o~ Capricorn.
o~ doubt OCGurs as to whether
The
two linking
spheres can be coru~ected by a cylinder, but this doubt is resolved by glancing under
the identi~ication
at the image o~ the diagonal map
u Sm-i x Sm-1 x I ~ Sm-1*Qm-1
To compute k(~1)p in the equator, bottom
x I
_ _ S2m-1
•
choose Sa to be one o~ the (m-1)-spheres
and choose Sb' Sc to be the top and
of the other collar.
Form B by joining Sb to
the north pole,
and C by joining Sc to the south pole.
Then L(Sa' SBC)
=
S
a
with an m-ball
(m-1)-sphere Meanwhile
Be
we can compute it by spanning
in the equator,
that meets the other
and hence also SBC' in exactly
=
0 because
k(~1) This completes gemapk
1, because
=
B, C are disjoint.
There~ore
L(Sa' SBC) + BC = 1.
the proof of Counterexample
o~ knott~d
5.
tor~.
Hudson has shown that i~ the codimension then his knotting
one point.
number
described
is even,
above is in fact
- 23 sufficient
to classify the knots, but if the codimension
is odd then an anologous Z is required.
knotting number
in the integers
More generally he has proved that
Iso(SP x Sm-p c S2m-p+1)
= {_Z2'
codimension
Z, codimension provided m ~ 3 and 1 ~ P < m - p. dimension
for knotting
higher dimensions, The case p
=
odd
This is the critical
tori, because
by Corollary
even
they unknot in all
1.
0 turns out to be exceptional.
Here
the torus Sa x 8m consists of two spheres, and the ambient isotopy classes of links of spheres in the critical dimension
are classified by their linking number, as
we shall show below:
=
Iso(80 x Sm c 82m+1) Z, m ~ 2. m . 1 2m Counter_~xample 6. ~lS.not~J1l __8 _x 8 .. JLm ~ 2. Notice that d = - 1 and 81 x 82m just fails to be (d+2)-connected. froo£' qf the kn2tting. We shall give two embeddings m 1 2m such that one bounds a disk and the other 8 C 8 x 8 doesn't.
Therefore
they cannot be ambient isotopic,
but they must be homotopic because homotopic.
any two maps are
It is trivial to choose the first one.
For the other choose the embedding after Theorem
19 in Chapter 7.
described
in Example 3
It consists of two little
- 24 -
linked m-spheres
1
connected by a pipe round the S •
We showed in Chapter 7 that this embedding
cannot
bound a disk. RemaI"k. -~-~
We shall shortly furnish a large class of
alternative
counterexamples
by giving conditions
, the SOIl'd torus Sr x E q-r . Sm to knot ln are given in terms of homotopy Corollary
for
The conditions
groups of spheres, in
2 to Theorem 25 below.
The simplest example
is that S4 knots in S3 x E4. We shall show there are an infinite number of knots, although
~4(83 ) =
Z2.
Notice that here d = 1 and 83 x E4 just fails to be (d+2)-connected.
- 25 This completes
the six counterexamples
designed to show the dimensional
restrictions
that were in
23 and 24 were the best possible. next We want/to classify the links of two disjoint
Theorems
spheres Smy sP in a larger sphere sqy up to ambient isotoDY.
The classical
situation of two curves linking
in 83 is somewhat deceptive because
knotting
with linkingy but it does illustrate
is confused
the three types
of linking that can occur.
L1J .
Homolog~callJJ}kin~.
Each curve is non-homologous of the other.
to zero in the complement
By duality this is a symmetrical
situation.
~
/
Here Ay Bare
homoligically
in the complement because,
of B.
unlil~edy
but A is essential
This situation
can be unsy~netricy
as we have drawn them, B is inessential
in the
- 26 -
complement
.01..
of A .
(}eC?Eletris~ .ljpking.
Two curves are geometrically ambient
isotoped
geometrically unlinked.
unlinked
with opposite
linked
if they can be
hemispheres.
We illustrate
curves that are homotopically
- 27 -
Summarising we have: (homologically\ \
)<
linked
In higher dimensions
(hOmotoPicallY) -
=)
-
linked
)1~eometricallY)
/ ~.\
linked. ~ 3~ so
we shall stick to codimension
as to separate knotting and linking and be able to concentrate on the latter.
Therefore
we shall assume
so that each of Sm~ sP is unknotted
in sq.
There are three cases. Then Sm~ SPare unlinked
by Corollary
1 to Theorem
geometrically
24.
In this case homological
linking
can occur~ and this is the only case in which it can occur. We shall show in Corollary
1 to Theorem
classified by the linking number9 more precise
25 that the link is
which is an integer.
To be
there are two linking numbers which differ only
by the sign (_)mp+1.
However we shall not bother
the 'homology linking nllinbers~because of the more general homotopy
linking numbers9
case.
2m + p ~ 2q -
linking can occur in
We shall define the homotopy
and show in Theorem
the link~ provided
they are special cases
linking numbers. Homotopy
both this and the previous
to define
4.
25 that one of them classifies
- 28 -
~
hom9topy
0t
linking numbers
Since each sphere is unknotted
_c.~~
S~Sp
we have
sP ~ Sq-p-1 x EP+1 Sm ~ sq-m-1 x Em+1• We assume that all three spheres Sm, sP, sq are oriented, are induced on Sq-p-1, sq-m-1 (we shall
and so orientations
examine these induced orientations Therefore
the linl: determines 0, E
f3 Notice
E
more carefully
in a moment).
homot~o£:¥_JiHkin~".nUIIlbers
1C (sq-p-i) m 1C (sq-m-1). p
that both these are in the (m+p-q+1)-stem,
and we
shall show in Lemma 62 that they have a common stable suspension (to within sign). homotopy
groups.
We call
f3 st~qle if they lie in stable
0"
Recall that 1C.(Sj) is stable if i ~ 2j - 2. 1
Therefore 0,
is stable if m + 2p ~ 2q
f3 is stable
• .t:>
1-1.
2m + p ~ 2q
4 4.
