ELEMENTARY DIFFERENTIAL TOPOLOGY BY
James R. Munkres Lectures
Given at Massachusetts Institute of Technology Fall, x g6 i
REVISED EDITION
Annals of Mathematics Studies Number
54.
ELEMENTARY DIFFERENTIAL TOPOLOGY BY
James R. Munkres Lectures
Given at Massachusetts Institute of Technology
Fall, 1961
REVISED EDITION
PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1966
Copyright © 1963, 1966, by Princeton University Press ALL RIGHTS RESERVED
Printed in the United States of America Revised edn., 1966 Second Printing, with corrections, 1968
To the memory of Sumner B. Myers
PREFACE
Differential topology may be defined as the study of those properties of differentiable manifolyds which are invariant under differentiable homecmorphisms.
Problems in this field arise from the interplay between the
topological, combinatorial, and differentiable structures of a manifold. They do not, however, involve such notions as connections, geodesics, curvature, and the like; in this way the subject may be distinguished from differential geometry.
One particular flowering of the subject took place in the 1930's, with work of H. Whitney, S. S. Cairns, and J. H. C. Whitehead.
A second
flowering has come more recently, with the exciting work of J. Milnor, R. Than, S. Smale, M. Kervaire, and others.
The later work depends on the
earlier, of course, but differs from it in many ways, most particularly in the extent to which it uses the results and methods of algebraic topology. The earlier work is more exclusively geometric in nature, and is thus in some sense more elementary.
One may make an analogy with the discipline of Number Theory, in which a theorem is called elementary if its proof involves no use of the
theory of functions of a complex variable-otherwise the proof is said to be non-elementary.
As one is well aware, the terminology does not reflect the
difficulty of the proof in question, the elementary proofs often being harder than the others.
It is in a similar sense that we speak of the elementary part of differential topology.
This is the subject of the present set of notes.
Since our theorems and proofs (with one small exception) will invoive no algebraic topology, the background we expect of the reader consists of a working knowledge of:
the calculus of functions of several variables
and the associated linear algebra, point-set topology, and, for Chapter II, the geometry (not the algebra) of simplicial complexes.
Apart from these
topics, the present notes endeavor to be self-contained. The reader will not find them especially elegant, however. vii
We are
not hoping to write anything like the definitive work, even on the most elementary aspects of the subject.
Rather our hope is to provide a set of
notes from which the student may acquire a feeling for differential topology, at least in its geometric aspects.
For this purpose, it is necessary that
the student work diligently through the exercises and problems scattered throughout the notes; they were chosen with this object in mind.
The word problem is used to label an exercise for which either the result itself, or the proof, is of particular interest or difficulty. Even the best student will find some challenges in the set of problems.
Those problems and exercises which are not essential to the logical continuity of the subject are marked with an asterisk.
A second object of these notes is to provide, in more accessible form than heretofore, proofs of a few of the basic often-used-but-seldomproved facts about differentiable manifolds.
Treated in the first chapter
are the body of theorems which state, roughly speaking, that any result which holds for manifolds and maps which are infinitely differentiable holds also if lesser degrees of differentiability are assumed.
Proofs of these
theorems have been part of the "folk-literature" for some time; only recently has anyone written them down.
([8) and [91.)
(The stronger theorems of
Whitney, concerning analytic manifolds, require quite different proofs, which appear in his classical paper [15].)
In a sense these results are negative, for they declare that nothing really interesting occurs between manifolds of class C°°.
C1
and those of class
However, they are still worth proving, at least partly for the tech-
niques involved.
The second chapter is devoted to proving the existence and uniqueness of a smooth triangulation of a differentiable manifold. follow J. H. C. Whitehead [14], with some modifications.
In this, we
The result itself
is one of the most useful tools of differential topology, while the techniques involved are essential to anyone studying both combinatorial and differentiable structures on a manifold.
The reader whose primary interest
is in triangulations may omit §4, §5, and §6 with little loss of continuity.
We have made a conscious effort to avoid any more overlap with the lectures on differential topology (4) given by Milnor at Princeton in 1958 viii
than was necessary.
It is for this reason that we snit a proof of Whitney's
imbedding theorem, contenting ourselves with a weaker one.
We hope the
reader will find our notes and Milnor's to be useful supplements to each other.
Remarks on the revised edition Besides correcting a number of errors of the first edition, we have also simplified a few of the proofs. In addition, in Section 2 we now prove rather than merely quote the requisite theorem from dimension theory, since few students have it as part of their background.
The reference to Hurewicz and Waliman's book was
inadequate anyway, since they dealt only with finite coverings.
Finally, we have added at the end some additional problems which exploit the Whitehead triangulation techniques.
The results they state have
already proved useful in the study of combinatorial and differentiable structures on manifolds.
ix
CONTENTS
PREFACE . .
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. vii
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Chapter I. Differentiable Manifolds 0. .
Introduction .
§2.
Submanifolds and Imbeddings .
.
.
§3.
Mappings and Approximations .
.
.
§4. §5.
Smoothing of Maps and Manifolds . Manifolds with Boundary . . . . .
§6.
Uniqueness of the Double of a Manifold .
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17
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25
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39
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47
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63
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Triangulations of
Chapter II.
Differentiable Manifolds §7.
Cell Complexes and Combinatorial Equivalence.
§8.
Immersions and Imbeddings of Complexes .
59.
The Secant Map Induced by f .
§10.
Fitting Together Imbedded Complexes .
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INDEX OF TERMS .
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REFERENCES . .
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69
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79
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109
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111
xi
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90
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97
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.
ELENEVTARY DIFFERENTIAL TOPOLOGY
CHAPTER I.
DIFVERENTIABLE MANIFOIDS
Introduction.
§1.
This section is devoted to defining such basic concepts as those of differentiable manifold, differentiable map, immersion, imbedding, and diffeanorphism, and to proving the implicit function theorem.
We consider the euclidean space Rm as the space of all infinite x - (x1, x2, ...),
sequences of real numbers,
euclidean half-space
i > m;
Then Rm-1 C Hm C Rm. max Ixi'j, lixUU
- 1;
by
1xI.
is the set with
We denote
of all m-tuples
((x1)2 + ... + (xIl1)2)
The unit sphere
jxj < r.
Sm-1
the set with
by
lixti,
and
is the subset of Rm with 1ixII < 1;
and the r-cube
Cm(r)
Often, we also consider Rm as simply the space where no confusion will arise.
(x1,...,xm),
1.1 Definition.
for
is the subset of Rm for which xm > o.
Fin
the unit ball Bm,
such that xi = 0
A (topological) manifold M is a Hausdorff space
with a countable basis, satisfying the following condition:
There is an in-
teger m such that each point of M has a neighborhood haneomorphic with an open subset of Hm
If h :U
or of
Rm.
fl (or Rm)
is a homeomorphism of the neighborhood U
of x with an open set in Hm or Rm, 6OOa'dinate neighborhood on M.
0-1 ,
If
h(U)
the pair
we say M is non-bounded.
is often called a
is open in IF and
then x is called a boundary point of
points is called the boundary of M,
(U,h)
M,
h(x)
lies in
and the set of all such
denoted by Bd M.
If Bd M is empty,
(In the literature, the word manifold is common-
ly used only when Bd M is empty; the more inclusive term than is manifold3
I.
4 with-boundary.)
The set M - Bd M is called the interigr of
denoted by Int M. also use
Int A
DIFFERENTIABLE MANIFOLDS
(If A is a subset of the topological space
in map
h2 IF,
and is X,
we
but this should cause no confusion.)
to mean X - CI(X-A),
To justify these definitions, we must note that if
and
M,
h1
:
U1 -. if
: U2 -+ IF are ho¢neomorphisms of neighborhoods of x with open sets and if
lies in RP-1,
h1(x)
so does
h2(x):
For otherwise, the
would give a homeomorphism of an open set in Rm with a neigh-
h1h21
borhood of the point p - h1(x) ly not open in Rm,
in
IF.
The latter neighborhood is certain-
contradicting the Brouwer theorem on invariance of
domain 13, p. 951.
One may also verify that the number m M;
it is called the dimension of
M,
is uniquely determined by
and M is called an m-manifold.
This may be done either by using the Brouwer theorem on invariance of domain, or by applying the theorem of dimension theory which states that the topo-
is m
logical dimension of M
[3, P. 46].
Strictly speaking, to apply the
latter theorem we need to know that M is a separable metrizable space; but this follows from a standard metrization theorem of point-set topology 12, p. 751.
Because M
is locally compact, separable, and metric,
paracompact [2, p. 791.
open covering Q of covering M (1)
ment of (2)
M,
is
We remind the reader that this means that for any there is another such collection
(B
of open sets
such that
The collection (A
M
(B is a refinement of the first, i.e., every ele-
is contained in an element of
The collection
(
a.
is locally-finite, i.e., every point of M has
a neighborhood intersecting only finitely many elements of
(g
.
In passing, let us note that because M has a countable basis, any locallyfinite open covering of M must be countable.
I
bounded
(a)
Exercise.
If M is an m-manifold, show that Bd M is a non-
m-1 manifold or is empty. (b)
Exercise.
Let MO . M x 0
and
Let M be an m-manifold with non-empty boundary.
M1 - M x 1
be two copies of
M.
The double of
M,
INTRODUCTION
§1.
denoted by D(M), fying
(x,o)
is the topological space obtained from M0 U M1
with
for each x 'in Bd M.
(x,1)
bounded manifold of dimension (c)
n,
5
Prove that D(M) is a non-
m.
M
If
Exercise.
by identi-
respectively, then M x'N
and
N are manifolds of dimensions m and
is a manifold of dimension m + n,
and
Bd(M x N) = ((Bd M) x N) U (M x (Bd N)).
1.2
If U is an open subset of
Definition.
is differentiable of class
fl,...,fn
functions class
If A Cr
of class
borhood
Ux
is any subset of (1 < r < oo)
x
of
which is of class If
Rm,
then
f
if for each point
U.
: U
If
f
is of
: A - Rn is differentiable
x
of
there is a neigh-
A,
a function
on U.
Cr
f : A - Rn
is differentiable, and f
x
is in
A,
we use
this matrix.
Now
for
must be extended to a neighborhood of
f
Df(x)
at x - the matrix whose general entry
We also use the notation
aij -
Rn
C°°.
may be extended to
fl(A n Ux)
such that
to denote the Jacobian matrix of is
are continuous on
r
it is said to be of class
r,
f
if the partial derivatives of the component
Cr
through order
for all finite
Cr
then
Rm,
x before these
partials are defined; in the cases of interest, -.he partials arc independenn.
of the choice of extension (see Exercise (b)).
We recall here the chain rule for derivatives, which states that D(fg) - Df
Dg,
where
fg
is the composite function, and the dot indicates
matrix multiplication.
I
(a)
Exercise.
Check that differentiability is well-defined; i.e.,
that the differentiability of ing space" (b)
A - Rn
Rm
for A
Exercise.
is of class
C1,
dent of the extension of chosen.
f : A -+ Rn
does not depend on which "contain-
is chosen.
Let U be open in apd f
x
is in
A,
Rm;
let U C A C U.
show that
to a neighborhood of
x
Df(x) in
If
f
is indepen-
Rm which is
ENTLAM MANIFOIDS
D
I.
6
Our definition of the differentiability of a map f: A -Rn is essentially a local one. We now obtain an equivalent global formulation of differentiability, which is given in Theorem 1.5. Remark.
There is a
1.3 Lemma. on
C(1/2),
f
is a Let
g(t) - 0
that
function c
is positive on the interior of f(t) . e-1/t
Let
Proof.
Then
C00
Cxx
for
: Rm -. R1
C(1),
and is zero outside
t > 0, and
f(t) - 0
function which is positive for
for
g'(t) > 0
t < 0,
for
for
I
C(1),
t < 0.
0.
Then g is a
g(t) - f(t)/(f(t) + f(1-t)).
which equals
C00
and
0 < t < 1,
function such g(t) . 1
for
t> 1. Let
h(t) .0 for
that
Then h is a
h(t) - g(2t+2) g(-2t + 2). h(t) > o
(t j > 1,
for
Iti <1,
function such
C00
h(t) .1
and
for
Itt < 1/2.
y
y
y=g(t)
y = h(t)
t I
Let
,(x1,...,xm) . h(x1)
(a)
Exercise.
be an open subset of Cr
I
Rm;
real-valued function
h(x2)
... h(xm).
Generalize the let Y
as follows:
C be a compact subset of
defined on Rm such that
and is zero in a neighborhood of the complement of
U.
Y
U.
Let U
There is a
is positive on
C
INTRODUCTION
S1.
7
Remark. Whenever an indexed collection (Cl) of subsets of X is said to be locally-finite, we shall mean by this that every point of X has a neighborhood intersecting CI for at most finitely many values of i. This convention is convenient; it implies that only the empty sat can equal Ci for more than finitely many i.
1.4 Lemma. space
(Ui)
be an open covering of the paracompact
There is a locally-finite open covering
X.
Vi C Ui
Let
(1)
for each
the normal space
for each
(1) Let (B be
Proof. each element B
as the union of those
..., there is a closed covering (Ci) of
a locally-finite refinement of J(B)
B for which
such that J(B) - J.
for only finitely many values of
(2) We construct the coverings
by induction.
(Ci)
Let
B C UJ(B)
De-
.
Each point has a this neigh-
(B;
J Let
W1
be an
whose closure is contained in
open set containing X - (V2 U V3 U ...), V1.
For
(Ui).
neighborhood intersecting only finitely many elements of
borhood intersects VJ
such that
i.
choose an index
of (B,
of X
be a locally-finite open covering of
(Vi)
X, where i = 1, 2,
X such that Ci C Vi
fine VJ
(2) Let
1.
(Vi)
C1 = W1 W1 U ... U WJ_1 U VJ U VJ+1 U ...
Suppose
- X.
Let WJ
be
an open set containing X - (W1 U ... U WJ_1 U VJ+1 U ...)
whose closure is contained in VJ.
Let
To prove that the collection
X lies in only finitely many sets VJ U VJ;1 U ...
.
As a result,
VJ.
CJ = 7J. (WJ)
covers
X,
Hence for some
x must belong to
note that any point x
J,
is not in
W1 U ... U WJ_1,
by
.the.yMuction hypothesis. Transfinite induction may be used to prove (2) for
aA bbitrary index set. Let A be a subset of
1.5 Theorem.
of class
Cr.
Then
in a neighborhood of
f A.
Rm;
let
f
:
A
Rn be
may be extended to a function which is of class Cr
I. DIFFERENTIABLE MANIFOLDS
8
By hypothesis, for each point x
Proof.
borhood Ux
of x
is of class
Cr
on
fIA n Ux may be eActended to a function which
such that
We choose
Ux.
A, there is a neigh-
of
Let M be the union
to be compact.
IIx
Ux ; it is an open subset of Rm. Let (Vi) finite open refinement of the covering (Ux) of M. Let of the sets
ing of
M by closed sets such that
for each
CI C Vi
function defined on Rm which is positive on borhood of the complement of
Vi.
borhood of any given point of E Wi(x) s
be a cover-
(Ci)
Let
be a
Ti
CO
and equals zero in a neigh-
CI
Then
Here we use Exercise (a) of 1.3.
M, since it equals a finite sum in some neigh-
is a C0 function on
E Y,(x)
I.
be a locally-
M.
Define
then
- 1'i(x)/E TJ(x);
cpi(x)
1.
For each. i,
if A n Vi
let
is empty, let
tended to be of class
Cr
fi
denote a
extension of
Cr
be the zero function.
fi
to Vi;
flA n Vi
may be ex-
Then Tifi
on M by letting it equal zero outside Vi.
Define f(x) = Ei cpi(x) fi(x)
.
This is a finite sum in some neighborhood of any point Cr
is of class for every
i,
on
Furthermore, if x
M.
is in
A,
x
of
and hence
M,
then cifi = (pif
so that ti
f(x) = E (Pi(x) f(x) = f(x)
f
Hence in
is the required
extension of
Cr
f
to the neighborhood M
of A
Rm.
1.6
m-manifold
M
Definition.
A differentiable m-manifold of class
and a differentiable structure
differentiable structure of class coordinate neighborhoods
(U,h)
Cr
on
on M,
The coordinate neighborhoods in
(2)
If
and
(U2,h2)
is differentiable of class (3)
The collection 5)
Cr.
Cr
on
M.
A
in turn, is a collection of
3) cover
belong to 3),
h1h21: h2(U1n U2)
of class
satisfying three conditions:
(1)
(U1 ,h1)
M,
S)
is an
Cr
M.
then
Rm e
is maximal with respect to property (2); i.e.,
if any coordinate neighborhood not in
1) is adjoined to the collection
5),
INTRODUCTION
§i.
9
then property (2) fails.
The elements of 3) manifold
M.
(a)
on M ture
are often called coordinate systems on the differentiable
Exercise.
Let 3)' be a collection of coordinate neighborhoods
satisfying (1) and (2).
3) of class
Cr
Prove there is .a unique differentiable struc-
containing 3)'.
(We call 3)'
a basis for
by
3),
analogy with the relation between a basis for a topology and the topology.) Hint:
Let
3) consist of all coordinate neighborhoods
M which overlap every element of 3)' means that for each element
hih a n d
h
Cr;
this
of 3)',
.: h(U n U1)
h11
on
differentiably with class
(U1,hi) 1
(U,h)
Rm
h1(U n Ud - Rm
:
are differentiable of class Cr. To show 3) is a differentiable structure, the
local formulation of differentiability is needed. (b)
Let M be a differentiable manifold of class
Exercise.
(we often suppress mention of the differentiable structure
Cr
where no
3),
confusion will arise).
Then M may also be considered to be a differenti-
able manifold of class
Cr-i,
in a natural way; one merely takes
basis for a differentiable structure that the inclusion 3) C 3)I
of
3)i
is proper.
class
Cr-i
on
M.
This proves that the class
3) as a Verify of a
Cr
differentiable manifold is uniquely determined.
M
We see in this way that the class of a differentiable manifold
may be lowered as far as one likes merely by adding new coordinate systems to the differentiable structure.
The reverse is also true, but it will re-
quire much work to prove. (c)
If M
Exercise*.
is a differentiable manifold, what are the
d'lfi'iculties involved in putting a differentiable structure on D(M)?
(D(M)
was defined in Exercise (b) of 1.1.)
1.7
Definition.
dimensions m and
n,
a subset of M and let
Let M and N
be differentiable manifolds, of
respectively, and of class at least f : A-* N;
then
f
Cr.
Let A be
is said to be of class
Cr
if
10
DIFFERENTIABLE MANIFOLDS
I.
for every pair
and
(U,h)
of coordinate systems of M and
(V,k)
res-
N,
pectively, for which f(A n U) C V, the composite kfh-1
is of class
: h(A n U) -. Rn
(Note that a map of class
Cr.
C2
though a manifold of class
is also of class
C2
is not one of class
C1
al-
C1,
until the differ-
entiable structure is changed.)
The rank of at
where
h(p),
open subset W
at the point p
(U, h)
and
(V, k)
of M is the rank of D(kfh-1)
p and
are coordinate systems about
This number is well-defined, provided there is an
respectively.
f(p),
f
of M such that W C A C W,
for if
and
(U1, h1)
(V1). k1)
were other such coordinate systems, we would have D(k1fh7,1) - D(k1k-1)
- D(kfh-1)
D(hhl1)
The requirements for a differentiable structure assure that
are both differentiable, so that D(k1k-1) D(kk,1)
as its inverse.
and D(kfh-1)
k1k-1
is non-singular, having
is non-singular, so D(k1fh,1)
Similarly,
have the same rank.
Exercise.
(a)
The standard r differentiable structure on Rm
is that having as basis the single coordinate system I : Rm -. Rm. ly for
If.
and
Similar-
Check that the definitions of differentiability given in 1.2
and 1.7 agree.
1.8 Definition.
Let
f : M - N be differentiable of class
let M and N have dimensions m and at each point p
of
f
M,
n,
If rank f - m
respectively.
is said to be an immersion.