Since m ~ p, we can have a. unstable while f3 is stable, and this will be a particularly
S3, S4 c
interesting
situation;
for example
Let L; denote the suspension homomorphism,
and
L;p-m the composite suspension 1C (sQ-P-1) -+1C 1 (sq-p) -+ .•. -+7\.(sq-m-i) . m m+ p When there is no confusion we shall abbreviate L;p-m to L;.
- 29 -
:t:f13_ iS2table~~en
a,
classitt~JL~ille=link.__ l_1f_~~t]1~_:r~?rd~
:!..:.:~
m
J so (8 • 8! c 8 ~ ~ ~ 2:1Cm.l9...~ A:L~()~.ii_:::-L:-lm+p+q+prn~~ Before we prove of examples
the theorem we deduce a corollar.y and a couple
and prove
If
Q£r9].la~J...:.
two lemmas.
=_
m .+ p
9 -:. 1..2.. the..n
ISOC§..m..J_8P~~sLqJ .. ~ 7Cm(8~1_~ Hm{8m)~~_~, and the )~0!10logi$a.)]..inldnK.number .9lClS~ ifi~ Two 50-spheres dimensions9
can be linked
but in 97~ 96 they become
linked again in 95~ 941
'0'
???
'00'
the linl~" in 101~ 100~ 999 98
unlinked9 52.
and then can be
The explanation
is
that the words link/unlink
~ nonzero/zero
of certain
stable homotopy
groups9
correspond
to the vanishing
and the unlinking
is 97~ 96
of the stable 49 5 stems, There are exactly two links of 89, 810
One .is geometrically
unlinked,
linked as in the second diagram
J 0,
i3
=
a, E
7C
816,
and the other is half-homotopically above 9 because
(85) ~ Z20 9 6 09 13 E 7C10(8 ) = 0,
a,
C
- 30 -
J!.2mm~_59~ •.
1~Sm
>
i..?_~unkn-?tt.ed in ~Cl,-!l1.~nan oriEglt_alio=n
S.:
k~e12:i-F$~~~lJJSeriis tS_9.to12 ..tS. l?£~s2FBnRJ"l.£,mE;0I!L-srJ2..h,J.§El_p! !-o_the iq,e,nJi ty keeping. Sr: f_i_xe_cl. proo(
by induction
the induction
on Cl, keeping
beginning
trivially
the codimension
with m = - 1.
h:SCl ~ SClbe the given homeomorphism. K1 L of SCl1 Sm and a vertex
x E L.
Choose Choose
fixed,
Let
triangulation
subdivisions
such
that h:K1 ~ K is simplicial. Let BCl9 Bm be the closed stars 2 of x in K19 L1• Then h maps BCJ.9Bm linearly into st(X9 K) 9 st(x. T.) and so by pseudo we can ambient kBCl
radial projection
isotop h to k? keeping
BCJ.. Now Bm is unknotted
=
induction
Sm fixed,
preserving,
we can isotop klBCJ.to the identity
By Alexander's the isotopy
trick
89 Chapter
3)
such that
in BCl, since Sm is locally
and klBCl is orientation
unknotted9
(see Lemma
(c.f. the proof
to each of BCJ.1SCl -
and so by keeping
Bm fixed.
of Lemma 16) we can extend
BCl keeping
Sm fixed9
and so
isotop k to the identity. For the next lemma we want to compare with knots S
Cl
=
rl-p-1
S~ ~
right-hand denote
in solid tori. 1"\
*S~.
Let g:S
P
links in spheres
Write rl
~ S~ denote
the embedding
onto the
end of the join, and let e:SCl-p-1 x El?+1 ~ SCl
the embedding
onto the complement.
f:Sffi~ SCl-p-1 x El?+1 determines
a link
Then any embedding
- 31 -
~P-in
o~~~~:~~":*~
Let ~ denote ambient
Proof.=~~.~
...
~:'c_s*c:J~
E~~~)_ : __Isol~9
x
If f ~ f', then we
isotopic.
-=-
can choose the ambient Theorem
12 in Chapter
Therefore
the ambient
keeping
sP sP
support 9 by
59 and it can be extended
to
sq.
rp(f) ~ rp(f'). Therefore
rp induces a map9 rp saY90fisotopy classes for, given rp(f) ~ cp(f'), then the end of
rp is injective~
keeping
isotopy to have compact
isotopy gives an orientation
preserving
fixed9 which by Lemma 59 is isotopic fixed.
The restriction
of sP gives f ~ f'. lilLk ~9 ambient
homeomorphism
to the identity
of this to the complement
Finally ~ is surjective~
for given a
of sP onto g (using
isotop the embedding
p :::;; q - 3), and hence k ~ (ef 9 g) for some f.
f£20f' ():f 'l'he_orem?5 ... Sm unknots in S q-p-1 x E p+1 because d + 2
=
(2m - q) + 2
~ q - p - 2 by the stability Therefore
Iso (Sm 9 Sp
C
sq)
~
Iso (m S
c
So-u-1 D+1) by Lemma 6a ~ ~ x E~
~ ~m (sq-p-1) by There remains
to show that ~
~ust be more explicit
=
of ~.
Corollary
2
to Theorem ~
(_)m+p+q+mPZa9 but first we
about our orientation
Suppose we are given orientations
conventions.
of Sm, SP,
sq.
We
- 32 -
define the induced orientation
on Sq-p-1 in
n-"O-1 "0+1 as follows. Sq - Sp '" S~ ~ x E~
=
choose a local coordinate linear subspace. orientation
At a point of sP
system in which sP appears as a
Choose axes 11
of sq so that 1,
."1
q to give the
P gives the orientation
".1
"0
of S~.
Then p +11 ..•, q determine an orientation in a transverse disk nq-p, and hence induce the require orientation
on
nq-p =
Sq-p-1 c sq-p-1 x EP+1•
topological
invariant definition,
To see that this is a observe that it can be
expressed homologically:
if x E H (SQ)1 Y ~ HP(SP) are the q 1 P then oy E H + (Sq1 SP) and the cap product
given orientations,
x n 6y E Hq-p-1(sq._ gives the induced orientation.