Cr;
If
f
morphism (into) and is an immersion, it is called an imbedding.
is a homeoIf
f
is a
haneomorphism of M onto N and is an immersion, it is called diffeomortsm; of course,
(a)
m - n in this case.
Exercise.
Note that Bd 11m - RP-1
RP-1 - Ea is an imbedding.
entiable manifold of class ture of class imbedding.
Cr
and the inclusion
Generalize this as follows: If M is a differCr,
then there is a unique differentiable struc-
on Bd M such that the inclusion Bd M
M is a
Cr
INTRODUCTION
§1.
(b)
11
Exercise.
Show that the composition of two immersions is an
Exercise.
Let M and N
immersion. (c)
let
Cr;
Cr differentiable structure on M x N
Construct a
bounded.
have class
natural inclusions of M and N into M x N are imbeddings. M
require
such that the
Why do we
to be non-bounded? Exercise*.
(d)
Construct a
so that the inclusion
H1 x H1
M be non-
i
:
C°° differentiable structure on
is differentiable.
H1 x H1 - R2
What
are the difficulties involved inputting a differentiable structure on in general?
M x N
Exercise
(e)
carries
S1
.
Construct a
into a figure eight.
immersion of (f)
B2
into
C°° immersion of
S1
into R2
which
Can such an immersion be extended to an
R2?
Prove that two differentiable structures of class
Exercise
on the topological manifold M are the same if and only if the identity
Cr
map from each differentiable manifold to the other is a Cr diffeomorphism. (g)
of class
CO0
Exercise*.
Construct two different differentiable structures
on the manifold
R1,
such that the resulting differentiable
manifolds are diffeomorphic.
a
1.9 R1
Problem.
Prove that any two differentiable structures on
give manifolds which are diffeomorphic. (It was long a classical problem to determine whether two differ-
ent differentiable structures on the same topological m-manifold always gave rise to diffeomorphic differentiable manifolds.
turned out that the answer is yes if m < 3 (s].
The answer is also yes for
Rn,
if n
Recently, it has
[10, 13], and no 4 [16].
if m = 7
Other results are
known, but much work remains to be done.)
1.1o
prove that f 1 Hint:
Problem.
If
f : M- N is a diffeomorphism of class
is a diffeomorphism of class
C1',
Cr.
Prove the result in the following steps.
Let
g map the
12
DIFFERENTIABLE MANIFOLDS
I.
open subset U of Rn into (1)
Rn.
exists if there is a matrix A
Dg(x)
such that
R(x1)
A -
g(x1) - g(x) (here 0
g, x1, x, as
x1
x.
R
and
g
is of class
R(x1)/11x1- xll
A
Dg(x).
C1
on U and
A
Dg(x)
- x) + R(x1, x),
is a compact subset of
then g(x1) - g(x)
R(x1, x) /11x1 - xll - 0
where in
(x1 - x) + R(x1).
are written as column matrices), and
It follows that
If
(2)
U,
and a matrix function
A.
-
uniformly as
(x1
- xll
11x1
for
o,
-
and x
x1
(For later use, this is stated in more generality than is actually
needed here.)
g
If
(3)
is of class
on U and has rank m
C1
g
satisfies a Lipschitz condition: There is a
M
such that
8 > 0
at
then
x,
and constants
m and
0 < m < Ilg(x1) - g(x) II/11x1 - x11 < M
for
0 < 11x1
- x11 < s. Let
(4)
an open set in Rm; Then D(g 1)
g be a homeomorphism of the open subset U let
exists at
g be of class
C1
Rm
onto
on U and have rank m at
and equals the inverse of
g(x)
of
x.
Dg(x).
Remark. Previously, we appealed to the Brouwer theorem on invariance of domain to prove that the boundary and dimension of a manifold are well-defined. If one restricts oneself to considering manifolds for which there exist differentiable structures (and it has recently been shown by Kervaire (17) that this is a real restriction), then this appeal may be
to some extent avoided by use of the following theorem, which resembles the invariance of domain theorem.
1.11 Theorem.
Rm
Let
be a
f
into Rm which has rank m
phism of some neighborhood of Proof.
C1
mapping of the open subset U
at the point x
x.
Then
f
is a homeomor-
onto a neighborhood in Rm
We may assume that
x
and
f(x)
of
of
f(x).
are at the origin, and
INTRODUCTION
§1.
Df(o) = I,
that
the cube
within x1
lies in
C(r)
of
1/251
x2
and
the identity matrix.
in
Choose
(Why?)
r
small enough that
and so that the maximum entry of Df(x) - I
U,
for
0
13
in
x
If we let
C(r).
g(x) - f(x) - x,
is
then for
C(r), it follows from the mean-value theorem that
Ig(x1) - g(x2) I
<
(1 /2) Ix1 - x21
If(x1) - f(x2)I >
(1/2)1x1 - x21
and
First, we prove that every point under
f
of a point
x
of
y
of
To do this, let
Int C(r).
.
is the image
Int C(r/2) x
- o,
0
x1
= y,
and in general, Xn+1 = y - g(xn)
providing xn
lies in the domain of
= Ig(xn) - g(xn-1)I <
,
Note that
g.
Ixn+1
- xnI
=
so that
(1/2)IXn - xn-11,
IXn+1 - xnI < (1/2)nIX1 - x01
(1/2)1IyI
=
Summing this inequality, we see that Ixn+1
for m < n.
defined for all
g
if
It also follows that the sequence
n.
Also,
Int C(r).
that
f(x) - y.
Then
x.
x = y - g(x),
xn
xn
so that
f(x) = y, x
necessarily is
is a Cauchy seso that
Ix - xoI < 21y1,
that there is only one point
Second,
xn+1
xc,..., x1_ do, so that in fact
Hence it converges, say to
quence.
Hence
In particular, Ixn+1 - xoI < 21yI < r.
lies in the domain of
is in
XI < (1/2)m 21y1
-
x
as desired. of
such
C(r)
For the contrary would contradict the fact that
if(X1) - f(X2)1 > (1/2)1x1 - x21
for
x1
and
Third, we note that the inverse map continuous, since
Iy1
in
x2
C(r).
Int C(r/2)
f-
- y21 > (1/2)If-1(y1) - f-1(y2)I,
for
C(r) y1
is
and
y2
in Int C(r/2). Hence Int C(r)
f
is a homeomorphism of the open set
onto the open set
1.12
Then
f
as desired.
Corollary (Inverse function theorem).
subset U of Rm into x.
Int C(r/2),
Rm;
let
f
be of class
is a diffeomorphism of class
a neighborhood in Rm
of
f(x).
f 1(Int C(r/2))n
Cr
Cr
Let
f
map the open
and have rank m at
of a neighborhood of x onto
T.
14
DTFFERENTIAB.LE MANIFOWS
Since we know
Proof.
is a homQOfnorphism on some such neighbor-
f
hood, we need merely restrict the neighborhood to a smaller one on which the rank of
is
C
This is possible because the set of those points
m.
which det Df(x) / 0
t. t? Corollary (Implicit function theorem). y',
..., yP)
denctc The sp_eral point of Rn x Rp.
U
a neighborhood ...,
1)f',
0
Sn
..., yp)
Rri
x RP
f'(x, g(x)) = o for Define
Proof.
x Ii'
inLc
o
in
Let
(xi, ...; xn,
f be a Cr map of
f(o) = 0
Rn 1nLo Rp
such that
g is of class
DF(x)
g(o) = 0,
Cr.
Op by the equation
F(x,y} _ (x',...,Xn, f}(x,y},...,fp(x,y)) Since
and let
There is a unique func-
0.
In, tMs neighbuPhcxxi; : U -
Let
let
RP;
bc; non-singular at
g mapping a neigl-foorhood of
tion
and
of
:'h/a(y',
for
r
is open.
.
has the form
f/ay1
Af/fix it is non-singular at the origin.
Hence F
If we are to have
then we must have
f(x,g(x)) = 0,
has a local inverse G near
= (x,0)
F(x,g(x)) _ (x,f'(x,g(x)))
so that
GF(x,g(x))
Hence we must define
(x,g(x)) = G(x,o)
g(x) = trG(x,O),
where
0.
n
.
is the projection of
onto R. We leave it to the reader to check that this definition of
Rnk RP g(x)
will satisfy the requirements of the theorem.
1. 18
Corollary. Let U be an open subset of
be a map of class = 0.
Rn
There 1s a
Cr Cr
such that
f'
Rrn,
let f
has rank m at the. origin, and,
diffeomorphism g
U - Rn f(o)
of a neighborhood of the origin in
onto another such that gf(x',...,xm) Proof.
We may assume the submatrix bf',...,frn/r6'...,m)
of
INTRODUCTION
§1.
Df f
15
is nbr.-singular at 0, since this condition may be obtained by following by a suitable non-sirtgulas' linear map. Define F : U x
Rn-m
Rn by the
equation
F : U x Rn"m -+ Rn by tho equation F(x1,...,xn)
Then F has rank n at
-
0,
for DF
has the block form 0.
1
Hence F
has a local inverse
I
Now p is a
g.
neighborhood of the origin in R
Cr
diffeomorphism of a
onto itself, and (X11 ...,x:n,o,...,o)
The preceding corollary also holds if U is an opus subset of B. One merely cxLebds f to an open subset of R' and applies the result just proved. Note.
1.15
a map of class
Froblom s Lot
Cr such that
There Is a Cr diffeomorphism cute another such that
be an open subset of Rm, id. f: U
Ti
has rank
('
h
it
at the origin, and
Rn be
r(o) - o.
of a neighborhood of the origin Lit
Rm
at (xi,..., xm) = (xl,..., xl').
RA U he an open subset of Rm, let f: U - R1)
1.16 problem.' be a map of class origin, and
f(o)
Cr such that -
There are
o.
horhoods of the origin in
gfh (2r
1
f
has rank k in a heighhorhood of the qr diffeomorphisms g find h of noigh-
R' and Rm m
1
respectively onto others such that
k
Submanifolds and Imbeddings.
S2.
In this section we define what is meant by a submanifold of a differentiable manifold, and prove that every differentiable manifold is a submanifold of some euclidean space.
2.1
let
N'
Let N be an n-manifold, of class at least
Definition.
denote the manifold of class
the collection of coordinate systems.
called a submanifold of N coordinate systems of
HP
or
RP,
of class of
N'
for some fixed
p.
(Ui,hi)
Cr
obtaining by possibly adding to
P be a subset of
Let Cr
such that
the inclusion
I
:
P
is
P by
is an open subset
hi(Uin P)
form a basis for a differentiable structure of class P,
Int N.
if there is a covering of
One shows at once that the coordinate neighborhoods
relative to this structure on
Cr;
Cr
on
P
N
(Uin P, hi)
and that
P,
is a
Cr
imbedding.
It is appropriate to note here that ours is a different usage of the term submanifold than that employed by differential gecmeters.
a submanifold is the image of a one-to-one immersion
f
:
P - N
For them,
of one mani-
fold in another, rather than the image of an imbedding.
Note that under our definition, the number termined by the
C°°
P.
The real line
manifold R2,
R1,
r
is not uniquely de-
for instance, considered as a subset of
Is a submanifold of R2
of class
Cr
for every r
between one and infinity.
F
(a)
Exercise.
(b)
Exercise*.
Show that Bm and
STII-1
are
Cm
submanifolda of
Rm.
Given r with manifold of
R2
1 < r < m,
but not a
Let R2 be given the natural
CO
structure.
construct a subset of R2 which is a Cr+1
submanifold of
17
R2.
:r
sub-
is
2.2
Theorem.
is a submanifold of
If
N
p; and let
tem about
and
kf(p)
is a
f : M - Int N
of class
(V, k)
Cr
imbedding, then
(U, h) be a coordinate sys-
let
M;
be a coordinate system about
Assume
f(p).
are at the origin (why is this justifiable?).
Using
Corollary 1.14, there is a diffeomorphism g of a neighborhood of
in Rn
onto an open set in Rn gkfh-1(x1,
kf(p)
such that
...,
_
XII1)
(x1,
..., xm, 0,
..., 0)
(V, gk) is the required coordinate system about
The coordinate system
it carries a neighborhood of
f(p);
f(M) onto an open set in IF or Rm.
f(p) in
Show that this theorem fails if
Exercise*.
(a)
f(M)
Cr.
Let p be a point of
Proof.
h(p)
DIFFERENTIABLE MANIFOLDS
I.
is only a Cr
f
homeo morphism.
2.3
Let
Problem*.
set of points x
for which
f : Rn _. R1
f(x) a o;
be of class
of
fold of Rn
and dimension n-1.
of class
Cr
P.
Prove that
let P be the
grad f = of/a(x1,...,xn)
suppose
non-zero at each point x
Cr;
P
is
is a non-bounded submani-
State and prove a generalization to the case in which f maps Rn into
Rm.
Remark. The preceding sections suggest that a very natural way in which differentiable manifolds may occur is as submanifolds of some euclidean space. At once the question arises as to whether every differentiable manifold so occurs. The answer to this question is yes, as we shall now prove. First we treat a special case.
2.4
there is a
Theorem.
Cr
Proof.
about p
Let M be a compact m-manifold of class
Cr.
Then
imbedding of M in some euclidean space.
For each p in
such that
h(U)
(Why is this possible?)
Int M,
contains
choose a coordinate system
C(1),
If p is in Bd M,
and
h(p)
(U,h)
lies at the origin.
we require
h(U)
to contain
SUPMANIFOIDS AND IM®DINGS
§2.
Let V denote
C(1)n Ui°.
The sets W cover cover
h-1(Int C(1)),
Let
and let W denote
h, U, and V's
accordingly.
For
be the function defined in Lemma 1.3.
c
h-1(Int C(1/2)).
choose a finite subcollection W1,...,Wn which
M;
index the corresponding
M;
19
i - 1,...,n,
define ci(x) _ q,(hi(x))
for x in
(Pi (x) - 0
for
is of class
Each map ci: M -. R1
open sets U and M - 0,
for it is of class
C°°,
Rn x (Rm)n by the equation
4P1(x) h1(x),...,(Pn(x) hn(x))
is defined only for
hi(x)
fined to be zero outside
it becomes a
Ui,
Let
((P1(x),...,mn(x)).
f : M - Rn X Rm x ... X Rm
(Pi(x)
on the two
Ca0
and is well-defined on their intersection.
f(x) Of course,
in M-7
x
0 : M -. Rn be defined by the equation (x)
Define
U;
x C°°
in
.
but if it is de-
U1,
function on all of
M;
this we assume done.
f f(y).
is clearly of class
Then
q)i(x) - cpi(y)
hi
is
it follows that
(pi(x)
1-1,
suppose
f(x)
so that if x belongs to Vi,
i,
hi(x) - ci(y)
hi(x) - hi(y),
hi(y)
and
mi(x)
contradicting the fact that
1-1.
Since M is canpact, that it has rank m. system
To prove it is
for every
From the fact that
y must also. - gi(y) > 0,
Cr.
(Uk,hk)
Let
belong to Wk,
We must show, finally,
and let us use the coordinate
f at
x0.
We need to show that
contains a non-singular m x m matrix.
- u and
in a neighborhood of
of D(fhk1)
is a homeonorphism.
for computing the rank of
the matrix D(f hk1) . (u1,...,um)
x0
f
hk(xo) - u0.
x0,
Now
mk(x)
Let
- hk(x) = hk(x)
so that the m x m submatrix
hk(x)
for
x u)/att
is the identity matrix when u - u0.
Remark. The proof just given generalizes to the non-compact case, but there are a few additional difficulties. One arises from the difficulty
of finding finitely many coordinate systems which cover M. The other arises from the fact that f need not be a haneomorphism just because it These difficulties are disposed of in the following lemmas and problems (2.6 - 2.9); we then leave the proof of the general theorem to the is 1-1.
student (2.10).
20
DIFFERENTIABLE MANIFOLDS
I.
Definition.
2.5
the carrier of
tions
is a non-negative function defined on
p
is the closure of the set of points
p
Cr partition of unity on M
A
> 0.
If
defined on
Cr
of class
pi
form a locally-finite covering of x.
for which
x
M,
o(x)
is a collection of non-negative funcsuch that the sets
M,
and such that
M,
Ci = carrier pi
Ei pi(x) =
1
for each
The same definition holds for a topological space, except differentiabil-
ity has no meaning.
2.6
Cr
manifold
for each
In the case
as always).
such that
(pi)
Exercise.
is said to be dominated by the covering
(pi)
this holds for a space
X
Let
(separable and metric,
be a locally-finite covering of the se-
(Ci)
by compact sets;
X
let
be a sequence of positive
ei
There is a positive real-valued function o(x) < ci
2.7
such that
Use 1.3 and 1.4.
parable metric space constants.
partition of unity
Cr
1;
r - o,
Hint:
(a)
be a locally-finite open covering of the
(U1)
There is a
M.
carrier pi C Ui (U1).
Let
Problem.
Lemma.
for
x
in
5(x)
defined on X
C1.
Let M be an m-manifold;
let
(i
be an open covering
There is a locally-finite open refinement of a which is the union
of
M.
of
m+1 collections (B0,...,
such that the sets belonging to any one la 1
are pairwise disjoint.
We apply a fundamental theorem of dimension theory:
Proof.
is an m-manifold, then every open covering (t
such that no point of M for the case
M
lies in more than
of
M
M
has a refinement e
m+1 elements of e.
compact may be found in [3,' p. 67).
If
A proof
The result may readily
be extended to the non-compact case, but it is just as easy to prove the theorem from scratch.
This is done in 2.12-2.15.
Let L° _ (U j);
that Vj C Uj by
for each
let j;
(Vj)
let
be a locally-f'_nite open covering such (pj)
be a partition of unity dominated
(V3), by 1.4 and 2.6. We now may assume the indices are positive integers
Given an integer
n
such that
0 < n < in, consider any set
io,
...,
trl
SUBMANIFOIDS AND IMBEDDINGS
§2.
of distinct positive integers. sisting of those
Let
W(io,
in)
21
be the open set con-
for which
x
p1(x) < min((Pi (x), ..., Pi (x)) n 0
for all values of
other than
i
The collection Mn is to con-
io,...,in.
sist of all such sets W(io,...,in). Let
We must prove the elements of Mn to be pairwise disjoint. j0,...,jn be different sets of indices;
and
io,...,in
to the first set and not to the second, and not to the first.
then
belongs
k
belongs to the second set and
I
If x is in W(io,...,in),
is in W(jo,...,Jn),
suppose
then
x
if
m?(x) < mk(x);
Hence these two sets are disjoint,
ek(x) < mp(x).
as desired.
We prove that
= (0U...U(Bm
(B
is,...,in be those indices for which since
ing
Uj.
Then
Ti(x) > 0.
there are at most
= 1;
E qi1(x)
cpi(x)
covers
If
is in
x
M,
let
There is some such index,
m+1 of them,
for all other indices
= 0
M.
i,
by choice of the coverx
so that
is in
W(io,...,in).
The fact that x Pi
Each point
IS is locally-finite follows similarly.
has a neighborhood U on which all but a finite number of the functions Let
are identically zero.
is not identically zero on
U.
intersect are those for which
io, ..., in be those indices for which Ti The only sets (jo, ..., jk)
which U can
W(jo, ..., jk)
is a subset of
(is, ..., in)
these are finite in number.
(a)
vertices
Exercise*.
v1,v4,...
.
Let M be a simplieial complex in the plane, with
Let Vi
be the (open) star of
-be the barycentric coordinate of x with respect to
x is in V1,
of course.)
vi, vi.
and let
mi(x)
(It is zero unless
Carry out the preceding construction in this case,
and sketch the resulting sets of the form W(io)
and
W(io,11).
This exercise should make clearer the idea of the proof of the preceding theorem.
2.8
Problem.
Let M be a
that M may be covered by
m+1
Cr
manifold of dimension
coordinate systems
((Ui,hi)).
m.
Prove
Indeed,
I.
22 choose
DIF)a'F.RENTLABLE MANIFOLDS
so that it is the union of a countable collection of disjoint
Ui
where
open sets Vij,
2.9
hi(Vij)
is a bounded subset of
Rm.
Let X and I be separable metric spaces; let
Definition.
f : X -+ Y be continuous.
The limit set of
f
is defined as the set of all
y in Y such that for some sequence xn in X having no convergent subsequence,
converges to
f(xn)
(a)
Exercise.
L(f)
(b)
Exercise.
Let
y.
It is denoted by L(f).
is a closed subset of
if X
Y,
is locally
compact.
is empty if and only if and only if
f
f : X -. Y be continuous and 1-1. is a hcmecmorphism;
f(X)
L(f)n f(X)
is closed in Y if
L(f) C f(X).
2.10
Problem.
Let M be a
Cr manifold of dimension m.
Prove
that M has a Cr imbedding as a closed subset of euclidean space of dimension
(m + 1)2.
Remark. This theorem shows that any differentiable manifold is a submanifold of some euclidean space. This fact will be of crucial importance to us in the next chapter. A natural question to ask now is whether one needs to use a euclid-
ean space of such a high dimension in order to imbed M. The answer is no. It is a classic result of Whitney that R21°+1 will suffice (151. Milnor's R2m will suffice, but the notes contain a neat proof of this. In fact, difficulties involved in proving this are much greater.
2.11
Problem*.
Let M be a Cr manifold of dimension m.
that if A is a closed subset of M and' h1
borhood U
of A in RP,
euclidean space
closed if
h1(A)
which agrees with is.
is a Cr imbedding of a neigh-
then there is a Cr imbedding f h1
on
A.
Show
f(M)
of M in some
may be chosen to be
S 2.
Cover M - A by coordinate systems
Hint:
2, ..., m+2.
Let
U1 - U;
let
Definition.
(Ui, hi)
for
Then proceed as in 2.4.
A'space A has covering dimension at most m
if every open covering of A has a refinement of order at most m means that no point of A ment).
1 -
be a Cr partition of unity
91, ..., IPM+2
dominated by the covering U1, ..., Um+2.
2.12
23
SUBMANIFOLDS AMID IMBIDDINGS
lies in more than
If A is a closed subset of
(which
m+1 elements of the refine-
this is equivalent to the state-
X.
ment that every open covering of X has a refinement whose restriction to
A has order at most
If
m.
covering dimension at most
2.13 Int An+1
m, so does
If
U An - X.
dimension at most m,
be a refinement of (B0
Let AO
have covering
let (Bo
X,
which intersects Ai
whose restriction to
be a refinement of
lies in Ai+1'
Let
has order at most
Al
m.
be empty for convenience.
Assume that (Bn has order at most
6A(An+1 - An)
Let 0 be a refinement of Mn whose restriction to
m.
m.
Define (n+l
(1)
If U is in ($ n and intersects
(2)
For each V in
U'
f(V)
L°
in (B
as follows:
An_1,
let U be in (B n+i
which intersects An but not An-i, which contains V.
Then for each U in
denote the union of those V in C for which f(V)
fined and equals (3)
whose restriction to An
is a refinement of (Bo
has order at most
choose an element
%, let
An C
X.
it such that any element of (3o (B1
such that
and each set 6 t(Ari+1 - An)
Al
so does
and A has
At.
Given an open covering of
Proof.
A
Let An be a closed subset of X,
Lemma.
and
is a closed subset of
At
U.
Let
Ut
is de-
belong to (Bn+1'
For each V in e which does not intersect An,
let V be-
long to (Bn+1'
It is easy to check that (Bn+l
to An+i
has order at most m.
belong to (Bn
Let ($
for all but finitely many
covers X and that its restriction consist of those open sets which n.
Then (B
has order at most
in.
I. DIFF i
24
If B1
2.14 Corollary.
have covering dimension at most Proof.
Let A
NTIABIE MANIFOLDS and
B2
are closed subsets of Y which
so does
m,
B1 U B2.
and let An = X = B1 U B2
= B1
for
1
Since C (A2 - A1)
the preceding lemma.
covering dimension at most
2.15
most
Theorem.
A2,
apply
it has
in.
Every m-manifold M has covering dimension at
M.
Proof.
at most
m:
Any compact subset B
(1)
B lies in some cube
C,
a simplicial complex K of dimension C,
is contained in
n > 2;
there is a subdivision K'
(2)
which in turn is the polyhedron of m.
Given any open covering Q
K'
are a refinement of d .
This re-
in.
Cover M by a locally-finite collection of sets
are homeomorphs of compact subsets of
given An = B1 U ... U Bq, Int (B1U ... U Bp),
dimension at most
of
of K sufficiently fine that the collec-
tion of stars of the vertices of finement has order
of Rm has covering dimension
Let
Al = B1.
choose an integer p > q
and let m,
Rm.
An+1 a B1 U ... U Bp.
by 2.14
Bi which
In general,
such that An C
Each An has covering
The theorem follows from 2.13.
U. Mappings and Approximations.
In this section we define what is meant by a strong
approxima-
C1
in order to
tion to a differentiable map of one manifold into another (3.5);
do this neatly, it is necessary first to construct the tangent bundle of a differentiable manifold.
We then prove (3.10) an important theorem about
approximations, which states that if
f : M
N
is an immersion, imbedding,
or diffeomorphism, then so is any sufficiently good strong
g
Lo
(In the case of a diffeomorphism, we
f.
g(Bd M) C
C1
approximation
to assume that
Bd N.)
Definition.
3.1
Let M be a Cr manifold of dimension m;
x
let
0
A tangent vector
be a point of M.
to M at
v
assigning to every coordinate system
of size m x 1 If
if a
and
(U, h)
and
(U, h)
a
(V, k)
are two coordinate systems about
f
f
:
a
The entries of
the coordinate system
R
a column matrix
x0
are the corresponding column matrices,
u0 = h(x0).
If
about
then
are called the components of v in
(U, h).
(a, b) - M
is a Cr map, where M.
define v as assigning to the coordinate system
then v is a tangent vector to M at
x0,
is an interval in
[a, b]
Let
f(t0) = x0.
ponents of v in the coordinate system
Let
a1,
(U, h);
let
f(t) = h-1(a1. t,...,am t).
if
x0
is in
about
0;
if
x0
[-e,e]
well-defined on f
at
[-e,o]
or
[o,e].
D(hf)(t0),
as the reader may verify.
Int M,
is in
M
...,
f
Bd M,
It
f.
We should note that every tangent vector v to velocity vector of some parametrized curve:
If we
(U, h) the matrix
is called the velocity vector of the parametrized curve
some interval
x0, and
a
is called a (parametrized) curve in
vector of
a
with the following condition:
D(kh1)(u0) where
is a correspondence
x0
at
x0
is the
am be the com-
h(x0) = 0.
Define
is well-defined on f
will either be
In either case, v will be the velocity
x0. 25
26
DIFFERENTIABLE MANIFOLDS
I.
form an m-dimensional
xo
Note that the tangent vectors to M at vector space;
we merely take the components of v and w in the coordinate
system
and carry out addition component-wise.
(U,h)
This space is called the
definition is independent of coordinate system.
tangent space to M at
x0.
Definition.
3.2
Let M be a
the set of all tangent vectors to
v at x0 and
it
into the point
x0.
let
M;
Let
manifold.
Cr n
T(M)
denote
T(M) - M carry the vector
:
is called the tangent bundle of
T(M)
M,
is called the projection mapp If
IF X Rm or
is a coordinate system on
(U, h)
define
M,
h : n-1(U) -
IF x Rm by the equation
h(v) where
One checks that this
- In(v) x (a1,
...,
CM)
are the components of v in the coordinate system
(at,...,a°1)
The requirement that
ately imposes a topology on
h be a homeomorphism immedi-
be open and that
n-1(U)
T(M)
(U,h).
which is Hausdorff and separable (see
Exercise (a)).
is a 2m-manifold with n-1(Bd M)
T(M)
pairs
(n-1(U),h)
If M,
as its boundary, for the
will be coordinate neighborhoods on T(M). and
(U,h)
(V,k)
are overlapping coordinate neighborhoods on
then k h-1(x, a) - (kh 1(x), D(kh 1)(x)
where a
on h(U n V) x Rm, is of class
a basis for a differentiable structure of class Whenever we consider
T(M),
tiable structure, if
r > 1.
The inclusion mapp, i
zero vector at
(a)
x,
nate systems on
M.
with k(U n V) x Rm.
Let
Thus k h
(n-1m, h)
i
serves as
Cr-1 on T(M).
we will assume it provided with this differen-
i : M - T(M)
which carries x into the
and the projection mapping
Exercise.
a)
is written as a column vector.
so that the collection of pairs
Cr-1,
.
and
(U,h)
Show that k h
1
(V,k)
n,
are maps of class
Cr-1.
be two overlapping coordi-
is a homeomorphism of
h(U n V) x Rm
Conclude that the topology on T(M) is well-defined.
MAPPINGS AND APPROXIMATIONS
S3.
27
Show also that it is separable and Hausdorff. Exercise*.
(b)
x of
Show that for some neighborhood V of the point
there is a diffeomorphism g: a-1(V) -.. V x Rm such that
M,
and
is the natural projection of V x Rm - V,
the tangent space at x
it g
1
g is a linear isomorphism of
onto x x Rm.
This shows that
: T(M) - M is a fibre bundle, with Rm as
it
fibre and the non-singular linear transformations of Rm as structural group.
(See [12].)
3.3 Cr;
T(N)
let
g : M -. N be a Cr- ',
of class
a vector at
g(xo)
map.
Cr
There is an induced map
defined as follows:
assigning the matrix a
fine
Let M and N be manifolds of class at least
Definition.
Let v be a vector at
xo
(U, h), and let w be
to the coordinate system
assigning
dg : T(M) -+
to the coordinate system (V, k).
p
We de-
w if
dg(v)
s - D (kgh-1) - ac Note that
it is called the
dg is a linear map on each tangent space;
differential of
(a)
g.
Exercise.
Check that dg is well-defined, independently of
coordinate system. (b)
Exercise.
Show that if
v as velocity vector, then
dg(v)
f :
(a,b] - M is a
C1
curve
having
is the velocity vector of the curve
gf : [a,b] - N. (c)
Exercise.
If
geometrically, noting that M.
f
:
[a,b]
M is a
C1 map,
f maps the 1-dimensional manifold
The preceding exercise then merely states that
ize this formula to the case where the domain of sion greater than (d)
interpret
d(gf) -
f
[a,b]
df into
General-
is a manifold of dimen-
1.
Exercise*.
If
g
is an immersion, an imbedding, or a diffeo-
morphism, how is this fact reflected by the map dg ?
3.4 Definition.
Let M be a
Cr manifold.
I
Suppose we have an
28
DIFFERENTIABLE MANIFOIDS
I.
inner product
defined on each tangent space of
[v, w)
ferentiable of class
which is dif-
M,
0 < q < r, in the sense that the map of
Cq,
into R which carries v into
is of class
[v, v)
product is called a Riemannian metric on
Such an inner
Cq.
As usual,
M.
T(M)
I(vil
is used to de-
note 4[v, v). Let us note that it follows from the identity
[v, W) = that the real function
fold
Rn;
E
(7,w)
such that
we simply take the components
the coordinate system
i
T(M) x
,(v)
=
n(w).
There is a standard Riemannian metric on the mani-
Exercise.
(a)
jr + w ii2 - war - vii2
is continuous on the subspace of
[v, w)
consisting of all pairs
T(M)
L
Rn
: Rn
of v and w relative to
ai and 91
and define [v,w) as their dot product,
ai0i
Let
M
M at
If v and w are tangent to dot product in
metric on
M. (b)
(U, h)
let
be a Cr manifold;
T(Rn) of
x, let us define
df(v) and df(w).
Why do we need Exercise.*
Iv,W)
Check that this is a Remannian
be a Riemannian metric on a
M, and let
be a coordinate system on
Cq matrix function
to equal the
[v,w)
and
defined on
G(x)
M;
let
be the matrices
9
corresponding to v and w under this coordinate system. unique
Cq+1 immersion.
to be an immersion?
f
Let
f : M - Rn be a
Show there is a
U which is symmetric and po-
sitive definite, such that
[v,wi Let
d(x)
(V, k);
tr
=
a
G(x)
.
be the matrix function corresponding to the coordinate system show that G(x)
=
D(hk 1)
G(x)
D(hk 1)
Any correspondence assigning to each coordinate system a matrix
G,
with this equation relating G and
G,
(U, h)
about
x0,
is called classically
a covariant tensor of second order.
3.5
ai
Definition.
If
v
is a tangent vector to Rn at
be the components of 7 relative to the coordinate system
I
x,
: Rn
let Rn.
MAPPINGS AND APPROXIMATIONS
S3.
29
gives a homeamorphism between T(Rn)
The correspondence v- (x, a)
Rn x Rn which is an isomorphism of the tangent space at
x,
we define
by the equation
dof(v)
Consider
x x Rn.
C1 map and v is a tangent vector to M at
f : M -. Rn is a
If
onto
with Rn x Rn for convenience, and we write 7".
We often identify T(Rn) (x, a).
x
and
FI(M, Rn),
- (f(x), dcf(v)).
df(v)
C' maps of M into Rn.
the space of all
We topologize this space as follows:
Choose a Riemannian metric on M. and the positive continuous function C1 maps
set of all
g : M
Given the on
8(x)
C1 map
f : M - Rn,
let W(f, 8)
M,
denote the
Rn such that
IIf(x) - g(x) II < 5(x) and
Ildof(v) - dog() II < 8(x) I1II for each x in M and each v , o tangent to M at x. then g is called a s-approximation to approximation to
f.
fine C1 topology on
The sets W(f, s)
Rn.
g
is in W(f, 8), C}
form a basis for what is called the
Ft (M, Rn).
Given a differentiable manifold in some
If
It is also called a strong
f.
N, let us imbed it differentiably
The fine C1 topology on FI(M, N) is defined to be that induced
from the fine C1 topology on FI(M, Rn).
This topology is independent of the
Riemannian metric on M and the choice of the imbedding of N (see Exercise
(b) of 3.6). To obtain the coarse C1 topology, one
alters the definition to re-
quire that the inequalities hold only for x in some compact set takes sets W(f, 8, A) as a basis.
A; one
We shall make no essential use of the
coarse C1 topology. (a)
Exercise.
Check that the sets W(f,s) form a basis for a
(b)
Exercise.
The fine C 0 topology on F0(X,Y) (the space of all
topology. continuous maps of the separable metric space space
Given
Y) is obtained as follows:
ous function
s
such that
p(f(x), g(x)) < 8(x) Y.
simply, a C a topology.
0
The map
g
for all
f.
f
x, where
is called a strong
s-approximation to
into the separable metric
f : X - Y and a positive continu-
on X, let a neighborhood of
metric on
X
consist of all maps o
g
is a (topological)
Co approximation to
f,
or
Check that these sets form a basis for
DIFFERENTIABLE MANIFOLDS
I.
30
a C1map.
subset of
Given
3.6
Lemma.
Let N be a submanifold of
Let
(U, h)
be a coordinate system on
let
U;
NP be
C be a compact
let
M;
f : Mm
let
be a coordinate system on N about
(V, k)
there exists a
e > 0,
Rn;
f(C).
such that if g : M-,* N and
s > 0
g(C) C V ,
(*) l
h 1(y) - kgh 1(y)
a
I <
,
ID(kfh-1)(y) - D(kgh-1)(y)I <
for all y e h(C),
s
,
then
Ilf(x) - g(x) II < e ,
(**)
Ild0f(v) - dog(v) II < e Ir1I for each non-zero v tangent to M at a point x of Proof.
gn
Suppose not.
and a vector vn
Then there exists for each n a function
tangent to M at a point xn of
inequalities in (*) hold for g - gn,
y - yn - h(xn),
the inequalities in (**) do not hold for g - gn, and all large
xn
n.
verge to
Ilf(n) - gn(xn) II
Then both
f(xn)
ID(kfh-1)(yn)
IId0k(dfrn)) -
and
and d0k(dgn(un))
dk(dgn(un))
df(un)
and
and v - n,
x - xn,
and
kf(xn)
and
converge to
gn(xn)
in Rm x Rm
kgn(xn) f(x),
con-
so that
n - D(kgnh-1)(yn) dok(dg(_un))
converge to
'
anI < mIanl/n
II < 4 mlldh(~un) 11 /n
dgn(un)
converge to dk(df(u)).
Then
T(Rm).
-
Since the right side approaches zero, so does the left. d0k(df(un))
but
s - 1/n,
0.
dh(un) - (yn, mn),
Let
and
u, a unit tangent to M at x.
-
Ikf(xn) - kgn(xn)I < 1/n, so that
kf(x).
such that the
C,
By passing to subsequences and renumbering, we may assume
x andn/Irnll Now
C.
Hence both
d0k(df(u)), so that
Since d(k1)
dk(df(un))
is continuous,
converge to df(u), so Ildof("un) - dogn(un) II -' o.
Contradiction.
(a)
there is an
Exercise. c > o
Prove the converse of this lemma:
such that if (**) holds, so does (*).
Given
5 > 0,
MAPPINGS AND APPROXIMATIONS
$3.
M by compact sets;
covering of
and
C1
nate systems about
N.
let
tem of
and
g(C1) C Ui,
y
in
is equivalent to the fine
f
Let X(f, 81)
Exercise.
kighil(y)I < 81,
-
hi(Ci).
Show this
C1 neighborhood sysis independent
F1(M, N)
of the Riemannian metric on M and the imbedding of (c)
consist
X(f, 8,)
Ikifhil(y)
for all
be coordi-
(V1, k1)
Let
Conclude that the fine C1 topology on
f.
be a locally finite
(C1)
(Ui, hi)
ID(kifhj1)(y) - D(kighj1)(y)I < ei,
neighborhood system of
Let
respectively.
f(C1)
such that
of all maps g : M -+ N and
Let f : M
Exercise.
(b)
31
N.
be defined as in the preceding ex-
ercise, except that the second inequality (involving the derivatives of Show the resulting neighborhood system of
is deleted. the fine
neighborhood system of
CO
pology on
above.
Conclude that the fine
Show that the topology on
Exercise*.
induced
F1(M, N)
The coarse C1 topology on
Exercise*.
C0 to-
N.
C0 topology is strictly coarser than the fine
from the fine (e)
is equivalent to
f
is independent of the metric chosen for
FO(M, N) (d)
f.
f)
C1 topology.
F1(M, N)
is defined
Formulate an alternate definition similar to that given in the
statement of Lemma 3.6, and prove the two definitions equivalent.
Consider
Exercise*.
(f)
F1(R, R)
in the fine and coarse C1 to=pologies,
and in the uniform topology, whose basis elements are of the form W(f, e), such that
where
and W(f, E)
e > 0,
f : R - R,
lub If(x) - g(x)I < E.
is the set of all
g
Describe the path-component of the zero
function in each case. 3.7
Problem.
(1)
Given
such that whenever
imation to
f
f
e(x) > 0, f
: M-+ N and defined on
g : N - P
be
C1 maps.
M, prove there is a
is a 5-approximation to
then
f,
e(x) >
gf is an E-approx-
gf. (2)
81(x) > 0
Let
and
If
f
imbeds M
82(y) > 0
and g is a 82-approximation to (3)
exists 8(x) > 0 morphism, then
f
If
closed subset of
such that whenever
g, then
f
gf
N, prove there exist
is a 81-approximation to is an e-approximation to 9-f.
is a diffeomorphism, show that given
such that if f is a 8-approximation to f-1
is an E-approximation to
f-1.
e(y) > o, there f
and is a diffeo-
I.
32
Exercise.*
(a)
I
phism, or if
f(M)
D
MANIFCIDS is not a homeomor-
is not closed.
3.8 Definition. Consider the set N.
f
Show that (2) may fail if
The definition of the fine
to defining the fine
F"(M,N)
of
Cr maps of M into
C1 topology given in Lemma 3.6 generalizes The same definition is
Cr topology for this space.
used, except that the second equality (which requires closeness of the first partial derivatives) is replaced by the statement that the partial derivatives of
kighjl, through order
derivatives of on
x in C. The fine
for
CO° topology
We shall have no occasion to use these topologies.
r.