SP)
We use the given orientation
on Sm and the induced orientation a E ~ (sq-p-1). linking nillfiber m
on sq-p-1 to define the
Similarly
define ~.
Sus~e~~eqltnk?
Suppose that we are given a link Sm, sP c Sq1 with linking numbers a1~. Let 2Sm1 sP c 2Sq denote the link formed by suspending
Sm and SQ1 while keeping sP the same. Orient the suspension 2Sq by choosing axes 0, 11 •••, q at a
point of sq so that 0 points
towards the north pole and
1, •.. , q gives the orientation
of sq.
Let a*, ~* denote the
linking numbers of the suspended link. LeIl)ma~2.h P~.oo!~.
First look at
~*.
At a point of Sm we can choose
- 33 -
axe s so that 0, 1, •.• , or~ents ~sq.
••. , q
Therefore m+1, •.. , q orients the same transverse
disk as in the unsuspended Therefore
link.
Meanwhile
sP is unchanged.
~ is unchanged.
Now look at a~. At a point of sP we can choose axes , q orients zsq.
so that 1, ... , P orients sP and 0,1,
...
By the prescribed
the axes so that
rule we must reorder
1, ••• , p come first.
(-)p
Therefore this introduces
into the orientation
0, P + 1, •.• , q. the suspension link.
determined
induced on the transverse
For the transverse
of the transverse
In the unsuspended by homotoping
a factor of disk by
disk we can choose ~Dq-P,
disk Dq-P in the unsuspended
link the class a E ~m ( S Q-p-1)
is
the embedding. Sm c sq - sP into a map
f·.Sm ~ D"q-P say.
In th e suspen d e d 1"Ink we can h omo t op ZSm c zsq - sP into the suspension ~f:ZSm -+ znq-p, which -r
the class ~a E ~m+1(sq-P).
determines
Adding in the factor
(-)p we have
The linlc consists
Proof. and g:S
m
-+
of disjoint
embeddings
SCl, where a is the class of f:Sm m 13 is the class of g:S
-+
sq
gSm
-+
sq
fSm.
f:Sm
-+
sq
- 34 "1 .t e SCl = Srn*SCl-rn-1 , and 1·et J:S m i'rl the left hand end of the join. and assume
denotes
f'rom now on that g = j. sq _ Sm Em+1 x sq-m-1. then ef:Sm
projection,
Sm x SCl-m-1 denote of the join. ef.
Ambient
=
complement
S Clbe the inclusion
-?
-?
-?
isotop g onto j,
Then f'Sm lies in the If e:Em+1 x sq-m-1 -? sq-m-1
SCl-m-1 represents
the torus half-way
Let r(ef):Sm
between
keeping
Therefore
gSm fixed.
we can ambient
ConseCluently assume
We have reached
a situation
Let
the graph
of
a, and since ~ is stable
a classif'ies the link, by the part of Theorem already.
a.
the two ends
Sm x SCl-m-1 denote
Then both f and reef) represent
proved
of'
25 that we have
isotop f onto reef)
f = reef) from now on.
where both
spheres are Gn~
embedded
in the complement
of the right hand!
More precisely f:Sm -? Sm x Eq-m is given by fx g:Sm
-?
Sm x ECl-m is given by gx
Let T be the antipodal h of' Sm x Eq-m,
= =
map of SCl-m-1.
isotopic
hf
=
(x, efx) (x, 0).
There
to the identity
Scl_ SCl-m-1 = Sm x ECl-m.
is a homeomorphism
such that
g j
hg = (1 x T)f.
If we were
content
_ -fI
" fx
to have a topological -
____ -: •
homeomorphism, describe: X
x
a-ill
then h would be easy to
for each x
E
E ....by the vector
Sm merely - fi.
translate
However
such
I/'\ -::::+=-
..•..
\
.
(C'.
I
---L
\<-L~
i
_
- 35 -
an h is not in general piecewise to construct
h is as follows.
(1 x T)f keeping are homotopic of a1
~
fS
m
First ambient
fixed, which
m
(1 x T)fS
ambient
isotopy
Theorem
12, it can be extended
~ = ~=
=
compact
Since the support, by
to sq, and so the link is
In the new position [ehg]
they
isotop f to g
fixed1 for similar reasons.
can be chosen to have
But the antipodal
because
and the stability
Then ambient
keeping
isotop g to
is possible m of fS 1
in the complement
ensures unknotting.
unchanged.
linear and so the best way
[e(1 x T)f]
hfSm,
we see that, removing
=
[Tef]
=
T[ef]
map T of S q-m-1 has degree
=
Ta.
(- )q-m .
Therefore
(_)q-ma•
Com1Ll~tion of the_proof
?5.
of ~e?rem
We are given Sm1 sp c sq with linking numbers the smaller sphere p - m times, with linking numbers
~*
= =
~.,'.'
=
a... .•.
a~:;9' ~ 'I'... say.
Then
(_)p(p-m)zp-ma ~
(_) (qp+-m)-pQ,~i: ~ is stable.
Therefore
~ = (_)q-m+p(p-m)zp-ma
= This completes
Suspend
to give a link Zp-mSm1
by Lemma 61 , and
by Lemma 62, because
as~'
(_)p+m+q+pmZp-ma.
the proof of the theorem.
sP c zp-msq
- 36 -
I~ neither
o~ a, ~ are stable then we can show
they have a common stable suspension
to within
the sign
(_)m+p+q+pm by suspending both sides of the link sufficiently
and applying Lemmas 61 and 62.