3.9 Definition. regular
si,
approximate the corresponding partial
is the one having as basis the union of the Cr topologies for
F°°(M, N)
all finite
within
kifhll
r,
Let
C1 homotopy between
and
fo
f
0
and
be
fl
C1 maps of M into
is a continuous map
f1
N.
A
ft: M x R -# N
such that
For some
(1)
for t> 1
ft = fo
e > o,
for
t < E
and
ft a ft
- E. is a
C1 map for each
(2)
ft
(3)
dft: T(M) x R - T(N)
The homotopy
ft
It is a differentiable If
fo
is continuous.
is said to be a differentiable
N is of class
ft: MX R
t.
and
Cl.;
this is stronger than conditions (2) and (3).
Cr hanotopy if it is of class f1
ft
is an immersion for each
a regular (or differentiable) homotopy is an imbedding for each
Cr.
are both immersions, we make the (standard) conven-
tion that a "regular (or differentiable) homotopy means that
C1 homotopy if the map
t.
ft
If
ft"
fo
between them always and
f1
are imbeddings,
is said to be an iso o y if
ft
t.
The two notions of regular and differentiable homotopies in fact differ only slightly.
It will appear later as a problem, to prove that the
existence of a regular hamotopy between of a differentiable one.
fo
and
f1
implies the existence
It is not clear, however, which notion should be
given preference as being more natural.
A regular homotopy is natural in
MAPPINGS AND APPROXIMATIONS
43.
that it is equivalent to the existence of a path the maps
and
fo
: R - F1(M,N)
0
between
C1 topology is used for the func-
where the coarse
ft,
33
However, the homotopies one constructs in practice are usually
tion space.
differentiable rather than merely regular.
This is the case with us, so we
shall have little occasion to use the weaker notion. (a)
Let
Exercise.
ft be a homotopy between
gt be a homotopy between go by the equations
and
let
g1;
ht(x) = f2t(x) ht(x) = g2t-1(x)
Show that
ht
f1 - g0.
for
t < 1/2
for
t > 1/2
and
f0
Define
ht: Mx R - N
C1 homotopy if -ft
is a regular (or differentiable)
let
f1j
and
gt
are. (b)
R
C1 homotopy which is not a
C1 homotopy.
differentiable (c)
Construct a regular
Exercise*.
Let
Exercise*.
ft
be a regular
C1 hanotopy.
by the equation
F1(M,N)
0(t)(x) = ft(x)
Show that
0
Define
is continuous, if the coarse
-
C1 topology is used for F1(M, N).
State and prove the converse.
3.10
I
f : M-* N be a
Let
Theorem.
sion or imbedding, there is a fine
C1 neighborhood of
of immersions or imbeddings, respectively.
there is a fine
C1 neighborhood
neighborhood and
Proof.
g
Let
[CI)
f
Int C1 C1
also cover and
is an immersion.
matrices of rank r by M(p,q;r); M(m,n;m)
f
f
is an immer-
consisting only
is a diffeomorphism,
then
M,
f(C1),
g is in this g
is a diffeomorphism.
and
(U1,h1)
and
(V1,k1)
respectively.
Then the matrix D(klfhj1)'(x)
rank m = dim M for all x in hi(Ci).
Now the set
If
be a locally-finite covering of M by compact
are coordinate systems about Suppose
f
such that if
f
of
If
carries Bd M into Bd N,
sets, such that the sets
(1)
C1 map.
has
We denote the set of p x q
it is considered as a subspace of Rp
is open in Rmn:
If one maps each matrix into the
sum of the squares of the determinants of its m x m submatricea, the set is
the inverse image of the open set R - (0)
under this continuous map.
34
I.
DIFFE NTIABIE MANIF'OIDS
where x
The matrices D(kifh.jl)(x), pact subset
M(m,n;m).
of
K1
C1 neighborhood of
the desired fine
Let
(2)
of
W1
f
C1 neighborhood W2
a fine
This choice of the
but
converges to
gn
gn(xn) - gn(yn)
f,
consisting of immersions.
specifies
si
Within this, there is
g which are 1-1 on each
consisting of maps
and
si-
as in 3.6.
f,
For otherwise, there would exist a number where
such that the
si
We have just proved that there is a fine
be an imbedding.
f
C1 neighborhood
Hence there is a
lies in M(m,n;m).
neighborhood of K1
is in hi(Ci), form a ccm-
i
dgn converges to
uniformly on
df,
for two distinct points xn and yn
of
f(xn) - f(x) , gn(xn) -, f(x)
similarly, gn(yn) -+ f(y) .
Ci,
By
C1.
passing to subsequences and renumbering, we may assume xn-+ x and
Since
yn - y.
Hence
f(x) - f(y); since f is 1-1, x - y. We refer
f
and g to coordinate systems:
xn - hi(xn)I
g - kighil,
and yn - hi(yn)
let
f - k1fhj1
zn i
for sane point
on the line segment Joining xn and
zn i - x,
(3)
Let
f be an imbedding.
neighborhood W2
with W.
o(x);
1.4).
f
Let
1-1;
on each
we shall denote this
We specify this neighborhood by a continuous function
be the distance in N from f(D1)
is a homeomorphism, this distance is positive.
function on M which is less than ei/2 Cc s-approximation to
1-1
C1
f whose intersection
C0 neighborhood of
p.
of M by compact sets, with Di C Int Ci ei
.
is non-singular.
thus we assume N is given a topological metric
covering Di
u
We have just proved there is a fine
consists of maps which are globally
intersection by W3.
By passing
,
of f consisting of immersions which are
Now we prove there is a fine
Ci.
u
contradicting the fact that Df(i)
i,
yn.
we have
0 - D?' (50 for each
,
(xn - yn) /Ilxn - 'in II
to a subsequence and renumbering, we may assume Then since
,
Then
0 - g(xn) - g(yn) - Dgn(zn,i) ' (i - yn)
on
f which lies in W2.
Ci
to
Let
(see
1
f(M - Int Ci); 6(x)
(see 2.6).
Suppose
There is a
for each
since
be a continuous
Let g be a
g(x) - g(y),
C1:
gn
and a sequence
where-
MAPPINGS AND APPROXIMATIONS
$3.
x
is in Di and y
is in D,,
and
35
Then
Ei < Ej.
p(f(x), f(y)) < Ei/2 + E,/2 < EJ. g
Since
is
on Cp x
1-1
is not in
so that
Cj,
p(f(x), f(y)) > Ej
by choice of E. Thus we have a contradiction. Finally, we prove there exists a fine
intersection with W3 phism,
L(f)n f(M)
consists of homeomorphisms.
is empty.
Let
g be a C0 s-approximation to
Let
First we show that
so that
g(xn) - y,
Hence
Let
(4)
fine
f(M)
L(g)n g(M)
so that
y if and only if
f(xn)
If x
is empty.
Note that
f be a diffeanorphism.
g(M)
is in
then
Ci,
which equals
L(g).
is a closed subset of N
Exercise.
There is a fine
if
g(Bd M) C Bd N,
Let
C1 neighborhood W4
The following lemma shows that there is a
such that if
f
and this neighborhood, and
(C1)
since
is.
Co neighborhood of
(a)
p(f(xn), g(xn)) -+ 0,
is not in L(f),
g(x)
consisting of imbeddings.
f
contains only finitely many
Ci
L(f) . L(g).
g is a homeaanorphism.
and only if
If xn is a sequence in M hav-
L(f) . L(g).
This means that
It follows that p(g(x), f(x)) < Ei,
Let
L(f).
for x in C
s(x) < Ei
if xn is in C. Hence
p(f(xn), g(xn)) < 1/i
be less than the
Ei
f which lies in W3.
ing no convergent subsequence, then each terms of the sequence.
whose
f
is a hameamor-
to the closed set
f(Ci)
be a continuous function on M such that
s(x)
f
Since
and let
Ei < 1/i
distance in N from the compact set
of
Co neighborhood of
g lies in the intersection of W. then
g
is a diffeomorphism.
f be a homeomorphism of X
into Y;
let
be a locally-finite covering of X by compact sets such that the sets
Int Ci cover
Prove there is a fine C0 neighborhood of
X.
g lies in this neighborhood and hcmeomorphism. (b)
regular
(X
is 1-1 on each
Exercise*.
then g is a
Let M and N be C1 manifolds; let
C1 hcmotopy between the maps t,
lar homotopy Ft between f0 t,
Ci,
and Y are separable metric, as always.)
is an immersion for each
for each
g
such that if
f
f0
and
f1
prove that for some and
f1
which is a
is also an immersion for each
t.
ft
of M into N. s(x) > 0,
any
be a If
ft
C1 regu-
s-approximation to ft
Prove similar theorems when
DIFFERENTIABLE MANIFCIDS
I.
36
is an imbedding and M is compact, or when
ft
ft
is a diffeomorphism be-
tween non-bounded manifolds.
(1) and (2) suffice for the
Generalize the proof of 3.10.
Hint:
case of an immersion or an imbedding; for the case of a diffeomorphism, (3)
and (4) may be applied to the map F : M x R - N x R defined by the equation
F(x, t) - (ft(x), t). Exercise*.
(c)
Show the hypothesis that M is compact is needed
in the preceding exercise when
is an imbedding.
f be a hcmeomorphism of M onto
Let
Lemma.
3.11
ft
and N are topological manifolds. such that if
g
There is a fine
lies in this neighborhood and
carries M onto
coordinate system
about
(Vi,ki)
prove that
for all
g(Ci)
We prove that
i.
f( i).
z
Sm-1
contains
of radius
1
+ si
f(Di)
cover
N.
kig(x) - kif(x) II < ei
II
g is onto;
z,
when
in particular, we
Said differently, the composite map
= Bd Bm under h lies outside
and the B(1 - si);
B(1 - 8i).
lies in B(1 - 8i)
radial projection from
of Rm - z
but not in h(Bm). onto
Sm-1.
Let
7.
be the
Then Xh carries Bm
Sm-1.
into
On the other hand, consider to the identity; we merely define This homotopy carries
Sm-1.
and
B(1 + 8i)
is a well-defined map carrying Bm into Rm,
h(Bm)
Suppose
such that for some
(see Exercise (a)).
Then the sets
contains'
image of the unit sphere we prove that
then g
and so that the sets Di = (kif)-1(B(1 - si))
(see 1.4).
h - (kig)(kif)-1
f
equals the unit m-ball
kif(Ci)
f(Ci),
Let g : M -* N be a map such that Ci,
Ci
small enough that the ball
si
is contained in ki(Vi),
is in
g(Bd M) C Bd N,
M by sets
also cover M
Int Ci
and the sets Choose
x
Co neighborhood of
We consider first the case in which Bd M is empty.
Choose a locally-finite covering of
still cover M
where M
N.
Proof.
Bm,
N,
x,
so
h(x)
Ft(x). lies outside
hISm-1
Ft(x) - t
:
Sm-1 _ Rm.
h(x) + (1-t)
It is homotopic x
for x
along the straight line between B(1-8i).
h(x)
Hence the map XFt is well-
in
MAPPINGS AND APPROXIMATIONS
S3.
Sm-1
xhle-1
defined, and is a homotopy between
Sm-1 and the identity.
Consider the homology sequence of the pair homanorphism of it induced by
o
Hm_1(sm-i)
(xh Sm-1)*
(xh)*
0 - ;(B, ;_1 (sm-1)
0 Sm-1),
is the zero-homomorphism of the infinite cyclic group H(Bm,
(xh),
because
and the
(Bm, Sm-1)
xh:
0 - Hm(Bm, Sm-1) -p
Now
37
xh maps Bm
into
Sm-1.
phism of the infinite cyclic group
And I
(xh ISm-1)
_1(Sm-1)
is the identity hommnorxhISm-1
because
is homoto-
This contradicts the commutativity of this diagram.
pic to the identity map.
(This argument is the only place where we use a bit of algebraic topology.)
Now let us consider the case in which Bd M D(M)
be the double of
if
: D(M)
and
be the double of
D(N)
There is a positive continuous function D(N)
and
p(f(x), i(x)) < E(x),
Now D(M) - MOU M1; 6 o
let
be the homeomorphism induced from
: D(M) -« D(N)
on D(N).
as in 1.1;
M,
E1
on
M.
a
Similarly,
is non-empty.
N;
Choose a metric
f.
e(x)
then
Let
g
on
let p
such that
D(M)
is onto.
determines two positive continuous functions p
induces two metrics
po
and
p1
on N.
I.
38
Let
g : M
if
N;
we require
g
DIFFERENTIABLE MANIFOLDS
g(Bd M) C Bd N,
f relative to the metric
to be an ee-approximation to
and an e,-approximation relative to and hence is onto.
(a)
g induces a map g : D(M) - D(N).
Exercise.
pi,
If p0,
then g is an E-approximation to
Then g is necessarily onto as well.
Prove the existence of the sets
Ci
used in the
preceding proof. (b)
fold
M;
let
Exercise.
Let A be a closed subset of the non-bounded mani-
B be a closed set containing A in its Interior.
there is a positive continuous function
f : B - M is a
C
0
B(x)
defined on B
8-approximation to the identity, then
Prove
such that if
f(B)
contains
A.
Smoothing of Maps and Manifolds.
§4.
We now approach the two main goals of this chapter.
The first of
our theorems states that if M and N are C00 manifolds, and is a
C1 immersion, imbedding, or diffeomorphism, then
f : M - N
f may be approxi-
C0° immersion, imbedding, or diffeomorphism, respectively.
mated by a
proof of this appears in 4.2 - 4.5, except for the case where
The
is a dif-
f
feomorphism and M has a boundary, which is postponed to Section
5.
The second theorem states that every differentiable structure of class
on a manifold M contains a
C1
C0D structure.
case where M is non-bounded appears in 4.T - 4.9;
The proof in the
the other case is treat-
in Section 5.
The fundamental tool needed for proving these theorems is the fol-
0
lowing "smoothing lemma."
where V CU.
a compact subset of the open set V, Cr map,
1 < r.
IF
Let U be an open subset of
Lemma.
4.1
Let
be a positive number.
8
Lot
or
Let A be
Rm.
f : U -' Rn be a
There is a map
f1: U -+ Rn
such that (1)
f1
is of class
(2)
f1
equals
CP,
for all
lDf1(x) - Df(x)l < 8,
on any open set on which
f
x.
is of class
There is a differentiable Cr homotopy ft between
such that Proof.
Rm,
V.
and
CP
A.
1
ft,
is of class
f1
in a neighborhood of
outside
f
If1(x) - f(x)1 < s
(3) (4)
CO*
ft
satisfies conditions (2)-(4)
borhood of U in Rm.
t.
f
is extended to a neigh-
We may also assume V is compact.
Let W be an open set containing A, a
C00 function on Rm which equals
0
outside
covering
and
It suffices to consider the case in which U is open in
since the other case reduces to this after
W.
for each
fo - f
1
such that W C V.
Let
T
in a neighborhood of A and equals
(Take a partition of unity dominated by the two-element
(W, Rm - A),
and let
7
be the function corresponding to W.) 39
be
DIFFERENTIABLE MANIFOLDS
I.
40
then
g(x) _ Y(x) f(x);
Let
Let
g
then
by letting it equal zero outside W;
g
We extend
g : U - Rn.
is of class
to Rm
Cr.
be a CO function on Rm which is positive on
q(x)
Here
and zero elsewhere.
c
Int C(E),
We
is a positive number yet to be chosen.
also assume that
9(x) dx= 1
S
,
C(E) since this may be obtained by multiplying
(p
by an appropriate constant.
Define
h(x) =
q, (y) g(x + y) dy C(E)
x
for
in
ment of
Choose -5E
Rm.
Then
V.
h(x) - 0
less than the distance from W to the complex
for
outside
(1 - v(x)) + h(x)
f1(x) = f(x) Since
h(x) = 0
and
`P(x)
for
Define
V.
condition (2) of the lemma
outside V,
x
.
is satisfied.
Now
f1(x) = h(x)
function of class
for x
in a neighborhood of
and
A,
h
is a
Cm:
h(x)
R(y) g(x + y) dy C(E)
(p(z - x) g(z) dz
x+C(E)
S 9(z
x) g(z) dz
Rm
Since
is a
c(z - x) - 0
tion,
h is clearly of class C. Thus condition (1) holds. Since
class of
f1
if
is not in x + C(E).
since
z
f1 = f - (1 - v) + h,
and
i
c
and h are of class
on any open set is no less than the class of
f.
C0 func-
C00,
the
Hence condi-
tion (4) is satisfied.
Now e
f1(x) - f(x) + (h(x) - g(x)),
small enough that
h is a 8-approximation to hi(x) i
and
ah
yi
and
yip
(x) -
are points in
g.
By a mean-value theorem,
gi(x + yi) (x + yi )
ax where
so that we need merely choose
C(E).
j
The functions
gi
and
agi/ax3
SMOOTHING OF MAPS AND MANIFOLDS
§4.
are uniformly continuous, so we need merely choose Igi(x)
-
and
gi(x*)I
41
so that
E
are less than
Iagi/axi(x) - agi/axJ(x*)I
if
S
x*I
then condition (3) holds.
< e;
Ix -
for
t < 1/3
for
a(t) - 1
and
t > 2/3
Define
(see 1.3).
ft(x) = a(t) fi(x) + (1 - a(t)) f(x)
Then ft f1 - f,
so that
Ift - fl - a(t) If1 - If
and
IDft - Df I
Finally,
ft
(a)
I
is of class
Exercise.
open in Rm and Iim
and
by
f1
- a(t)IDf1 - DfI < 8
on any open set where
then
and
f1
are.
Show that if U
is
Show that if U is open in
ft(U) C Fin.
ft(U) C Hn.
it need not be true that
f
so that this relation will hold.
(Hint:
Modify the con-
Replace
C(e)
throughout the proof.)
C(e) n FIm (b)
V,
<8
Consider the proof just given.
f(U) C Fin,
f(U) C Fin,
struction of
CP
Outside
f1.
Likewise,
also.
ft - f
.
and
f
Cr differentiable homotopy between
is a
a(t) - 0
Co' function such that
a be a monotonic
Finally, let
Exercise.
Consider the preceding theorem in the case
Show that the conclusions still hold, except that
r . 0.
no
IDf1(x) - Df(x)I < e
longer makes sense (since Df does not exist), and
ft
is of class
C
0
and
hence only a homotopy.
4.2
Theorem.
Let M and N be manifolds of class
f : M-+ N be a map of class is a Cp map h : M -+ N
Cr
(1)
h is a 8-approximation to
(2)
There is a Cr
Proof. (Ui, hi)
Let
Let
differentiable homotopy
for each
f,
be so
There
ft
between
and
f
t.
(Ui) be a locally-finite covering of M, Ui compact,
(Oil ki)
on
N.
with Wi contained in the open set 8
e(x) > o.
f.
is a coordinate system on M and
the coordinate system M,
Let
Let
such that
h which is a 8-approximation to
where
(1 < r < p < oo).
Cp.
Let vi,
(Wi)
and Vi
all that any 8-approximation to
f
f(Ui) is contained in be an open covering of contained in
carries Vi into
,1T
Q.
Let OP
DIFFERENTIABLE MANIFOIDS
I.
42
and approximates
on W1U ... U Wi_1
gi_1 - k1 fi_1 h11,
sider
of Rn or
lci(p 1)
hi(Ui) - Rn
which carries
Con-
8(1 - 1/21-1).
within
f
into the open subset
hi(Ui)
We apply Lensna 4.1 to obtain a Cr map gi
Hn.
which equals
Hn,
or
f:,_1: M -+ N is a Cr map which is of class
Assume
fo - f.
and is of class
hi(Vi)
outside
gi-1
:
hi(Wi).
We need Exercise (a) of 4.1 if both M and N -have bound-
aries'.
Let
be a close enough approximation to
h1(U1)
into
C00
on
gi
k1( 01).