Consider
Sm c Sr x Eq-r•
embeddings
us plot against
coordinates
different
types o~ knotting
attention
to q ~ m +
Keeping m fixed, let
q and r the three regions can occur.
in which
As usual we restrict
3.
r
! rn+3
unstable range
;>
<:
metastable range
> ~------stable
range
- 37 -
C'f
1. UllJ.a1,ottJ-J1..E.. Region which
is the condition
Theorem
24 Corollary
are classified
by g +r ~ 2m + 3, in Sr x Eq-r by
is bounded
by Sm to unknot 1. r
Therefore
the classes
by ~ (8 ). m
C0
Region 2. Homotop¥-~ ta.Q).enlcl1,?ttill.E.. q + r < because ~m(Sr)
2m + 3 and r ~ m/2 + 1. the condition to be stable.
homotopy-stable
r ~ m/2 + 1 is exactly In the next corollary
Region
the condition
for
we classifY
range.
in the following
Sm c s2m x E1 described
there are no analogous
that region
Therefore
corollary
situation
is the complement.
6 above.
example
in the unstable
~
like the knotted
It is interesting
in spheresy
by
knots.
Here we can tie knots
smooth
is bounded
I'Vecall these homotopy-stable
3. [~~2t02y-un~tabl§kr~t;in£.
in Counter
of embeddings
(g) lies entirely
all the knots
lie in the unstable
results
in the smooth
is complicated
range. category;
by the presence
and it is difficult
to disentangle
that we classify So far the
of sphere knots them from the
situation. Ham 0t
Iso(Sm
c Sr x Eq-r)
e 7C
d~§J.N.lill!.
1'"
(Sm-r-1 ) ..-
~""
~
(Sr).
..-7 m
q-r-1
There:t~£.. s~ u~n~o_tf3 gL§r __x-.m.~_:~tt_.1?-nd ..Q!lIY iL the susJ?ensJ_C?ll
L~.§..a_T!l0E2m0.rl?.h~ sIT!.~
- 38 -
Pro.£.f' •
Notice
that conditions
for region
QL) imply
<1- r - 1 ~ (2m - r + 2) - r - 1 9 be caus e 2m - r ~ <1- 2 < m9
because
m ~ 2r -2.
Also it is easy to show r ~ 2.
Therefore
Iso(Sm c Sr x E<1-r) ~ Iso(Sm9
S<1-r-1 c S<1) by Lemma
60
~ Iso(S<1-r-19 sm c S<1) putting the smaller sphere first. ~ ~ Define
by Theorem Finally
e
the isomorphism 259
~e
= ~
we can write
<1-r-
1 (s<1-m-1) 9 by Theorem s t abl e.
= (_)(<1-r-1)+m+<1+(<1-r-<1)ma• Then~
and so the diagram ~
=
A~
[Sm c Sr x E<1-r] A injective
in Sr x E<1-r <
Sm unknots
is commutative.
where
Iso(Sm c Sr x E<1-r) ~ 7 surjective Therefore
25 since ~
»>
'T-
(Sr).
m
~ ~ injective
~;.c~
~
injective
<----.~ ~ monomorphism.
This completes
the proof
of the Corollary.
isotopy
classes
homotopy
classes
sphere c torus of embeddings epi
Z ---7
x E10 I
of maps
Z2
z ---- --) 0 2 --+ ---------------- -------.---------------.-I i
i
---- ...-.------
- 39 In the last example there are three knots all homotopically trivial, and no realisation homotopy
as an embedding
of the other
class.
It would be interesting
(i)
th e
to extend the results
. (---3 " reglon ",,--j.
(ii) knots of Sm in arbitrary neighbourhoods
q-dimensional
regular
of Sr, rather than just the product neighbourhood.
(iii) knots of Sm in an (r-1)-connected is not big enough for unknotting. of an obstruction
manifold,
where r
This would be the beginning
theory.
We now return to the task of'proving which occupies
to:
Theorems
23 and 24,
the rest of this chapter.
We are given a continuous map f:M ~ Q which we have to homotop into a piecewise
linear embedding
in the interior,
and we are
given that m :;-;; q - 3 M is d-connected Q is (d+1)-connected, where d
=
2m - q.
The first step is to make l'piecewise approximation. follows.
Next homotopy
linear by simplicial
l' into the interior
Let Q1 be a regular neighbourhood
of Q as
of fM in Q.
Since
- 40 -
M is compact shrinking
so is Q1' and therefore
this collar
Q1 has a collar.
to half its length
(the inner
o
homotop
By half)
0
Q1 into Q1'
This homotopy
carries
fM into Q19
0
c Q.
o
Now homotop Corollary
f into general
1 of Chapter
position
6. Therefore
in Q by Theorem
singular
18
set s(r) or r
will have dimension dim s(r) ~ d, the double is contained
point
dimension.
in the following
The next main
step of the proof
lemma. o
I;~rruna_63. TE~..rEL_~~..Lst co11aJ?_sibl$=subt3J2.§tc~s _C2 D of.M,4 Q re~.l?estJVE?)~1l.c:q.J;haL~i.fJ,.c
c. =:..- r-J2..
D f
c
>
- 41 -
The main idea of the proof' is to use the engulfing
Proof. Theorem
7
20 or Chapter
several
Since M is d-connected, a collapsible
times in an inductive
we can start by engulfing
process. S(f) in
C1d+1
subspace
S(f) c C1 c M. Oi' course when C1 is mapped by f into Q it no longer remains collapsible 9 because bits of S(f) get glued together to i'orm non-bounding
cycles.
we can engulf fCi
Nevertheless,
since Q is (d+1)-connected,
in a collapsible c Q.
1 yet9 because
although
it may contain
other stuff as well.
as to minimise
the dimension
it.
More precisely
the ith induction
D~+29
o
fC1 c D We are not finished
subspace
f
-1
D1 contains
C1' The idea is to move D1 so
of this other stuff and then engulf
we shall define an induction statement
on i9 where
is as follows:
There exist three collapsible
subspaces
C. in M and l
o
D. :) E. in Q9 such that l
l
S(1') c C.
( 1)
l
-1
E. c C.
(2)
f
(3)
dim (D. - E.)
l
l
l
The induction
l
begins
above, and choosing ends at i S (l') c C. l
= =
~ d - i +3•
at i
l
19 by constructing
E1 to be a point of £'e1•
d +49 because f-1 D.