Then
is well-defined by the equations
fi
for x
fi(x) - fi_1(x) kit
fi(x)
is of class
class
Cp
on any open set where and
Igi(y) - gi_1(y)I
a
fi
(Recalling that
to be sufficiently small, we
IDgi(y) - Dgi-1(y)I
Also,
h(x) - lima , . fi(x);
h is of class
Cp
(Here we use Lemma 3.6,
fi_1.
h
is well-defined because
fi(y) - fi+1(y) - ... on some neighborhood of large.
for
x,
sufficiently
i
and is a s-approximation to
To construct the differentiable homotopy between preceed as follows: gi_1
and
between
and
fi_1(x)
for each fixed
fi(x);
Fi
t;
Fi
will be a
is constant in
there is a number n Define
(a)
Cr
for
such that
outside
for
i < t < i +
F(x, t) - h(x)
for
f1_1
Define
I
t.> n and
for
t < 0
ft(x) - F(x, tan (at))
for
o < t < 1/2
ft(x) - h(x)
for
1/2 < t
A.
Vi.
t < 1
ft(x) - f(x)
Exercise.
U of
x
on Mx R; and for any compact subset B of x
f
in
M, B.
.
Strengthen the preceding theorem as follows:
A be a closed subset of M and let borhood
we
h,
8/21 approximation to for
t,
F(x, t) - Fi+1(x, t-i)
is of class
and
f
Cr differentiable homotopy Fi(x, t)
F(x, t} - FVx, t)
Then. F
f.
Lemma 4.1 gives us a Cr differentiable homotopy between
From this we obtain a
gi.
is of
gi
Furthermore, if we require
is.)
g1_1
in U1
x
for
s(x)/21 approximation to
Define
of course.)
outside V1
on W1U...U Wi.
f i
can make
Cp
gi hi(x)
Also,
gi_1 that it carries
already be of class
Cp
Let
in a neigh-
Then we may add the following to the conclusion of the
theorem: (3) fi(x) - f(x)
and
ft(x) - f(x)
for each x
in A.
SMOOTHING OF MAPS AND MANIFOLDS
S4.
In the above proof, take
Hint:
covering
of
(U, M - A)
to be a refinement of the
(Ui)
Let B be a closed subset of the open set U of such that h is of class
map h : M - N
outside
f
equals
M.
Co 8-approximations to
Corollary.
4.3
f : M - N be a
Let M and N be manifolds of class
Cr differentiable homotopy
between
ft
and
f
fi.
We remind the
is required to be an immersion for
t.
4.4
Corollary.
f : M - N be a
f1: M
let
f
f1;
(a)
ft
There is a
(1 < r < p < co).
N be a
Cr diffecmorphism and a
let
Cp imbedding
and
f1.
(1 < r < p < oc).
There is a
Cr differentiable isotopy
3s a diffeomorphism for each
Exercise*.
f
Cp;
Let M and N be non-bounded manifolds of class
Cp diffeomorphism )f1: M -' N,
f and
N be manifolds of class
Cr differentiable isotopy between
Corollary. : M
Let M and
Cr imbedding
and a
N,
4.5 Cp;
ft
let
Cp;
Cp immersion
There is a
(1 < r < p < co).
reader that by our convention (3.9), each
are, of course,
ft
This follows from Theorems 3.10 and 4.2.
Proof.
r - 0.
f.
Cr immersion
and a
f1: M-+ N,
and
the preceding theorem hold.
Consider the preceding theorem in the case
Show that the results still hold, except that h and only
Prove there is a Cr
in a neighborhood of B
Cp
and (1) and (2) of
U,
Exercise.
(c)
if Ui C U.
gi = gi_1
Assume the hypotheses of the preceding theorem.
Exercise.
(b)
and choose
M,
43
ft between
t.
State and prove the stronger forms of these three
corollaries, obtained by applying the results given in Exercises
(a) and
(b) of 4.2. (b)
f0
and
f1
Exercise*.
be
Let M and N be manifolds of class
Cr maps of M into
homotopy between f0
and
f1.
Let
N;
let
8(x) > 0.
ft be a
Cr;
let
C1 differentiable
Prove there is a
Cr
44
DIFFERENTIABLE MANIFOLDS
I.
differentiable homotopy Ft between
C1 maps of M into
be
f1
Let
tween them.
suppose
Rn;
t.
N;
let
ft
let
C1;
C1 differentiable hanotopy Ft for each
t.
Consider first the following problem:
For each
t,
is a 8-approximation to
Rm into
C1 regular homotopy between
is a
a compact subset of
let
U;
be as in 4.1;
7
fo
C1 regular homotopy be-
be a
ft
C1 map of the open subset U of Hm or
be a
is a
ft,
between them such that Ft
ft
Ft
such that
f1
Prove there is a
8(x) > 0.
Outline:
and
0
Let M and N be manifolds of class
Problem*.
4.6
and
for each
ft
8-approximation to
f
let
and
fo
let
or into
Hn f1.
Let A be
gt(x) = Y(x) ft(x).
Define ht(x) =
for suitably small
where
c,
cp
S p(s) gt+s(x) ds is positive on
(-e, e)
and
0
outside,
E
and S
Let
- 1.
Ft(x) = ft(x) .
differentiable hanotopy between and equals
Let
Lemma.
with WC V and V C U. Rn
: U
8(x) > 0,
of
fo
Then Ft
is a
Let
it
(2)
h is a s-approximation to
(3)
h(W)
(4)
If
is a U1
C
f
outside
h(U1)
Rn
onto
Rm.
is an imbedding.
Rm,
Suppose
Given
such that
V. f.
C00 submanifold of R.
is also a
0 of
Rm;
f(U1)
is a
let
a
to g : 0
f(U)
is a haneanorphism of
g(x...... xm) = (x1..... xm, gin+1(x),...,gn(x))
since it equals
f(U)
Rn be the inverse of this map.
Then
Cr,
C°° submanifold
C°° submanifold of R.
The restriction of
g is of class
A,
A.
is an open subset of U and
onto an open subset
and
is a
on sane neighborhood of
of : U _ Rm
Cr imbedding h : U -
h equals
Proof.
f1
be the projection of Rn
Cr map such that
there is a
then
and
and W be bounded open subsets of
U, V,
(1)
Rn,
- T(x)) + ht(x).
outside some neighborhood of
ft
4.7
f
(1
E
f(xf)-1.
Further, if of
g
on
f(U,)
f(U1)
is a
C00 submanifold of
Co structure, and note that
its induced
relative to this structure.
The map g
followed by the inclusion of
C00,
go: 0
Consider
Co
4.1, choose
on
be of class
C0D
go
af(V).
on any open subset of 0
.
which is of class
g0
We further require that go
where
go
is C. Let g :
g(x) - (x, g)(x)); it is an e-approxima-
h : U _ Rn by the equation
is properly chosen,
4.8
Let M be a non-bounded
Theorem.
Rn be a Cr imbedding.
submanifold of Proof.
an imbedding. n
of
h(x) - gaf(x).
h will be a B-approximation to
Let
s(x) > 0.
h : M - Rn which is a s-approximation to
jection
which is C.
g.
Define
f : M
Rn,
1 (x),...,gn(x))
outside
0 - Rn be defined by the equation tion to
into
to be an e-approximation to
and equals
nf(W)
Co map
defined by the equation
Rn-m,
fo(x)
is a
alf(Ui)
is the inverse of this, which is
f(U1)
go(x) _ (g
Using
the restriction
Rn,
(The converse is easy (2.2).) For impose
nf(U1) is of class Co.
to
45
SMOOTHING OF MAPS AND MANIFOLDS
S4.
f,
By 3.7, if
e
(1 < r).
Let
f - gnf.
Cr manifold
There is a such that
Cr imbedding
h(M)
is a
C0D
Rn.
Let
s
be small enough that any 6-approximation to
For each x in Rn
M,
df(x)
has rank
m.
f
is
Then for some pro-
onto a coordinate m-plane, the map d(af)(x)
has rank in,
46
MANIPDIDS
I.
is a Cr diffecmorphism of a neighborhood of x onto an open
xf
co that
Choose a
set in the coordinate m-plane, by the inverse function theorem.
covering of M by sets about
Ci,
cover
M,
xiflCi
approximation to
xifICi
for x in
8(x)< bi
is an imbedding; let
is a 8-approximation to
xif'
Bi-
be small enough that
a (x)
f': M - Rn is a 8-approximation to
Then if
Ci.
of Rn onto a coordinate
x1
si be a number such that any
Let
is an imbedding.
(Ui, hi)
Int Ci
equals the unit m-ball, such that the sets
hi(Ci)
and such that for some projection
m-plane,
then
such that for some coordinate system
Ci
Let Wi be an open covering of M,
xif,
xif'ICi
so that
f,
is an imbedding.
with Wi contained in the open set Vi,
and Vi C Int Ci. Let
is a Cr map which is a (i - 1/23-1)8(x) f,_1(Wk)
fj_,: M - Rn
As an induction hypothesis, suppose
f0 - f.
is a CO* submanifold of
Rn,
approximation to for
f
such that
We then construct a map
k < J.
Apply the preceding lemma to the map
f3.
f3_1h1 1
: h3(Int C3)
Rn
to obtain a nap fM Rn which is a a/23 approximation to equals
f3_1
outside V3,
f(W3)
and such that
f3_,, which
is a C0° submanifold of
Condition (4) of the preceding lemma guarantees we can choose
Rn.
that
f3 (Wk)
is
a CO* submanifold of Rn for k < j
As before, we let
h(x) - lim1
00
f3 so
.
and note that h satis-
f3(x),
fies the conditions of the theorem.
4.9
Corollary.
non-bounded manifold
(a)
If 1) is a
M, 1) contains a C° structure.
Exercise.
Let M be a non-bounded
a non-bounded C°° manifold. ding, prove there is a to
f,
such that
Let s(x) > 0.
Cr manifold;
If f
let N be
: M-+ N is a Cr imbed-
Cr imbedding h : M-+ N which is a 8-approximation
h(M)
entiably isotopic to
Cr differentiable structure on the
f.
is a
Cc* submanifold of
N,
and h is
Cr differ-
Manifolds with Boundary.
§5.
There are additional technicalities involved in proving our two main theorems 4.5 and 4.9,
for manifolds with boundary.
First, we need to prove the local retraction theorem (5,.5),
which states that a non-bounded- Cr submanifold
M
a neighborhood which is retractable onto M by a it is easy to find such a retraction of class
plane normal to M at tion of class
Cr requires more work;
mann manifolds (5.1 - 5.4).
Cr retraction.
Cr-1;
and collapses it onto
x
of euclidean space has If
r > 1,
roughly, one takes the To construct a retrac-
x.
one needs first to study the Grass-
One also needs a topological lemma (5.7).
From this, we can prove the product neighborhood theorem, which
states that Bd M has a neighborhood in M which is diffeomorphic with This in turn is used to construct a differentiable structure
Bd M x [0, 1).
on D(M),
the double of M The theorems for
(5.8 - 5.10).
M
then reduce to the corresponding theorems for
the non-bounded manifold D(M)
5.1
The Grassmann manifold Gp,n
Definition.
n-dimensional subspaces of Let
M(p,q)
(5.11 - 5.13).
denote the set of all
notes those having rank
r.
the same element of Gp,n nations of
n x n matrix
p x q
matrices;
The rows of any matrix A
a set of n independent vectors of Gp,n which we denote by
is the set of all
Rn+p.
)..(A).
Rn+p,
of
M(p,q;r)
de-
M(n,n+p;n)
are
so they determine an element of
Further, two matrices
A
and B determine
if and only if the rows of each are linear combi-
the rows of the other;
i.e., if A - CB
for some non-singular
C.
M(n,n+p;n)
is an open subset of
has a natural topology and
Rn(n+p)
(see 3.10),
C0D differentiable structure.
Let
so that it
Gp,n be given
the identification topology: V is open in Gp,n if and only if
>v-1
(V)
is open in M(n, n+p; n). We note that a.: M(n, n+p; n)
be open in
M(n, n+p; n).
For any
Gp,n
C
47
is an open map.
in M(n, n; n), the map
For let C
U
: A- CA
48
MANIFOLDS WITH BOUNDARY
53.
x-1(x(U))
M(n, n+p; n).
is open.
Theorem:
Gp,n
5.2
map
is the union of the sets
X(U)
M(n, n; n), so
Proof.
(1)
is a non-bounded
is of class
: M(n,n+p;n) - Gp,n
x
Gp,n
(P
n-tuple of vectors
the
dim pn;
Let us explain first the
onto a coordinate n-plane
Let U be the set of all planes on
.
which lie in it and project under
(v1,...;vn)
in
Each plane of U uniquely determines an
is a linear isomorphism.
it
C
COO.
of Rn+P
it
which is a linear isomorphism on
which
for all
l9 is an n-plane through the--origin in
If
there is some projection
Rn+p,
C(U)
C°° manifold of
is locally euclidean.
geometric idea of the proof.
C(U) is open in
with itself, so that
is a homeomorphism of M(n, n+p; n)
U.
Thus, each vector vi has p
ponents which may be chosen arbitrarily;
into
Conversely, each
the natural unit basis vectors for the coordinate n-plane. such n-tuple determines a plane of
s
since there are n
com-
such vectors,
U is homeomorphic with Rpn. More precisely, let
submatrix of A has rank
n x n
M(n,n+p;n).
Hence V . X(U)
Let [I Q]
of
M(n,n+p);
into
Gp,n.
It is
Now
q,
m
:
To . XT.
since
U--* M(n,p)
means that
Hence
the inverse of homeomorphism.
W
Then
To,
.(I
only if
P-1Q] =
TO(P-1
whence vo
M(n,p) 011 Q21;
[I Q,] Q).
P-1
q,([P Q])
since the equation
[P1 Q1] = C[P2 Q2],
since
is
into the matrix
is a continuous map of
be defined by the equation
induces a continuous map
P
and hence in
M(n,p)
of
To
x[P Q] -
x-1((?),
where
[P Q]
GpIn.
since MI Q1] = x[I Q2]
1-1,
is constant on each set
x([P2 Q2]) P21Q2.
let
onto V,
M(n,p)
Let
is open in
map the arbitrary matrix Q
Y
Some
Gp,n.
suppose it consists of the first n
Than U is open in Rn(n+p)
and non-singular.
it maps
n;
Let U be the set of matrices of the form
columns. n x n
be the general element of
x(A)
x([P1 Q1])
PT1Q1 . (CP2)-1(C%) _
of V into
q,o,o(Q) - q,((I Q]) . I-1Q = Q.
M(n,p);
Hence
T.
q,o
is
is a
49
DIFFERE 'IAB1E MANIFOLDS
I.
M(n,p)
U C M(n,n+p;n) m
V C Gp'n
(2)
We need only prove
Gpn has a C°° differentiable structure.
that two coordinate neighborhoods
of the type just constructed have
(V, (Pe)
cpN,NT,U,VN
class
Let
CO overlap.
be the corresponding maps and open sets
Then Y
which determine another coordinate neighborhood.
takes the columns
of the matrix Q and distributes them in a certain way among the columns of the identity matrix is of class Hence
C°°
(poT0 I = cp
so that Y is certainly of class C. Likewise R
I,
on the open set U of M(n,n+p;n),
-
is of class
C°°
since
9((P Q)) = P-'Q.
(on the open set (U n U)
where it is
defined).
Gp,n is Hausdorff.
(3)
so that any
e > 0
Choose
Then the matrix
Gp,n.
spaces of
rank at least
matrix within
B, in M(n, n+p; n).
e-neighborhood of neighborhoods of
X(B)
be distinct sub-
must have rank at least
( B I
2n x (n+p)
and
)L(A)
n+t.
of this one has
a
Let U denote the e-neighborhood of A, and V, the
n+1.
and
%(A)
Gp,n
(4)
For let
Then
and
X(U)
).(V)
a(B), respectively, since
has a countable basis.
.
Gp,n
Indeed,
are disjoint
is an open map.
is covered by
coordinate systems of the type constructed above.
(n+p):/n!pi
(a)
Exercise*.
Show that Gp'n is compact.
(b)
Exercise*.
Show that there is a
5.3
Lemma.
C°° diffeomorphism of
Gp,n
onto G
Let
f : M - Gp,n be a
there is a neighborhood U of x that
).f* = f.
Proof.
such that
f(x)
(f*
Let
and a
Cr map
is called a lifting of (V, qt0)
lies in V.
Given x
Cr map.
f
f*: U - M(n,n+p;n) over
cpo: V - M(n,p)
M,
such
U.)
be a coordinate system on
Now
in
is a
Gp,n,
as in 5.2,
Cm diffeemorphism.
MANIFOLDS WITH BOUNDARY
45.
50
Simply define
f*(x) - Y(co(f(x))) - [I 1pof(x)I (a)
Let
Exercise*.
onto V. and
be another coordinate system on Gp,n,
:p0)
xg-1 is
such that
with V x M(n, n; n)
the natural projection of V x M(n, n; n)
Let
Show that there is
be as in 5.2.
(V, 4V 0)
,-1(V)
of
a diffeomorphisn g
.
g
the
ti
let a map hx, carrying
For each x in V n V,
corresponding diffecmorphisn.
into itself, be defined by the equation
M(n,n;n)
(x, hx(P)) - a g _'(x, P)
Show that
and
is merely multiplication by a non-singular matrix Ax,
hx
show that the map x - Ax
is continuous.
This will show that
: M(n,n+p;n) - Gpn is a principal fibre bundle with fibre and structural group
(See [12].)
M(n,n;n).
5.k
tangent to M at If we identify 3.5.
Let
Definition.
then
x,
Rn be a
f : M
df(v)
is a tangent vector to Rn at
with Rn x Rn, then
T(Rn)
f(x).
df(v) . Mx), do fnv ), as in
As v ranges over the tangent space at
m-plane through the origin in Rn.
If v is
Cr immersion.
x,
d0f(v)
The tangent map
t
ranges over an
of the immersion
will be the map t : M - Gp,m which assigns this m-plane to the point (n . m + p,
be a coordinate system on
(U, h)
is given by the equation evaluated at t
x.
of course.)
Let
Hence
f
t(x) . 7.(D(fh1)),
and
h(x),
is of class
where the matrix D(fh
is the projection of
X
Then locally the map
M.
M(m,m+p;m)
onto
1)
t
is
Gp,m.
Cr-1.
Similarly, one has the normal map of the immersion, n : M - Gm'p, which carries D(fh-1)
x
into the orthogonal complement of
is non-singular;
is m x m and non-singular. 7.([R(x) I]),
where
suppose it is of the form
t(x).
Now the matrix
[P(x) Q(x)]
where
Then locally n is given by the equation
R - - (P-Q) 1
.
Hence n is also of class
P n(x)
Cr-1 .
a
(local retraction theorem).
Theorem
5.5
and a Cr retraction r : W -+ f(M).
y in
for
(We recall this means that
r(y) . y
Consider the normal map n : M - Gm,p and the tangent map
of the imbedding
Gp'm
There is a fine
and
i(x)
Cr-1.
Both are of class
(p - n - m).
f
such that for any map $ which
neighborhood of n
Cc
lies in this neighborhood,
are independent subspaces of
t(x)
i.e., their intersection is the zero vector (see Exercise (a)).
Rn,
a map
of class
: M -. Gp..m
Pi
(In the case
Theorem 4.2.
n is only of class
r - 1,
Let
t is also a map of class
?1(x);
Co,
and we need to
be the orthogonal complement
t(x)
Cr.
Let E be the subset of M x Rn consisting of pairs that v lies in the subspace
(x, v)
that E is a Cr submanifold of Mx Rn of
in a neighborhood of t*(x)
be
t*
linearly independent set. n*
and
T*
Let
(U, h)
We prove
Cr liftings of
1'I
is a p x n matrix,
and
t
and
jointly form a
be a coordinate system about x
are defined on U;
let
h(x)
(u1,
..., um).
g : U x Rn,.. Rm x Rn by the equation
Define
rA(x)
g(x,(v1,...,vn))
Than
$,
n.
is an m x n matrix; the rows of A and B
- B(x)
such that
let k and
dimension
Then Z 1 (x) - A(x)
x.
such
(If we used the map n instead of
P1(x).
E would be what is called the normal bundle of the imbedding.)