=
then D.l
as required.
=
C19
D1 as
The induction
E.l and so we have
- 42 -
There remains to prove the inductive aSSllffie the ith inductive
statement
Then fC. U E. c D. by (2).
=
is true, where i ~ 1.
Let
111
F
step, and so
D. - (fC. U 111
E.).
Then dim F ~ d - i + 3 by (2).
Using Theorem
15 of Chapter 6 ambient isotop D. in 1
Q keeping fC. U E. fixed until F is in general position 1
1
respect to fw.
with
Then
dim (F n fM) ~ (d - i + 3) + m -'1 ~ d - i, because m ~ '1 - 3. dim f-ip ~ d - i,
Therefore because
f is non-degenerate,
being
Let Ei+1 denote the new position Then E. 1 is collapsible 1+
Since M is d-connected, 0f
the closure
",-1 F )
I
because
in general position.
of Di after the isotopy. it is homeomorphic to D .. 1
we can engulf f -1F ( or more precisely
by pushing
out a feeler from C .• 1
That
is to say there exists a subspace C. 1 of M such that f -1FcC.
1+ 1 ......,., C.
1+
1
dim (Ci+1 - Ci) ~ d - i + 1 . Then C. 1 is collapsible because C. 1~ C. ~ 0; l+
1+
S(f) c C.1+l' because
S(f) c C., by induction 1 C
1 f- E.
1
C.1+1.
1 c C.1+l' because f-1E.1+1 = C.1 U f-1E.1 U f-1F
1+
=
C. U f-1F, by induction 1
c Ci+1' by engulfing.
- 43 -
Since Q is (d+1)-connected
we can now engulf f(C. 1 - C.) 2+ 2 That is to say there by pushing out a feeler from E.2+ 1" o
exists a subspace D.2+ 1 of Q such that f( C. 1 - C.) cD. 1 --'--;l E. 1 1+ 1 1+ 1+ dim (D. 1) ~ d - i + 2. 2+ 1 - E·2-t Then Di+1 is collapsible
because Di+1 ~
Ei+1~
O.
We have
constructed the three spaces, and verified all the conditions of the (i+1)th inductive
fCi c E + , since the isotopy kept fC i 1 i
This follows because fixed9
statement excent C.2+ 1 c f-1D i+1· .t;'
and so fC.1+ 1
= C
=
fC.1 U f(C.2+ 1 - C.) 2 E. 1 u D. 19 by engulfing 1+ 2+ Di+1"
This completes the proof of the inductive
step, and hence the
proof of Lemn~ 63. We return to the proof of Theorem 23" submanifold
Q~ of Q containing 'l'
Triangulate
M9
subcomplexes.
Q~;;
Choose a compact
fM u D in its interior.
such that f is simplicial
If we pass to the barycentric
and C9 Dare second derived
complexes
then f remains simplicial because f is non-degenerate (being in general position). Let Bm, Bq denote the second
derived neighbourhoods
of C, D in M, Q* respectively;
are balls by Theorem 59 because Lemma 63 implies S(f)
C
Bm
=
these
C, D are collapsible. Then f-1Bq• In fact the lemma implies
- 44 -
more: it implies that f Om I maps B
f'
,JI
embeds
-+
Bm
0q B -+
-
\ embeds M - B
Bq m
-+ Q -
q B .
Now we see our way clear: we have localised singularities them out. that g .mB
of f' inside balls, where
More precisely
let g~Bm
all the
it is easy to straighten
Bq be an embedding
-+
such
= f 10m B
, obtained by joining the boundary to an ' point in some linear representation of' Bq. Extend interior m g to an embedding g:M -+ Q by making g equal to f outside B . Then g ~ f.
Notice
that the homotopy is global, but takes place inside the ball Bq• This completes the proof of Theorem
23
in the case M closed. prs>of.. ()t~_TA~or.emS2_ wht:;n Lt§..._boulld~d. We are given a continuous piecewise
linear embedding
map f:M of M in
Q such that flM is a
-+
Q.
First make f'piecewise
o
linear keeping M f'ixed by relative
simplicial
The next thing to do is to straighten boundary. .
If g
-1
0
Call a map g:M •
Q = M.
-+ Q
proJ2~
approximation.
up the map near the (as in the case of embeddings)
- 45 -
•
0
E fixed, such that S(g) c Q. =-~··"_m.~~·~~"""""_==",,,,,
-.,.-
",r',-" .. ~",
•••....•. ", ...,__
After
Once the lemma is proved we can apply the same arguments in the unbounded
case to eliminate
working
in the interior of Q.
entirely
the singularities
as
of g,
The proof of the
theorem will therefore be complete. Pro~f 91'L.§mma
64.
Since M is compact we can choose a to Lemma 24, that is to sayan
collar by the Corollary c
M
embedding
:111 x I ~ M,
a) = x, for all x E r~1. Let Ma = closure (M-im ~~)• ,).be a homotopy Let ht:M ~ M/that starts with the identity and finishes with a such tha t
C~ff(X,
map hi that shrinks the collar onto the boundary homeomorphically by stretching particular
onto M.
Such a homotopy
and maps Ma
can be easily defined
the inner half of a collar twice as long.
In
- 46 -
u)
h1CM(X9 for all x E M9 U E I.
. M
f ~ fh
1 keeping
=
x9
Then ht keeps
M
fixed9 and therefore
fixed.
Now let Q1 be a regular neighbourhood
of fM in Q.
Since
M is compact so is Q1' and we can choose a collar cQ
such that c(y, 0)
=
:Q.1 x I ~ Q1
y for all y
Let kt:Q1 ~ Q1 be the homotopy inner boundary
a ~
E
.
Q1'
Let QO
shrinking
and keeping QO fixed.