Given x in M;
Choose
Cr which lies in this neighborhood, using
use Exercise (c) of that section.) of
f(M)
f(M).)
Proof. t : M
f : M - Rn be
Let
There is a neighborhood W of
M non-bounded.
Cr imbedding;
51
DIFFF.EENTIABLE MANIFOLDS
I.
(U x Rn, g)
_
l-1
(h(x), [v1... vn]. L B(x) J
is a Cr coordinate system on M x Rn.
Now if
(x, v)
lies in E. then v is some linear combination of the rows of A(x), so that
v . [y' ...
for some choice of yt, ..., yp. (x, v)
g(x, v)
lies in E.
As a result,
yp o
... o
A(x)
B(x)
Conversely, if v is of this form, then (x, v)
lies in E
if and only if
lies in h(U) x R. Hence the coordinate map g carries the in-
tersection of E with U x Rn onto the open subset
so that E
is a Cr submanifold of M x Rn.
h(U) x Rp
of 0+1),
52
I.
DIFFERENTIABLE MANIFOLDS
z
y
M OF
f(c)
Consider the f(x) + v;
Cr map of M x Rn -. Rn which carries
(x, v) into
let F be the restriction of this map to E.
We prove that coordinate system constructed above.
dF
has rank n at each point of Mx 0. (u',...,um,y',...,yp,o,...,o)
g(x, v)
on
Use the
E which was
Then
Fg '(u, y) = fh-'(u) + (y'... yp]
A(h-'(u))
so that
and
A(h-'(u))tr
a(Fg ')/ay
a(Fg') /auk
=
,'
a(fh') /auk + ((y' ...yp] . a(Ah ' ) /auk) tr
The last term vanishes at points of
g(M x o),
since y - 0
at such points.
Hence
D(Fg') at points of g(M x 0).
_
[D(fh')(u)
A(h'(u))tr]
The columns of D(fh ') span the tangent plane t(x);
MANIFOLDS WITH BOUNDARY
$5.
the rows of A
span the plane
53
by choice of A these spaces are
A(x);
independent.
F : E - Rn is a homeomorphism when restricted to the sub-
Now,
furthermore, each point of M x o
space M x 0;
has a neighborhood which
F maps homeomorphically onto an open subset of Rn
(by the inverse func-
in E
It follows that there is a neighborhood of Mx o
tion theorem).
which F maps homeomorphically onto an open set in Rn
in E
There is also a neighborhood of Mx 0
(see Lemma 5.7).
on which dF
has rank n;
let V be the intersection of these neighborhoods and set W - F(V). There is a natural projection a
Cr diffeomorphism of V onto W,
tion of W onto
F it
F-1
- r
M.
Since
F
is
is the required retrac-
f(M).
Construct the fine
Exercise.
(a)
of M x Rn onto
it
Co neighborhood of the normal
map n required in the preceding proof. Use the fact that
Hint:
hoods of
n(xo)
5.6
and
is an open map to construct neighbor-
consisting of subspaces which are independent.
t(x0)
Let
Corollary.
x
There is a neighborhood W of
M non-bounded.
f : M -. N be a Cr imbedding; f(M)
in N and a Cr retraction r : W -
f(M) . Let
Proof.
hood U
of
(a)
is the required retraction,
Exercise.*
M and N non-bounded. simply when N ric.
Let
f : M
There is a neighbor-
of U onto
gf(M).
defined on W - g 1(U).
N be a Cr immersion, with r > 2;
The normal bundle E
of
f
may be described most
is a submanifold of RP having the derived Riemannian met-
In this case, let E be the subspace of M x RP consisting of pairs
(x, v)
such that v is a p-tuple representing a vector tangent to N and
normal to case,
N - Rq be a Cr imbedding.
in Rq and a Cr retraction ro
gf(M)
Then g Irog
g:
E
df(w)
for each w tangent to M at
is described as a subset of M x T(N).)
manifold of M x RP.
x.
(In the more genera).
Then
IR
is a
Prove the tubular neighborhood theorem:
Cr-f
010.
I. DIFFERENTIABLE MANIFOLDS
54
If f : M - N C RP is a Crimbedding, there is a neighborhood U
of M x 0
in the normal bundle E and a Cr-1 diffecenorphism h of U
with a neighborhood of
in N such that hIM x o
f(M)
Let F : E -y RP be the map
Hint:
space
X.
let
r
h . rF.
Let A be a closed subset of the locally compact
Lemma.
Let
let
F(x, V) - f(x) +-V*;
be a retraction of a neighborhood of N onto N;
5.7
f.
.
X -+ Y be a homeanorphism when restricted to A;
f:
of A has a neighborhood Ux which
pose each point x
phically onto an open subset of Y.
neighborhood of A
onto an open subset of
maps homeomor-
f
is a homeomorphism of some
f
Then
sup-
Y.
(X and Y are separable met-
ric, as always.)
Let U be the union of the neighborhoods Ux.
Proof.
locally 1-1 at each point of such that
fIB
is
1-1,
fIB
where xn and x are in
f(x),
(1)
If
prove there is a neighborhood of be the E/n-neighborhood of
and yn
from some
f(xn)
If
U.
where
C,
f is
of Un such that
on no set
1-1
f(xn) - f(yn).
Then
C.
locally
f(x)
f(y);
-
1-1
at
x,
(2)
If
C
since
is
f
so we could not have
Un be the a/n-neighborhood of
f
is
1-1
is
f
fin,
on no set
of A - Un such that
1-1.
Let Un
U1
is can-
there are points xn
x and y will be points on C,
1-1
c
and so that
f is
1-1
But
x - y.
for large
f(xn) - f(yn)
where
f
on
1-1
is
1-1
n. C U A,
on V U A.
is small enough that
on U1
is
f
U1
(using (1)).
there are points xn of Un - A and yn
f(xn) - f(yn).
numbering, we may assume xn -' x,
we
0,
By passing to subsequences and
C,
U1UA,
on
1-1
is a compact subset of U and f is
is compact and lies in U, If
is
f
we prove there is a neighborhood V of C such that Let
Since
onwards.
is small enough that
c
so it con-
x.
renumbering, we may assume xn -+ x and yn -. y; of
n
on whose closure
C
f(xn) -+
is open in Y,
f(Ux)
is a compact subset of U and
C
pact and lies in
Now
B.
xn must converge to
is a homeomorphism,
For let
is a homeomorphism.
tains all elements of the sequence fjUx
is
f
Furthermore, if B is any subset of U
U.
then
Then
By passing to subsequences and re-
where x
is a point of
C.
Then
f(yn) -
S 5. MANIFOIDS WITH BOUNDARY
Now
f(x). 1-1
at
fIC U A is a homeomorphism, so that yn -. x. so we could not have
x,
such that Vn is a compact subset of U and we may choose a neighborhood Vn.1
f
is a compact subset of U and
neighborhood V of
is
f
such that
Then
is
f
on the
1-1
There is a non-negative
let
Let
(U.).
be a partition of unity dominated by the
(mi)
dim M - m.
For each
x in Ui n Bd M.
is zero for
positive at
set in Hm about so that
k(x)
he (x)
at points of Let
mth coordinate function
Furthermore, if
onto an open set in
a(hY k-')/auJ - 0
D(hik ') is non-singular,
the
i,
k(x) - (u',...,um)
then
x,
k-')/sum
a(I
is
is a non-singular transformation of an open
For hi k-1
k(x):
g(x) - 0
be a locally-finite covering of M by
((Ui, hi))
Let
is another coordinate sustem about the point
Bd M.
By (2),
1.
coordinate systems;
since
on VnUA.
1-1
on Vn+tU A.
i-1
Let M be a Cr manifold.
has rank
Proof.
0,
is
An,
Cr function g on M such that for each x in Bd M,
dg(x)
hi(x)
Define VO
A.
Lemma.
real-valued
covering
is empty.
of the compact set VnUAn+I
Let V be the union of the sets Vn.
and
n.
In general, suppose Vn is a neighborhood of
to be the empty set.
5.8
is locally
f
Let A be the union of the increas-
of compact sets, where A0
AOC A1C...
ing sequence
But
for large
f(xn) - f(yn)
We now prove the lemma.
(3)
Vn.1
55
Hm,
and
at points of FP-l,
h
k-'(u...... u1°-',0)
for
j < m.
Since
a(bY k_1)/21u m must be non-zero at points of
RD1-';
this derivative must be positive
is non-negative for all x, RTII-' .
g(x) - F.i Ti(x)
Furthermore, if
q(x).
Then
k(x) - (u1,...,um)
g(x) - 0
whenever x
is in
is a coordinate system on M,
then
(gk 1)/sum - Ei hi (,ik-.')/sum + L Whenever x
is in Bd M,
the first term vanishes, because
second term is strictly positive. Bd M.
a(hY k-')/sum
Hence
dg(x)
has rank
hT(x) - Oj I
the
if x is in
I. DIFFERENTIABLE MAIIFOIDS
56
5.9
phism p
There is a
Cr manifold.
of a neighborhood of Bd M with Bd Mx (0, is in Bd M.
x
whenever
(x, 0)
Let M be a
Theorem.
Cr diffeomor-
such that
1)
p(x) _
Such a neighborhood is called a product
neighborhood of the boundary. Proof.
Cr retraction
g on M
r : W
By 5.8, there is a non-negative
Bd M.
such that if
is in Bd M,
x
is in Bd M,
then
g(x) - 0
by the equation
f : W - Bd M x(0, e)
Define
f(x) _ (x, 0) and
and
has rank
dg(x)
is non-singular:
df(x)
Cr function
f(y) _ (r(y), g(y)).
be a coordinate system about
k(x) _ (u1,...,um)
Bd M in M and a
there is a neighborhood W of
By 5.6,
if
1.
x
For let
let h - (klBd M) x i.
x;
D(hfk 1)
one uses Theorem 1.11 to prove that each x in Bd M has a neighborhood
Ux which Since that
f
carries homeomorphically onto an open subset of Bd M x to, o).
is a homeomorphism when restricted to
f
is a homeomorphism of some neighborhood U of Bd M in W onto
f
a neighborhood V of Bd M x 0
in Bd M x to, oa).
2.6, we may construct a Cr function (x, t) of
a restriction of
5.10
t < 8(x).
Then p(U)
contains
(x, 0)
and
PO: U0 - Bd %x[o, 1)
of the boundary in MO
and
and let p1.
A
such that the point
Let
be the diffeo-
s
(x, t)
into
Bd M x to, 1),
and
Let M be a Cr manifold with non-empty boundary.
Definition.
M,
D(M),
(x, 1)
is the union of MO - M x0
identified whenever x
may impose a differentiable structure on D(M)
D(M),
on Bd M
p will satisfy the demands of the theorem.
Recall that the double of
with
Using Exercise (a) of
onto itself which carries
oo)
and let p = of.
(x, t/8(x)),
8(x) > 0
lies in V if
Bd M x to, a*)
morphism of Bd M x (0,
M x1,
Bd M, it follows from 5.7
and
and
M1.
M1 a
We
Let
be product neighborhoods
Let U be the union of
P : U -. Bd M x(-1, 1)
is in Bd M.
as follows:
Bd Mix(-1, o]
p1: U1
and
UO
and
U1
in
be the homeomorphism induced by po
Cr differentiable structure on D(M)
is well-defined if we
§5. MANIFOLDS WITH BOUNDARY require (1)
and
to be a
P
in D(M)
M1
57
Cr diffeomorphism, and (2) the inclusions of
MO
Cr imbeddings.
to be
Now the differentiable structure on the choice of the product neighborhoods
p0
D(M)
and
depends strongly on However, the differ-
p1.
entiable manifolds arising from two such choices are diffeamorphic, as we shall prove later (6.3).
(a)
D(H2)
Exercise*.
Impose two distinct differentiable structures on
by using different choices for the product neighborhood of
Bd H2.
Show that the resulting differentiable manifolds are diffeomorphic, but that
the diffeomorphism may not be chosen as an arbitrarily good tion to the identity on each copy of (b)
H2.
If M and N
Exercise*.
C1 approxima-
are
Cr manifolds, show that Mx N
has a Cr differentiable structure such that each inclusion M - M x y and N-, x x N is a Cr imbedding.
Theorem.
5.11
manifold
M,
then
Proof. empty.
If
9) is a
5) contains a
Cr differentiable structure on the
C0D structure.
We have proved the theorem in the case where Bd M
Otherwise, let us consider the non-bounded manifold D(M),
provide it with a
Cr differentiable structure, as in 5.10.
structure contained in this, and let D'
is
and
Choose a
Co
denote the corresponding differen-
tiable manifold.
Consider the map defined in 5.10.
The Cr structure on Bd M
note the resulting differentiable manifold by diffeomorphism of the
where
P-1: Bd M x (-1, 1) -D(M)
P
is the map
contains a C0D one; let us de(Bd M)'.
C00 manifold (Bd M)' x (-1, 1)
Then
P-1
is a Cr
with an open subset
U of the C0 manifold D' I. Now there is a Cr diffeomorphism h U, which is C0D in a neighborhood of (Bd M)' x (-1/2, 1/2). h(Bd M x 0)
of
(Bd M)' x 0,
(Bd M)' x (-1) 1)
and equals
(Here we use Exercise (b) of 4.2.)
is a C00 submanifold of
D'.
P-1
with outside
Then the set
58
I. DIFFRRENTIABIE MANIFOLDS
h induces a
Now merely define i
f(x) - hP(x)
f
if x is in U,
and the set
fi(M)
f(x) - x
C°° submanifold of
otherwise.
If
Cr imbedding of
is a
fi
we
onto itself;
D'
D'.
Let M and N be non-bounded
Lemma.
5.12
is a
of
and
is the inclusion mapping, then
: M - MOC D'
in D',
Cr diffecanorphism
M
Our result follows.
Cp manifolds;
let
f : M - N be a Cr diffe miorphism. Identify N with N xo in N xR for Given a positive function
convenience.
function
on M
s
mation to
Cp imbedding of M into N xR which is a e-approxi-
Then there is a
f.
there is a positive
on N x R,
such that the following holds:
g be a
Let
a
Cp diffeomorphism h
of N x R
onto itself,
such that (1)
h carries
(2)
h
g(M)
into
N.
is an e-approximation to the identity, and equals the identity
outside N x (-2/3, 2/3). Let
Proof.
and equals
t < 1/3
Let
k > 3.
enough that
P(t)
for
0
be a monotonic C0 function which equals
be the projection of
it
g(M)
lies in
N x
is a diffeamorphism of M onto
N x R
(-1 /3,
N x 0.
for all
k >I 0'(t)l
onto
Let
N x 0.
then
t;
be small
8
ng
1/3), and small enough that (By 3.7,
as closely as desired by taking
of - f
imate
Let
t > 2/3.
s
for
i
ng may be made to approxsmall enough.)
Then the general point of g(M)
is of the form
where
y
is in N and
function on <
si nce
1/-1
g((ng) -1 (y, 0)), of
with
N, (
f
(y, (p(y)),
T
is a real
'I
9
-1/3 < (p(y)
Y, R Y NN (
m(Y)
X
is a Cp f
s
y,0)
9(x) = (9,Vt9))
y.
Define
h : NxR - NxR by
R
the equation h(Y, s)
I
(y, T(Y, s)),
where
VY, s) - s - v(y)s(I5I).
M
is N 1'
k
IT
55. MANIFOLDS WITH BOUNDARY so when
Now I(p(y)I < 1/3,
59
h carries
Hence
s - p(y), 0(18I) . 1.
g(M) in-
to N x 0. If Isl > 2/3, 9(IsO - 0 and h is the identity. 5, we may make
By proper choice of
For this, we note that
tion to the zero-function as desired.
using 3.7, we see that if g approximates
g(ag)-1(y, o);
of - f,
ng approximates
enough,
as close an approxima-
c(y)
approximates ff
1
approximates f 1,
(ag)-
Let
. identity.
g(ag)
is an
q)(y)
so(y) - min e(y, a)
N must be imbedded in some RP before this makes sense;
-1 < s < 1.
is imbedded in Rp+1.
then N x R
closely
and
be small enough that
8
e0(y)/2k approximation to the zero function, where for
f
(y, (p(y)) -
Now
I(p(y)I < eo(y),
so
II h(y, s) - (y, s) II - IY(y,, s) - sI < E0(y) Let v be a unit tangent to N xR at
as desired.
v2 be its components tangent to II
and y xR,
N x s
- v II < II dh(v1) - 'vi II
dh(
eo/2, since
The second term is less than
+
II
let v1
(y, s);
and
respectively.
dhCC2) - v2U
Ia!/as - 11
I1P(y)A'(1s1)I
The first term equals
eo/2.
HdY(v,) (I
-
IIo(IsOdrv(1) II < eo/2
as desired.
5.13
f : M - N
Theorem.
be a Cr diffeomorphism
is a Cp diffeomorphism Proof.
tion to
Let M and N be manifolds of class
f
and
Let
8
be small enough that if
then f
then
f1
is a
f1
is onto.
81
and
s2
to
as a sub-
By Prob-
f,
re-
and h : D(N) -
is a 82-approximation to the identity, then bg is a 8-approximation f.
Let g : M - D(N) to
Consider N
on M and D(N)
spectively, such that if g is a 51-approximation to D(N)
f.
Co 5-approxima-
is a Cr imbedding of M into D(N).
lem 3.7, there are positive functions
let
5(x) > 0, there
f1: M -+ N which is a 8-approximation to
f1(Bd M) C Bd N,
manifold of D(N);
Given
(1 < r < p < a).
Cp;
f.
If
81
be a Cp imbedding which is a
is small enough,
neighborhood Bd N x(-1, 1)
81-approximation
g will carry Bd M into the product
which is used to give
D(N)
its differentiable
I. DIFFERENTIABLE MANIFOLDS
60
Then we may apply the preceding lemma, to obtain a
structure.
phism h of D(N)
which carries
enough, we may make sure
Then
h
is a
maps M
hg - f1
D(N).
Since
into
5.14
there is a
Problem.`
N
onto
between
ft
for each
which is a 8-approximation to entiable isotopy t,
Bd M
into Bd N;
f
and
f1,
N
such that
t.
You will need to generalize Lemma 5.12 as follows:
Cr differentiable isotopy between two
each
and carries
our result follows.
Cp,
Cr differentiable isotopy
Hint:
be a
D(N)
Generalize the preceding theorem to prove that
is a diffeomorphism of M
ft
small
s1
M must be carried into the subset
is of class
fl
By making
Bd N.
s2-approximation to the identity.
a connectivity argument shows that of
onto
g(Bd M)
Cp diffeomor-
f
for each
gt
Let
Cr imbeddings of M into N x R t.
Then there is a
ht which is a diffeomorphism of N x R
which satisfies (1) and (2) for each
Cr differ-
onto itself for
t, and in addition satisfies
the conditions: for some
(3)
If
gt(M) C N
(4)
If
Fat is of class Cp
5.15
Problem*.
Let
t,
then
for some
is the identity map.
ht t,
f : M - N be a
so is
ht.
N
Cr map;
Prove that any sufficiently good strong approximation to ably homotopic to f. Specifically, prove that given such that for any Cr map
Hint:
a
Let
f,
for each
ft
between
f
and
f,
g which is an
t.
Cr retraction of a neighborhood of
N.
is differenti-
there is a s(x) > 0
N be a submanifold of'euclidean space
g along straight lines in RP, in
f
g : M - N which is a 8-approximation to
there is a Cr differentiable homotopy e-approximation to
e(x) > o,
non-bounded.
N
in RP,
and then apply
onto r
N.
RP;
r
let
Deform
f
be into
to make the image remain
MANIFOLDS WITH BOUNDARY
55.
Generalize Problem 5.15 to the case where
Problem*.
5.16
draw the additional conclusion that whenever both
a boundary;
g(x) lie in Bd N,
so does
ft(x)
for each
Imbed Bd N in Rp;
Hint:
it
of D(N)
is the projection of RP+1
f(x)
and
extend to an imbedding h : U - RP+1, so that
onto RP.
xh(U)
h(N),
Finally, choose an imbedding
in some euclidean space which agrees with h in a neighborhood using 2.11.
of Bd N,
traction r to D(N)
N has
t.
where U is a neighborhood of Bd N in D(N), where
61
at
Then apply the techniques of 5.15, choosing the re-
so that it carries a neighborhood of x in the plane normal
x
onto
This implies that r
x.
joining two nearby points of
h(Bd N)
carries the line segment
into h(Bd N).