=
closure(Q1-im
the collar onto its
More precisely
for
t ~ 1 define t + u),
a ~
u ~ 1- t } Y E Q,1
1),
We now use k to construct into QO and sketches c ' Q
a homotopy
gt:M ~ Q that moves MO
the collar cM out again compatibly with
More precisely gtc~:I (x , u )
1-t~u~1
for
= {
a ~
t ~ 1 define
cQ (fx , u), a ~ u ~ t
•
X E
cQ(fx, t), t ~ u ~ 1
1i
gtlMO = ktfh1lMo' Notice
that gt keeps M fixed.
Define
g
=
g1' and we have
f ~ fh1 = go ~ g1 = g, and the collars Meanwhile gMo c Q09 all keeping ~l fixed. cM' cQ are compatible with g in the sense that the diagram
cQ ).
- 47 -
000
•
is commutative.
Therefore
So g is proper.
Also the restriction
an embedding,
gM
C
Q1 c Q and gM
=
fliI c
Q,
and
of g to the collar is
and so
S(g) c closure Qo = QO
0
C
0
Q1 c Q.
This completes the proof of Lemma 64 and Theorem
23.
Proof of Theorem=-- 24 when M is__.,-~~ closed. ,~~_.~-.... ... ~ :=lI;-."
""
. ..-..-...-.
+
-
o
We are given homotopic
embeddings
f, g:M ~ Q, we have to show
they are ambient isotopic keeping Q fixed, prOVided m ::;; q - 3 M is (d+1)-connected Q is (d+2)-connected. Without
loss of generality
because
if we prove the result for the unbounded
f, g are ambient isotopic possible
o
in Q.
case then
Then, by Theorem 12, it is
to choose an ambient isotopic with compact support,
which is therefore
Q fixed.
we can assume that M is unbounded,
Therefore
extendable
to an ambient isotopy of Q keeping
assume Q unbounded.
R~marko If \lITe had Hudson t s concordance
. :>isotopy result available,
then the theorem could be deduced immediately as follows.
The homotopy
gives a continuous
from Theorem map
23,
- 48 -
F:M x I ~ Q x I
I)" is an embedding in (Q
I)""
such that FI(M
x
connectivities
o~ M~ Q have to be increased by one each~
because the double-point 2 (m + 1) -
dimension
(q + 1)
=
x
The
o~ F is
(2m - q) + 1 = d + 1 "
By Theorem 23 homotop F into an embedding
G:M x I ~ Q x I keeping
UI
x
I)" ~ixed"
between ~ and g"
In other words G is a concordance
There~ore
they are ambient isotopic.
as we have not proved the concordance
>isotopy
However
in these notes~
we give a separate proo~ o~ Theorem 24~ similar to that o~ Theorem
23 above.
Begin the proo~ by ambient isotoping with respect to ~M~ by Theorem continuous
15.
g into general position
The given homotopy
is a
map h:l1 x I ~ Q
which we can make piecewise by relative
linear keeping
(M x I)" ~ixed~
simplicial approximation.
fJ_~_~.d. '_we__c_aJl~in_d.col~.§l..1?.~:!:J:L~_~=f2.1.l.9_~_ge s.G..9 D __ ol'Jk_.~~h S(ll.l~_
C~"
=
~p_at
1
h- D.
We prove the lemma in two stages in order to make the proo~ more translucent. by assuming
In the ~irst stage we prove a weaker result
the stronger hypotheses
- 49 -
m~q-5
M is (d+2)-connected
Q is (d+4)-connected. The proof follows the pattern of the proof of Lemma 63. In the second stage we show how to sharpen the proof as various points in order get by with the correct hypotheses of
Theorem 24, namely
m ~ q - 3 M is (d+1)-connected Q is (d+2)-connected. We achieve the sharpening by using the piping techniques
of
the last chapter. Proo~ of First S~age.
Let ~~M x I ~ M denote projection.
Notice that hl(M x I)· is in general position because we have already isotoped g into general position with respect to fM. By Theorem 18 homotop h into general position fixed.
keeping
(M x I)·
Therefore dim S (h) ~ 2(m + 1 ) - q
= There is an induction
d + 2.
on i9 as follows.
There exist collapsible
subspaces C. in M and D. ~ E. in Q such that 111
( 1)
S(h)
(2 )
1 h -1 E. c C. x I C h- D.
(3)
dim (D. - E.) ~ d - i + 6.
C
Ci x I
111 1
1
- 50 -
=
The induction begins at i hypotheses
1.
Since under the stronger
we are assuming 1\1 to be (d+2)-connected,
engulf
~S(h) in a collapsible
subspace c~+3 of M.
to be (d+4)-connected9
engulf h(C1 x 1) in a collapsible
d+5 subspace D1 of Q.
and the conditions
Since Q is assumed
Choose E1 to be a point of h(C1 x 1),
=
for i
For the inductive
1 are satisfied.
steP9 assume true for i.
Obtain E.1+ 1
by ambient isotoping D. in Q keeping h(C. x 1) U E. fixed9 III until the cODplement F is in general position with respect to
1).
heM x
Then dim ~h-1F ~ dim h-1F dim [F n heM x 1)]
=
~ (d - i + 6) + (m + 1) - q ~
d
-
i + 2,
because we are assuming m ~ q - 5. Engulf in M AA-1F c C. 1 ~ C. 1+ 1 C. ) ~ d - i + 3. dim (C. 1 1+ 1
-
Therefore dim (C. 1 x 1 - C. x 1) ~ d - i + 4. 1+
Engulf in
1
Q
h( C 1+ . 1 x 1 - C.1 x 1) cD. 1+ 1 ~
E.1+ 1
dim (D.1+ 1 - E.1+1 ) ~ d - i + 5. Verify the (i+1)th induction The induction S(h)
C
ends at i
C. x 1 ::; h 1
-1
D.,
1
=
statement as in the proof of Lemma 63.
d + 7 with D.
as required.
1
=
E'9 and consequently 1
- 51 -
When homotoping positiony
the full strength of Theorem
now use the additional
information
for the pair M x I, (M x I)·. that the (d+2)-dimensional interior
18 was not used.
We
that h is in general position
In particular
this implies
stuff of S(h) all lies in the
of M x I, at places where exactly two sheets of M x I
cross one another. top dimensional
7.