Exercise*.
Provide an alternate proof for Problem 5.14.
(a)
*
5.17 Problem .
A space X is locally contractible if for each
point x of X and each neighborhood U of
x,
V of x
U
stant map
such that the inclusion map c
: V
is hcmotopic to the con-
: V - x.
Let M and N be Fr(M, N)
i
there is a neighborhood
Cr manifolds;
M
compact.
is locally contractible in the C1 topology.
Prove that Prove also that the
following spaces are locally contractible in the C1 topology: (1)
The space of all
Cr immersions of M into
N.
(2)
The space of all
Cr imbeddings of M into
N.
(3)
The space of all Cr diffeomorphisms of M
onto
M.
Uniqueness of the Double of a Manifold*.
§6.
Let M be a non-bounded Cr manifold, let W be a
Lemma*.
6.1
neighborhood of M x 0
in M x R+, where R+ - [0, a').
f be a Cr
Let
imbedding of W into M x R+ which equals the identity on M x 0. is a Cr diffeomorphism
f
of W onto
f(W)
which equals
f
There
in a neighbor-
hood of the complement of W, and equals the identity in a neighborhood of
M x 0. Let
Proof.
denote the general point of M x R+. We may as-
(x,t)
sume that W is the set of points is a positive
0(x) < 1
Step 1.
Since
and
Cr function on M. f
Let
is non-singular,
Cr function
Choose a positive
for which
(x,t)
is positive.
such that
aT(x,t)/at > 0
for
o < t < E(x)
1 > E(x)IaT(x,t)/atI
for
o < t < 0(x)
We define a diffecmorphism g of W with itself by the equation
Y(x,t) - (1 - a(t/0(x))) E(x) - t + a(t/0(x)) a(t)
t < 1/3 cause
g(x,t) -
where
(x, Y(x,t)),
Here
where
f(x,t) - (X(x,t), T(x,t)).
aT(x,0)/at
on M
c(x) < 1
0 < t < 0(x),
is, as usual, a monotonic
and equals
C00 function which equals
for
2/3 < t.
g(x,o) - (x,o),
and
g(x,t) - (x,t)
Let f1 which equals
f
1
for
t > 20(x)/3,
- (Y)E(X) + a + (1 - E(X))(1/0(X))
fg. Then near W -W.
f1
0
for
The map g is a diffecmorphism, be-
1
aT(X,t)At =
t
and
CO > E(X) > 0
.
is a Cr diffeamorphism of W with
Furthermore, if we set
f(W),
f1(x,t) . (X1,T1),
then 0 < aT1(x,t)/at < 1
for
t < 0(x)/3
This follows from the fact that T1(x,t) - T(x,E(x)t) that
aT1(x,t)/at - E(x) aT(x,E(x)t)/at,
by choice of
for
t < 0(x)/3,
so
which is positive and less than
E.
Step 2.
Now we define a diffeomorphism h of W with itself by
the equation h(x,t) - (x, cp(x,t)), V(x,t) - a(2t/0(X)) Again,
.
where t + [1 - a(2t/0(X))) T1(x,t)
h is a diffecmorphism because 63
c(x,0) - 0,
and
q,(x,t) - t
for
1,
64
DIFFERENTIABLE MANIFOLDS
I.
and
t > 9(x)/3,
This last inequality
t < 9(x)/3.
for
acp (x,t)/at > 0
follows from the following computation:
tt
acV(x,t)/at - a + (1-a) aT1(x,t)/at + (1 - T1(x,t)/t) aT1(x,t)/at > 0
which is positive because
,
aT1(x,t*)/at < 1,
T1(x,t)/t
0 < t* < t.
where
Let
f2 = f1 h-1.
which equals
f(W)
and
a'/13
near W - W.
f
is a
f2
Furthermore, if we set
f2(x,t) _ (X2,T2),
in M x R+.
If M is compact, the completion of the proof is easy.
Step 3.
small enough that Mx [0,s]
8
Cr diffeomorphism of W with
in a neighborhood Y of M x o
then T2(x,t) a t
Choose
Then
where
f3(x,t) _ (X3,T3),
T3 - T2
is contained in Y,
and define
and
X3(x,t) = X2(x, a(t/b).t) f3
M
is a diffeomorphism of
is a diffeomorphism because the map x -. X(x,t0) for
and
to < s,
T3(x,t) - t
for
t < 8.
If M is not compact, more work is involved.
We need the follow-
ing lemma:
Lemma*.
Let U be an open set in Rm whose closure is a
6.2
ball.
Let
be a compact subset of U; let V be a neighborhood of
C
whose closure is contained in U.
Let
f be a Crimbedding of U x [o,a] into
Rmx [o,a) which equals the identity on U x 0, such that f(x,t)
There is a
Cr diffeomorphism
f1
_ (X(x,t),t).
with
of U x[o,a)
- (X1,t)
such that
f(Ux (O,a))
is the identity on U x o
(1)
f1
(2)
X1(x,t) = X(x,t) If
(3)
C
f
and on
for some
C x[o,s],
8 > 0.
outside V x(0,a/2).
is the identity on some set x x (o,b),
then
is the
f1
identity on this set also. Proof.
and equals
outside
0
which equals Let
Let
1
for
(p(x) be a C°° function on Rm which equals V.
Let
t < 1/3
r(t)
Define
0
for
E < a/2
C
2/3 < t.
f1(x,t) = (X1(x,t),t),
X1(x,t) = X(x, t(1 - .(t/E) q)(x)]) Here
on
be a monotonic C00 function on R
and equals
P(x,t) _ (X(x,t),t).
1
is a positive number yet to be chosen.
where
Now for x
V,
outside
in
and
and
C
X1(x,t) - X(x,0) - x,
t < E/3,
Condition (2) is clear, since
is satisfied.
tion (1)
for
y(t/E) - 0
identity on x x[o,b).
If
[o,b],
then ht that
is the
so that
be an imbedding of U
By Exercise (b) of 3.10, there is a function 8(x) > 0
t.
such that if
for x
is a diffeomorphism, and for
f
ht(x) - X1(x,t)
this it will suffice that the map
on U
f
t(1 - y(p),
so is
so condi-
Thus condition (3) holds.
All we need now to do is to show
for each
m(x) - 0
Finally, suppose
t > a/2.
is in
t
X1(x,t) - X(x,t(1 - y@)) - x.
in Rm,
65
UNIQJENESS OF THE DOUBLE OF A MANIFOLD
§6.
gt(x) - X(x,t),
is a 8-approximation to the map
ht
Then there is a positive constant
is an imbedding.
e0
such
ht will be an imbedding if 1X1(x,t) - X(x,t)f < 80
and i
X1 (x,t) /ax - ax(x,t) /axl < 80
for all x in U. By uniform continuity of X and such that
if x is in j and
faX(x,t)/at ap(x)/axi
minimum of
co
and
IaX(x,to)/ax - aX(x,t)/axl < 80/2
for
x
in
it, y(t/E). W(x)l < E,
where
so that
The first term is within
so/2
to - t(1 - yT).
of
of Ui with a ball in Rm.
that (Ci) covers
aX(x,t)/ax,
as desired.
Also, .
and the second is within
which equals
f
Let Ci be a compact subset of Ui such
fi - (Xi,Ti)
near Y - Y,
(1)
T1(x,t) - t
on Y,
(2)
Xi(x,t) - x
for x in
c
Cover M by a locally-
for which there is a diffeomorphism
Ui,
For convenience, assume
M.
Induction hypothesis:
Choose
It - tot -
Hence our desired result follows.
finite collection of open sets
f(Y)
Hence
t y(t/E)[aX(x,t0)/at a(p (x)/ax]
-
Completion of the proof of the theorem.
with
to be the
e
IX1(x,t) - X(x,t)I < 80,
6X1(x,t)/6x = aX(x,t0)/ax
hi
Choose
Ti.
so /2M.
Now X1(x,t) - X(x,to),
of zero.
co
Let M be the maximum value of the
It - tol < co.
entries of
80 /2
and
IX(x,t0) - X(x,t)l < 80
there is a constant
aX/ax,
U1
and U2
is a
are empty.
Cr diffeanorphism of Y
such that
and Ci
and
so that Uii x(o,c)
o < t < 8j,
for
is contained in Y.
J < i.
By &"I"
DIFFERENTIABLE MANIFOLDS
I.
66
the lemma, we may obtain a diffeamorphism outside
for some choice of
ei+1.
we will have Xi+1(x,t) = x
lemma,
on
extend it to U by letting it equal that
?
on Ui+1x o
and on
for all
Ci x[0,81)
f2
fi
Because of condition (3) of the
?(x,t) - limy _. . fi(x,t)
We then define
which equals
° (Xi+,,t)
such that Xi+1(x,t) - x
Ui+1 x(0,c),
Ci+1x[0,81+1)+
fi+1
outside
for
j < I.
in Y;
(x,t)
we
It is easily seen
Y.
satisfies the requirements of the theorem.
6.3
The double of a Cr manifold M is uniquely deter-
Theorem*.
mined, up to diffeomorphism. Proof.
and D'(M)
Let D(M)
be two differentiable manifolds ob-
tained by using different product neighborhoods of Bd M to define the differentiable structure.
Than P : V -. Bd M x(-1,1)
are diffecmorphisms of the open subsets V and V' respectively, with Bd M x(-1,1). Bd M xo
Now
PIP-'
into Bd M x (-1,1);
in Bd M x ( - 1 , 1 )
P': V'- Bd M x(-1,1)
and of
and
D(M)
maps a neighborhood W of it equals the identity on
and is a diffeomorphism when restricted to the subsets
Bd M x o,
(Bd M x [ o,1)) n W = W.I.
and
(Bd m x (-1 , o)) n w = W_
of
By the preceding
W.
which equals
theorem, there is a homeomorphism g of W with P'P 1(W) PIP-1
near Bd W,
and W_,
is a diffeamorphism when restricted to the subsets W.
and equals the identity in a neighborhood of Bd M xo. f - (P')-1g P is defined on the neighborhood
Then
Bd M in D(M),
of D(M)
with D'(M)
(a)
Hence
f may be extended to a homeomorphism
is a diffeomorphism of D(M)
f
Exercise*.
onto D'(M).
If
f
and
g are in Diff M,
f
weakly diffeotopic to g if there is a Cr diffeomorphism F
relation.
F(x,t) - (f(x),t)
in a neighborhood of M x(-o,o)
in a neighborhood of M x Let
i(M)
P-1(W).
Let Diff M denote the space of all Cr diffeomor-
phisms of M onto itself.
such that
of
near the
by letting it equal the identity outside
One checks readily that
I
P-1(W)
and equals the identity map of D(M) - D'(M)
boundary of this neighborhood.
(g(x) ,t)
D'(M),
is said to be
of M x R and F(x,t) -
Show that this is an equivalence
denote the equivalence classes.
The composition of two
UNIQUENESS OF TEE DOUBLE OF A MANIFOLD
§6.
diffeomorphisms is a di£feomorphism, so that Diff M this group operation makes
between
ft
g
are weakly diffeotopic. Exercise*.
(c)
Prove that
(g(x),1).
Exercise*.
(d)
group of
f
or
and
f
let G be a
Let M be non-bounded;
G(x,o) a (f(x),0)
and g
Cr diffeo-
and
G(x,1)
are weakly diffeotopic.
Let M be orientable;
let
r(M)
denote the sub-
generated by orientation-preserving diffeomorphisms.
r(M)
r(M)/r(M) 2gZ,
that
then
t,
(The converse is an unsolved problem.)
with itself such that
morphism of M x I
show that
into a group.
T(M)
g which is a diffeomorphism for each
and
f
is a group;
Show that if there exists a differentiable isotopy
Exercise*.
(b)
67
0
Show
according as M possesses an orientation-
reversing diffeomorphism or not. Exercise*.
Show that
(e)
Hint:
is a diffeomorphism of
f
If
r(Rm)
the linear map which carries
into Df(o)
x
Rm,
first deform
into
f
then deform this into the
x;
identity map. *
Exercise .
(f)
of the set of all
x
If
the support of
f : M -+ M,
for which
Let
f(x) / x.
ments of Diff M having compact support; let
f
is the closure
Diffc M denote those ele-
rc(M)
denote the weak diffeo-
topy classes of DiffcM, where the support of the diffeotopy is required
to have compact intersection with Mx I.
If M
is orientable, let
rc(M)
denote the subgroup generated by orientation-preserving diffeomorphisms. Milnor has proved that that in general
is non-trivial (see [5) and [10)).
Prove
is abelian.
rc(Rm)
Given
Hint:
rc(R6)
f
and
g,
Let
f
deform them so their supports are dis-
joint. Exercise*.
be a diffeomorphism of
Bd M
onto
M U£N denote the non-bounded manifold obtained from
M U N
by identify-
(g)
Let
ing each x in
Bd M with f(x).
Bd N.
Put a differentiable structure on M U1N
such that the inclusions of M and
N
are imbeddings.
Show that the re-
sulting differentiable manifold is unique up to diffeomorphism. Exercise*.
(h)
Bd N.
Let
For any diffeomorphism
element
f
1
0
f
of Diff(Bd M).
f0 f
be a fixed diffeomorphism of Bd M
of Bd M
onto
Bd N,
let
onto
f denote the
Show that up to diffeomorphism, the
68
I.
DIFFERENTIABLE MANIFOLDS
differentiable manifold M Uf.N
of f.
depends only on the weak diffeotopy class
CHAPTER II.
TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS
Cell Complexes and Combinatorial Equivalence.
§7.
In this section we prove the theorem that two finite polyhedra in euclidean space may be subdivided into simplicial complexes in such a way that their intersection is a subcomplex of each of them (7.10).
For this
purpose, it is necessary first to define what is meant by a rectilinear cell complex, and to study some properties of cell complexes.
7.1
If v0,...,vm are independent points of
Definition.
the simplex
a - v0...vm they span is the set of points
x = E bivi,
where bi > 0
and
barycentric coordinates of center of
a
E bi a 1.
The point
x.
and is denoted by
v.
the interior of
Bm;
If
A
a
a.
such that
The numbers bi E vi/(m+1)
are called the
is called the bar -
A face of a simplex
spanned by a subset of the vertices of to
x
The simplex
is the simplex
a
a
Is homeomorphic
is called an open simplex.
and B are two subsets of
Rn,
the loin A * B
B is the union of all closed line segments joining a point of point of
Rn,
B,providing no two of these line segments intersect
sibly at their end points. Then
of A and A and a
except pos-
a - v0 * (V1 * (v2 * ... )).
A (simplicial) complex K is a collection of simplices in Rn, such that (1)
Every face of a simplex of K is in
(2)
The intersection of two simplices of K is a face of each of them.
(3)
Each point of
many simplices of
K.
JKJ
K.
has a neighborhood intersecting only finitely 69
TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS
II.
70
Here
denotes the union of the simpliees of K and is called the
IKI
polytope of
K;
sometimes it is called a polyhedron.
(It would be more
general to let the simplicea of K lie in R°° = Un Rn, will suffice for our purposes.
but this definition
We are in fact restricting ourselves to
finite-dimensional complexes.)
A subdivision each simplex of
of K is a complex such that
K'
is contained in a simplex of
K'
K.
and
IK'I = IKI
A subcomplex of K
is a subset of K which is itself a complex.
If x is a point of the interiors of all simplices subset of fine J
denoted by
IKI,
St(S,K)
the star of x
IKI,
such that
at
If
St(x,K).
S
to be the union of the sets
is a subcomplex of
x
in K is the union of
lies in
Is any subset of IKI, we de-
for all x in
St(x,K)
it is convenient to write
K,
It is an open
s.
If
S.
for
St(J,K)
St(IJI,K) Let
a
be the simplex v0...vm.
f(x) = f(E bivi)
plexes, a map f each simplex of
for all x
E bi f(vi) e
IKI
A map in
a.
K linearly into a simplex of L.
the phrase - f : X - L is linear. K'
is linear if
e - RP
If K and L are com-
is linear relative to K and L if it carries
ILI
if for some subdivision
f
of
The map
K,
f
:
We often shorten this to
f : K - L is pi8oewise-l132ear
K' - L is linear.
a linear isomorphism if it is a homeomorphism of
IKI
onto
: K - L is
f
ILI
and carries
each simplex of K linearly onto one of L. It was long a famous unsolved problem (often called the Hauptvermutung)
whether the existence of a homeomorphism between
IKI
plied the existence of a piecewise-linear homeomorphism of Yes, if dimension K = 2 [11]; yee, if
Partial answers were: manifold
T1, 71;
the answer is no if dimension K > 6
(a)
St(x,K),
IKI
im-
onto
ILI.
IKI is a 3-
[6].
Recently, Milnor has shown that It is still unknown if the ans-
is a manifold.
Exercise.
Let K be a complex.
denoted by 5t(x,K),
We sometimes use
IKI
ILI
yes, if K and L are smooth triangulations of diffeo-
morphic manifolds (this we shall prove).
wer is yes when
and
Show that the closure of
is the polytope of a finite subcompleX of
St(x,K) to denote the complex as well as the polytope,
where no confusion will arise.
K.
S7. CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE
Show that if
Exercise.
(b)
homeomorphism of
f : K- L is a piecewise-linear
ELI, then they have subdivisions
onto
IKI
71
K' and L'
f : K' -- L' is a linear isomorphism.
such that
Let
7.2 Definition.
of all points x
c
consisting
be a bounded subset of Rn,
satisfying a system of linear equations and linear in-
equalities Li(x) - Ej such a set set of
c
Rn.
xJ > bi
aij
1 - i,...,p ;
;
is called a (rectilinear) cell. (Convexity of a set
It is a compact convex sub-
means that
c
contains each line seg-
c
ment joining two of its points.)
The dimension m of al plane
d'
containing
points, but not m + 2.
c;
is the dimension of the smallest dimension-
c
this means that
Since
contains m +
c
homeomorphism of
c
independent
must contain the simplex spanned by these
c
m + 1 points, it must have interior points as a subset of (P.
denote these points; let Bd c
1
be the remainder of
Int c
We show there is a
c.
with the m-ball BF carrying Bd c
Let us adjoin to the system defining
Let
Sm-1:
onto
a set of equations for I?
c
Some of the inequalities of the system may now be redundant; let L1(x) > bi,
i - 1,
..., p,
be a minimal system of inequalities which, along with
the equations for 6), serve to determine
p, the hyperplane
Li(x) - bi
c.
intersects 61
for otherwise, this hyperplane would contain
We note that for
in a plane of dimension m-1; (P and the corresponding in-
equality could be discarded without changing the set
Furthermore,
c.
equals the set A of those points for which each of these inequali-
Int c
ties is strict:
if xo bj,
i - 1, ...,
Clearly,
A is contained in Int a.
is a point of the intersection of
than arbitrarily near x0
so that
x0
Z.
d' with the hyperplane
are points of
La(x) -
(P for which Li(x) < bj,
does not lie in Int c.
Let c be a point of ning at
On the other hand,
Int a
and let
The intersection of r with
and convex; i.e., a closed interval.
necessarily a point y
of Bd c.
c
r be a ray in P beginis non-degenerate, compact,
One end point is
Each point x
c;
the other is
of the open line segment
72
II. TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS
cy necessarily lies in so that
1 - 1, ..., p,
for
Int c,
Li(c) > bi Hence
Li(x) > bi.
c
and
L1(y) > bi
for
is equal to the join of
c
with Bd c. To complete the proof we need only show Without loss of generality we may assume
Sm-1.
The map
origin.
x
carries Rm- 0
x/IIxlI
homeomorphic with
Bd c
is Rm and c is the
(P
continuously onto the unit
sphere; it is necessarily a homeomorphism when restricted to
Let
Lemma.