The trick now is to punch holes in this
stuff, by piping
one of the sheets over the
More precisely
we use the piping Lemma 48 of
o.
free-end M x Chapter
h into general
The triple M x 0, M x 1 c M x I is "cylinderlike"
in the sense of Chapter 7, and Q has no boundary Therefore
by assumption.
by the piping lemma, we can homotop h keeping
(M x I)·
fixed, and then find a subspace T of M x I such that (1 )
S(h) cT
(2)
dim T ~ d + 2
(3)
dim
(4)
M x I ~(M
[(M
x 1 ) n T] ~ d + 1 x 1)
u T '-:ol.M x 1 .
By being a little more precise
in the proof of Lemma 48 at
one J.)oint,we can factor the first of these collaJ.)ses
(5)
M x I ~
(M xi)
u (1tT x I)
~
(M xi)
u T.
engulf
it is J.)ossible,using (3), to d+2 of M x 1. (M x 1) n T in a collapsible subspace R
Define
C~+2
Since M is (d+1)-connected
I
=
~(R
U
T).
Notice
that comJ.)aredwith the dimension
of C1 in the J.)roofof the first stage, we have scored an
- 52 -
improvement
of 1.
Then
=
C1 x r
(~R x D)
U
(~T x r)
'~R u (-JeTx I), cylinderwi se ~ RuT Therefore
by
(5).
heR u T) by Lemma 38 and (1) above.
h(C1 x I)~
But dim h(C1 x I) ~ d + 3 dim heR u T) ~ d + 2. h(C1 x I) can be "furledil in the sense of Chapter 7. Since Q is (d+2)-connected engulf h(C1 x r) in a collapsible d+2 subspace D1 of Q, of the same dimension, by the furling Corollary to Theorem 20. As before define E1 to be a point of
Therefore
h(C1 x I).
Notice
the dimension
that we have scored an improvement
of 2 in
of D1•
We can write this improvement statement by replacing
into the ith inductive (3) by
condition
dim (D. - E.) ~ d - i + l
2
4.
We now have to do some more piping and furling for the inductive step.
As before
obtain E. 1 by ambient 2+
isotoping D. keeping
h(C. x I) U E. fixed, until the complement l
position
2
with respect
to h~M x I.)
Let W
2
F is in general
=
dim ~W ~ dim W
=
dim [F n heM x I)]
~ (d - i + 4) + (m + 1)-q
~ d - i + 2
-1 closure (h F).
Then
- 53 -
because m ~ q - 3.
We now want to engulf
with a feeler of the same dimension, is to furl ~W. dimensional piping
heM
x
We furl
D)
of
heM
x
punching
top dimensional
holes in the top triangulation),
piece of F off the end
After "
l\iIxI
pipe
J~
-
c.~
Notice
by
I).
Before
M
C but 1 and the way to do this
(of aome suitable
simplexes
the relevant
~w by
~w from
,
l'
hole
that S(h) c C. x I, and F docs not meet h(C. x I), 1
and so that the self-intersections
1
of heM x I) do not get in
the way of the pipe. More precisely,
we can adapt Lemma 48 to give the following
- 54 -
result.
We can find an ambient isotopy of D. keeping 1
h(C. x I) U E. fixed, such that in the new position 1
~Wd-i+2
1
can be furled to a subspace Xd-i+1 relative to C., and 1
(Ci x I) U (~W x I)~(Ci By the Corollary
x
I) U (X x I) U W.
to Theorem 20, engulf in M
~w c
C. 1"--; 1+
C.
1
d - i + 2. dim (C. 1+ 1 - C.) 1 ~
Let yd-i+3
=
closure
(C. 1 x I-C. 1+
1
Then Y can be furled to Z relative
(C. x I) U Y 1
=
X
I), Zd-i+2
=
(X xl)
UW U (C. 1+1
x
to C. x I because 1
C.1+1 x I
... ~ (C. x I) U (~W x I) U (C. 1 x 1) cy 1ind erwi S E 1 1+ ~(C. x I) U Y by above. 1
Therefore hY can be furled to hZ relative Lemma 38, because to hZ relative
S(h) C Ci x I.
Moreover
to h(C.1 x I) by hY can be furled
to E.1+ l' because
hY n E.1+ 1
C
heM x I) n E.1+ 1
=
h(C.1 x I) U hW
C
hZ n E.1+ l'
and therefore hY n Ei+1 = hZ n Ei+1• hY
C
D.1+1 ~
Therefore
engulf in Q
E.1+ 1
dim (D. 1 - E.1+1) ~ d - i + 3. 1+ Verify the (i+1)th inductive
statement as in the proof of the
first stage, and the proof of Lenma 65 is complete.
- 55 Lemrna 66.
1:;h9. t
Tl?-e1Z.e_~~ts.:LEa~].J32m c ~L__ l?q c~suc]l
f t
13m x
WDS
h
~_Bq
I
0
embeds
Bm
~:beds
1M- Bm.).,,, I
BCl
x I ~
.:'._Q _
B'l,
The obvious way is to take derived neighbourhoods o~ the collapsible
subspaces
run into the technical triangulations
66.
C, D o~ Lemma
di~~iculty
However we
o~ not being able to ~ind
such that both maps h
M x I ----?Q
1~ M
are simplicial o~ Chapter
(as is illustrated
1).
Therefore
in the Example
~irst choose triagulation
M x I, M such that 1C:K ~ L is simplicial, subcomplex
o~ L.
such that A-10 speci~ied
=
later.
Choose
De~ine
8,
Bm
=
it is a regular neighbourhood
~ Q1 is simplicial.
deriveds
then h:K
non-degenerate.
A-1[O,
K
1
simplicial
map
< 1 o~ a smallness
to be
8] which is ball, because 14.
o~ K, and a triangulation
Q1
o~ heM x I) in Q such that
o~ a regular neighbourhood 1
8
o~ C by Lemma
Now choose a subdivision
h:K
0 <
K, L o~
and C is a ~ull
Let A:L ~ I be the unique C.
at the end
I~ K2, Q2 are barycentric
2 ~ Q2 remains Now choose
B
simplicial
such that
8
because
~irst h is
< A1CV ~or all vertices
Call a simplex o~ K2 e~ceptionaJ. i~ it 2 not in C x I. meets C x I, but is not contained in C x I. Then (A1C)-18 meets V E K
- 56 -
only exceptional
simplexes~
and meets each exceptional
simplex
in a hyperplane. Let K3 be a rirst derived or K2 obtained by starring all exceptional
on (A~)-18~
simplexes
and the rest barycentrically.
Since S(h) c e x I~ no exceptional any other simplex by h.
simplex is identiried with
Thererore we can derine a rirst derived
Q3 of Q2' such that h:K3 ~ Q3 remain simplicial, by starring images of exceptional simplexes at the image or the star-point, and the rest barycentrically. Derine Bq == N(D, Q3)~ which is a ball~ being a second derived neighbourhood subspace D.
of the collapsible
Then
h -1B q
==
I, K3 ) == q m x I h-1B == B
N(e
x
S(h)
C
Bm
x I.
The proor of Lemma 66 is complete. eon~inJ1J1J.Jt the l?.1~o~()fco:r~T.l1.e.9r_em _24. So far we have the picture:
III
B
x I.
- 57 -
~ ~
Me
BrnXI
~
Bq
B~
Since hiM -
Bm
is a proper embedding
this means that by Theorem
~IM - Bm
is isotopic
Om
of M - B-
0q
in Q - B
Om
to g I M - B.
9
Therefore
1 of Chapter 5 they are ambient Extend the ambient isotopy of Q - q arbitrarily 12 Corollary
isotopic. over Bq to give an ambient
B
isotopy of Q.
The latter moves
Om
m
~ to f' 9 saY9 where f' agrees with g except on B 9 and r' IB 9 glBm are proper embeddings of Bm in Bq that agree on the Since m ~ q - 39 by Theorem 9 Corollary 1 of Chapter 4 we can ambient isotop Bq keeping Bq fixed so as to
boundary.
move f' IBIDonto glBm• to an ambient are ambient
This ambient isotopy extends trivially
isotopy of Q9 moving f' onto g.
isotopic.
when M is closed.
This completes
Hence f and g
the proof of Theorem
24
- 58 -
We are given embeddings
. . Q
keeping
f, g:M ~ Q that are homotopic
M fixed, and we have to show they are ambient
keeping
fixed.
isotopic
The first thing to do is to make them agree
on a collar. Let Q1 be a compact
submanifold
image heM x I) of the gi7en homotopy
of Q containing h.
the
Choose a collar
By Theorem 10 of Chapter 5 choose two collars cfQ'
of M.
f
Q1 such that c ' cQ are compatible M compatible
with g.
k of Q1 keeping fixed, k extends ~ fixed.
13 there is an ambient isotopy . f g . Q1 fixed, moving cQ onto cQ' Since Q1 is kept
M
O
C
to an ambient
if we replace
on the collar
wi th both f and g. Then fMO' gM
By Theorem
trivially
Therefore
then f, g agrees
g with f, and cM' cQ are
QO'
isotopy
f by ~f,
of Q keeping
and write
cQ
=
c~,
im c ' and cM' cQ are compatible M
Let MO = closure The picture
(M - im cM), Qo = closure (Q1- im CQ)
now looks like:
- 59 -
J:,~I£1Ela_ 67.
ThC:;; ... 2T1fl:PSY-1Mo?~LMO ~MO
::"Q.o
are hQmo_toJlJc __:ij1~Qo
?ee~ing MO fixed. Let et~MO ~ M be a homotopy inclusion
and ending with a homeomorphism
collaX' of Mo over that of M. all x E M? tEl.
is obtained by
of the three homotopies
jfet, jhte1? jge1_t· the proof of the lemma. of what we have done so far is to push the
singulaX'ities of the homotopy that the boundaries
homotopy
that shrinks
jCQ(Y? t) = cQ(y? 1)
Then the required homotopy
composition
The purpose
However
etcNr(x? 1) = cM(x? 1 -t)?
Let j:Q1 ~ QO be the retraction
all Y E Q1? tEl.
This completes
that stretches a
In particular
the collar cQ onto its inner boundary?
timewise
starting with the
into the interiors
of M? Q so
do not interfere with the engulfing.
there is the trivial technical
difficulty
that a constant
of the collar is of cour se a singular map of (collar) x I.
Nor can we ambient
isotop glim cM away from flim cM for two reasons~ firstly we have got to keep M fixed? and secondly there is an obstruction in H2q-2m(Q). Therefore we get round this
difficulty
by defining
the homotopy
to be a map of a reduced
product M # I? obtained from M x I by shrinking point for each x
im cM. More precisely? identify
x x I to a
E
(but not the complements
the collars
of M x O? M x 1
of the collaX's) and define M # I to be
- 60 -
the relative
mapping
Mx0--7Mx1.
cylinder
Let 7\.# :M:I+ 1
of the homeomorphism M
--7
denote the projection.
collar
M
=
If X c M, define X:I+ 1
=
M:l+I
7\.;1 x.
Then u (collar).
0,1.[0#1)
Define h# :M# 1--7 Q by.rapPing
110 # I by Lemma
lembedding
67
the collar by f7\.#.
0
Then 7C#S(~) c Mo c M. 0
0
h#S(~) Therefore
c Qo c Q1 c Q.
we can apply engulfing
-
)
case, and obtain balls B
Om
map s B :1+ I
embeds
On
(M -
Ell
--7
Bm) # 1
Therefore,
as before,
By Theorem
12 they are ambient
proof of Theorem
0
c M,
B';1.
--7
embeds 13m #: 1
l
in the interiors m
of M, Q, as in the unbounded Bq c Q, such that 1
arguments
--7
Eq .
Q -
f and g are isotopic isotopic
24 is complete.
.
keeping M fixed.
keeping
Q
fixed.
The