7.3
finite number of
be an m-cell.
c
m-1 cells, each the intersection of an m-1 plane with
Let (P
Proof.
is the union of a
Then Bd c
These cells are uniquely determined by
Bd c.
Bd c.
by equations for 6,
c.
be the m-plane containing
let
c;
c be given
along with a minimal set of inequalities
L1(x) > bi,
i . 1, ..., p.
di
Let
then di
denote the set of points
is a cell, and Bd c
of dimension m - 1. the subset of for all
S
with
Furthermore,
(P is an m - 1 plane
is precisely di.
denote
L,(x) > b
is convex
S
for which L1(x) >
It also contains a point y Li(y) < bi,
for which Li(x) = b1;
in particular, it
Int c;
contains a point x
that
Let
Then
c
is a cell
di
(P for which i.
j
and contains
bi.
Li(x) = bi
whose intersection with Bd c We prove that
of
is the union of these cells.
the intersection of the hyperplane T1,
x
such
since otherwise
would lie entirely in the region Li(x) > bi,
inequality the set
Then 611.
so that discarding the L1(x) > bi
c.
S n 61 Since
from the set of inequalities for
By convexity,
S
is not empty; s n Mi C di,
contains a point since
di
S
is open in
z
c
would not change
such that (P,
S n 6) 1
L1(z) = bi.
is open in
must be a cell of dimension m - 1.
V.
CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE
The uniqueness of the cells
is easy:
di
union of p planes of dimension m-1.
Bd c
73
is contained in the
Hence the intersection of any other
m-1 plane with Bd c lies in the union of finitely many planes of dimension less than m-1, and hence has dimension less than m-1.
7.4
Definition.
into which Bd c di
If
decomposes is called a face of
is called a m - 2 face of
is also a face of
7.5
is an m-cell, each of the m -
c
as is
c,
Lemma.
Let
c
And so on.
c.
cells
1
di
each m - 2 face of a
c;
By convention, the empty set
itself.
c
be a cell given by a system of linear equations
and inequalities. Replacing some inequalities by equalities determines a c, and conversely.
face of
We proceed by induction on
Proof.
m - 0, the lemma is trivial.
m > 0, let
If
c; adjoin to the system determining
c
m, the dimension of
equalities for minimal set. If
a set of equations for
face let
e
Lj(x) = b,
if it does not intersect be the intersection of
the intersection of
e
let
q
a
e
of
y
(P
Otherwise,
it is a cell. 1
Now
plane, so that ar-
for which
L,(y) < b,.
c.
is convex, it must lie in some m-1 face
be the smallest integer such that
be a point of dq.
e
lies on the boundary of Since
it gives the empty face.
are points
Lj(x) _
this gives the trivial
(P with this hyperplane is an m -
bitrarily near each point of Hence
c,
d',
satisfying
c
with this hyperplane;
c
form a
< i < p
as we have just proved.
dj of c,
contains
be the in-
La(x) > bi by an equality.
consider the subset of
j > p,
If the hyperplane
1
This
(P.
Li(x) > bi
Then let us replace some inequality
In the case
c;
Let
c, so numbered that those for which
j < p, this determines an m-1 face
bj.
be the m-plane containing
60
we clearly may do without loss of generality.
If
c.
not in d1U...U dq_1;
e
let
lies in y
d
i
otherwise,
:
d1U...U dq.
be a point of
Then LI(x) > bi
for
i < q
and
Lq(x) - bq
Li(y) > bi
for
i < q
and
Lq(y) > bq
a
Let x not in
II.
74
lies in
z e (x + y)/2
The point that
TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS but
e,
does not lie in d1U...U dq,
z
Li(z) > bi
adjoined, determine the
m - I
Furthermore,
e
an equality.
By the induction hypothesis,
cell
di
c,
with the equation
which contains
e.
is obtained from this system by replacing La(x) > bi e
is a face of
by
and of
di,
c.
A (rectilinear) cell complex K is a collection
Definition.
7.6
so
i < q,
contrary to hypothesis.
The system of equations and inequalities for Li(x) e bi
for all
of cells in Rn such that (1)
Each face of a cell in K is in K.
(2)
The intersection of two cells of K is a face of each of them.
(3)
Each point of
many cells of K.
JKJ
(Here
IKI
has a neighborhood intersecting only finitely denotes the union of the cells of
K.)
In order that this definition be non-vacuous, we need to note that a single cell
along with its faces, constitutes a cell complex.
c,
This follows
from the preceding lemma (see Exercise (a) below).
A subdivision of a cell complex K is a cell complex and each cell of
(K'l - IKI
K'
is contained in one of
K.
of K is the dimension of the largest dimensional cell in K. ton of
K,
denoted by Kp,
dimension at most
(a)
such that
The dimension The p-skele-
is the collection of all cells of K having
One checks that Kp
p.
K'
is a subcomplex of
K.
If K consists of a single cell, along with its
Exercise.
faces, show that K is a cell complex. (b)
Show that the simplex
Exercise.
a - vo ... vm
is an m-cell,
and that the notion of "face" is the same whether one uses the definition in 7.1 or the one in 7.4.
Shaw that a simplicial complex is a special kind of
cell complex.
7.7 I.
Then
Lemma. K1
and
Let K2
K1 and
K2
be cell complexes such that
have a common subdivision.
JK11I
It is clear that the intersection of two cells is again a
Proof.
cl
L. be the collection of all cells of the form
Let
cell.
is a cell of
and
Ki
and
is a cell of
c2
where
c1n c2,
Then L is a cell complex,
K2.
1K1! _ IK21 - ILI.
Exercise.
(a)
where
eln e2,
(b)
Exercise.
7.8
Lemma.
is of the form
and conversely.
ci;
Show that L is a cell complex.
Any cell complex K has a simplicial subdivision.
K is already a simplicial complex.
L is a simplicial subdivision of is an m-cell of
K,
the m -
Km-1,
I
a1 * c,...,ap * c.
of
skeleton of
ai * a
(Recall that
denotes the join of
K'
of
K,
which is a subdivision of
K.
It is often convenient to choose be canonically defined.
K'
the barycentric subdivision of
Exercise.
division of
c
as the centroid of If
K
c
c,
ai
the
in
is already simplicial, the
centroid is the same as the barycenter, and this subdivision
(a)
If
K.
adjoin to the collection L the aim-
c;
result will be a simplicial complex
order that
If
K.
In general, suppose
If we carry out this construction for each m-cell
c.)
of
al,...,ap be the simplices of L lying in Bd c.
let
Choose an interior point c plices
c,n c2
We proceed by induction on the dimension m
or m - 1,
m = 0
Show that any face of
is a face of
ei
Proof.
and
75
CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE
§7.
K'
is called
K.
Check that
K'
is a complex, and that it is a sub-
K.
7.9
1K1! _ JK21,
7.10
Corollary. K,
and K2
Theorem.
If
K,
and K.
have a common simplicial subdivision.
Let
K,
and
K.
in Rn. There are simplicial subdivisions respectively, such that
are simplicial complexes such that
K,1 U K2
be two finite simpliciat amgzlexes
K,
and
K2
is a simplicial complex.
of K 04.
TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS
II.
76
L such that plex of
We prove that some subdivision of
ILI . Rn.
a rectilinear triangulation Let
angulation
of
Rn:
a1,...,ap be the simplices of
K1.
L1
of Rn which contains
J1
affine transformation the collection
is a subcom-
K1
Choose a rectilinear tri-
One way to construct such a
a1.
of Rn which carries one of its simplices onto
h1
will be the required complex.
h1(J)
(An affine trans-
formation is a linear transformation composed with a translation.) let
Ji be a rectilinear triangulation of Rn containing ai;
a common subdivision of J1,...,Jp tope of a subcomplex of
J1
of Rn and find a non-singular
is to take any rectilinear triangulation J
a1;
is a simplicial complex
A ,rectilinear triangulation of Rn
Proof.
Then
(using 7.9).
1K11
Similarly,
let
is the poly-
and this subcomplex is a subdivision of
L1,
be
L1
K1.
Similarly, let L2 be a rectilinear triangulation of Rn containing some subdivision of
vision of
and
L1
having polytopes
and
K2
and
IK1I
of the sets
IK11
IK2I,
is the set
Let
St(K1,K).
in R2
[0,1] x o
and
IK2I
is the union
.
be a subcomplex of the simplicial complex
K1
K1
The proof of Lemma 7.8 generalizes to the follow-
be a simplicial subdivision of
tending K
be the subcomplexes of L
respectively.
for n . 1,2,...
Definition.
ing situation:
K2
L be a common subdi-
You will need some further hypothesis to avoid
[0,1] xl/n
7.12
and
Kj'
Let
Generalize this theorem to the case in which
are not finite.
the case where
K;
and let
L2,
Problem .
7.11
as a subcomplex.
K.
to a subdivision of
let
We define a canonical way of ex-
K1. K,
K;
without subdividing any simplex outside
We will call it the standard extension of
K,'
to a subdivision of
K:
Every simplex of
K;
plex of K which is outside in A . St(K1,K) - IK1I
ces of K lie in A.
belongs to
K',
of course, as does every sim-
The simplices whose interiors lie
St(K1,K).
are subdivided step-by-step as follows: Each 1-simplex
a
No verti-
of K whose interior lies in A
§7.
CELL COMPLEXES AND COMBINATORIAL EQUIVALENCE
is subdivided into two 1-simplices, the barycenter of
Let
For any m-simplex
its boundary has already been subdivided.
a,
a be its barycenter, and for each simplex
s
a itself.)
join of the empty set with a is
Exercise.
(a)
Problem.
K.
IKI.
Let K be a simplicial complex;
locally-finite collection of subsets of
IKI.
Let
quence of simplicial subdivisions of K such that This means that any simplex of
simplex of simplices integer
Ki+1 a
Na.
(a)
as well.
such that
a
K1
belongs to
equals
K1+1
Ki
positive continuous function on such that for any simplex
a
minimum of
in
for
x
SKI.
of a.
K',
be a
be a se-
K1, K2, ...
Ki
Ai
outside is a
denote the collection of those for all
i
greater than acme
Let K be a simplicial complex;
Exercise.
(Ai)
which does not intersect
limi,y m Ki
Let
let
Show that this collection is a subdivision of
8(x),
the
Check that the collection of simplices obtained in
this way is a complex whose polytope is
7.13
(Convention:
a.
After a finite number of
steps, we will have the required subdivision of
Ail
of the subdivision of
s * a be a simplex of the subdivision of
let
Bd a,
being the extra
a
In general, suppose the m - 1 simplices have already been subdi-
vertex. vided.
77
K.
let
There is a subdivision the diameter of
a
8(x) K'
be a
of K
is less than the
Immersions and Imbeddings of Complexes
S8.
In this section, we define the notion of a Cr map
f : K- M,
where K is a simplicial complex and M is a differentiable manifold, and we develop a theory of such maps analogous to that for maps of one manifold In particular, we define (8.2) the differential of such a map
into another.
and use this to define (8.3) the concepts of immersion, imbedding, and triAs before we define
angulation (which is the analogue of diffeomorphism). (8.5) what is meant by a strong
C1 approximation to a map
f : K
M,
and
prove the fundamental theorem which states that a sufficiently good strong
C1 approximation g
to an immersion or imbedding is also an immersion or
imbedding, respectively.
(The theorem also holds for triangulations, with
the additional hypothesis that
g carries Bd JK)
into Bd M.)
From now on, we restrict ourselves to simplicial complexes and subThe integer
divisions, unless otherwise specified.
r
(t < r < co)
will
remain fixed, for the remainder of this chapter.
8.1
Let K be a complex.
Definition.
relative to K if
The map
f
:
differentiable of class
Cr
each simplex
a
We usually shorten this to the phrase,
is of class
Cr.
of
K.
The map
equal to the dimension of
f a,
flu
is of class
is said to be non-degenerate if for each
a
in
IKl - M Cr,
is
for
f : K
flu
M
has rank
K.
We wish to generalize the notions of immersion and imbedding to this situation.
take a
As an analogue to a
Cr imbedding of a manifold, one might
Cr non-degenerate homeomorphism of a complex.
79
That is, until one
80
II.
TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS
looks at the homeomorphism
The crucial
of the accompanying illustration.
f
property of imbeddings - that any sufficiently good strong C1 approximation to an imbedding is also an imbedding - fails here, since arbitrarily close to
f
are maps like
This example shows us that we must seek further to
g.
find the proper generalization.
Definition.
8.2
b
of
Let
define the map
a,
f
a
x
and
Given the point
Cr map.
dfb: a -. Rn by the equation Df(b)
dfb(x)
Here
Rn be a
(x - b)
.
are written as column matrices, as usual; and we choose
b
some orthonormal coordinate system in the plane in which to compute
The map
Df.
in the plane of Let
dfb(x)
is independent of this choice of coordinates
dfb
a:
be some ray in
R
curve in Rn,
a
beginning at
Then
b.
fIR
is a
which we suppose parametrized by are length along
is merely the tangent vector of this curve at
11 x - b U.
in order
lies,
a
Hence
f(b),
Now
multiplied by
does not depend on the choice of coordinates,
dfb(x)
since it involves only the distance function in the plane of more, it follows that
R.
Cr
depends only on
dfbjR
fIR,
Further-
a.
not on any other values
of f. Now if ed for each
a
two simplices of
f : K
Rn
is a
Cr map, we have maps
dfb: a -+Rn defin-
in 3f(b,K).
These maps agree on the intersection of any
a''E(b,K),
since either (1) one is a face of the other, or
(2) their intersection lies in the union of rays emanating from
b.
Hence
the map
dfb: 'ST(b,K) -y Rn is well-defined and continuous.
By analogy with the situation for differen-
tiable manifolds, we call it the differential of
8.3
Definition.
submanifold of
Rn.
Let
The map
f f
: K- M be a
f.
Cr map, where M
is a
Cr
is said to be an immersion if
dfb: 5(b,K) - Rn is one-to-one for eaqh b.
An immersion which is a homeomorphism is called
§$.
m4msioris AND D Ef3DlivGS OF CGMPL cES
81
an imbedding; if it is also a homer fiorphism Onto, it is called a
Cr triangulat' on of
M.
If x and
hslorig to the simplex
b
of 1,
a
then the require-
ment that dbja be one-to-one implies that the matrix D(fla) b
equal to the diuensio<: c°
The cpnv nse is not true in general, as the
-degenerate map.
8.1 shQws.
has rank at
Hence an immersion is automatically a non-
or.
However, in the aaae where
f
example in
is a hoineoinc)rphism onto, there is
a convera; as the following theorem shows.
K in the mtu_.ifold
let
M;
manifold i, Show that Exercise.
(b)
I - M be a Pr i zme-rsion of the ccispleX
Lot
L3xorcise.
N -+ N be a Cr immersion of M in the
.g
gf is a
Cr immersion of the complex K in N.
Show that the definitions of
Or immersion, imbed-
ding; aid triangulation are independent of which particular imbedding of M in euclidoan space is chosen.
8.++
Theorf m.
is a balteomrrphis&l of
Let IKj
f : K - M be s, Gr non-degenerate map which onto
Then
M.
Ws:y it could fail zo be 1-1 is for it to map rays
at b
into a single ray, in Rn beg_nning at are
fiR2
anti
Since
E(b,K)
is 1-1.
simplex of
b
Rentce no neighborhoods of
This means that
fIR1
f(b).
to any simplex of
in RI and
No lie in the same
St(b, V).
and yn be sequences of points of b;
sib
of
Choose a coordinate system
to
f(b).
Ra beginning
and
R1
Cr cu:'vcs In N which are tangent to each other at
is non-degenerate, the
f
Gr triangulation of M.
is 1-1 on M(b,K). The only
We neec1 to prove that dit.
Proof`.
is a
f
let
xn, yn, bI
(U,h) R1
in M about
and
R2,
f(b).
xn
.Let
respectively, converging
be their respective images under
hf.
Let
a
be
extends; let V be the ope:: c-.t 21-U(7,K) n 1' 1(U) . !Vow hf(V) Ss opF-r_ in RM: it coLi,a.ins xn, but not
the simplex of
yn.
Tf n -s large,
hf(V - V)
into whase interior
ca" (b,Ic)
xn
lies in V,
'n at past one
between xn - b'
'.et
x;:
sna
yn - b'
zn;
so the line segment, further,
approaches
o,
;ienote the point of V - V
z, not
b' e,
ltnyn
intersects
because the angle-
as n
corresponding to zn.
By
82
TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS
II.
passing to subsequences and renumbering, we may assume and that the unit vectors
simplex of K to the vector
Let R3
U.
Now Ri though
and
hfIR1
tion that
f
and
R3
hfIR3
II
(zn - b)/II zn - b
converge, say
II
be the ray along which u lies.
lie in a single closed simplex of are tangent at
is non-degenerate.
(xx - b')/II xn - b'
lies in a single
zn
al-
(b,K),
this contradicts the assump-
b';
The computation is as follows:
approaches the unit tangent vector to
hf(R1)
at
b'.
Further,
zn - b II zn-b II
D(hf) (b) II
Since the angle between xn - b'
and
(zn - b) zn - b
+
zn - b II)
II zn-b II
II
goes to zero,
zn - b'
of this equation approaches a vector tangent to
side approaches D(hf)(b) u,
of II
hf1R1
which is tangent to
at
hfIR3
the left side
The right
b'.
at b'.
(The picture is, unfortunately, a bit misleading, since it shows zn
lying on
R3,
which we can't guarantee.
The proof would be easier if
this were the case.)
(a)
Exercise*.
combinatorial n-manifold. the link of
s,
Assume the hypotheses of 8.4.
Show that K is a
This means that for each k-simplex
Lk s - ITE s - St s, is a combinatorial
s
of
K,
n-k-t sphere; that
is, it has a subdivision isomorphic with a subdivision of the boundary of an n-k simplex
a.
§8. IMMERSIONS AND IMBEDDINGS OF COMPLERES
v with a complex L in Rn,
isomorphism h of and a'
Find a subdivision
ILI Do.
a vertex cf
s - v,
Hint: Prove it first when
of
a'
83
h(v) - a
such that
such that the inclusion
a
Radial projection from a of Bd a'
L is linear.
Find a linear
K.
onto
I
:
will
Bd ILI
then be a linear homeomorphism.
Definition.
8.5
f : K -' Rn be a Cr map.
Let
5-approximation to
For same subdivision
(1)
(2)
II
(3)
of
K'
f K,
g : K' -' Rn is a
IKI -
Cr map.
IKI.
for all b
dfb(x) - dgb(x)II < s(b)II x - b II
II
g :
if
for all b in
f(b) - g(b)II < s(b)
be a
8
By analogy with 3.5, the map
positive continuous function on K.
Rn is said to be a
Let
in
and all
IKI
x in n(b, K'). Such a map g is also called a strong
CI approximation to
f.
Note that this definition is independent of coordinate systems, de-
pending only on the distance functions
in the euclidean space
Note also that if K is finite,
and in Rn.
containing K,
Q x - y II
B
may be re-
quired to be constant, without changing the situation; we shall always assume
s
is constant in this case. As in 3.6, condition (3)
close to each other, since
where p
D(gIa)(b)I,
Let subset of
IKI,
g
II dfb(x) - dgb(x)II/II x - b
g :
K' - Rn be
is said to be a
ditions (2) and (3) hold for all b non-degenerate on
U,
g is said to be an
U if the angle between the vectors
for all b
in U and all x # b
8.6
a > 0.
an
If b
Lenm;a.
is in
Let
dfb(x)
Supposing
f f
and
a angle-approximation to
dfb(x)
ID(fIcy) (b) -
a
containing x.
on U if conand
g to be f
is less than
on a,
In St(b,K').
there is a neighborhood of b f.
g are
[-n
a angle-approximation to
f : K - Rn be a non-degenerate
IKI,
and
If U is an open
Cr maps.
5-approximation to
in U.
f
-
II
equals the dimension of the simplex
Rn and
f : K
holds if the Jacobians of
Cr map.
Let
on which dfb
is
84
II. TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS
It will suffice to prove this in the case where K is a
Proof.
single simplex the plane of
Df(b).
is a point of
and u is a unit vector in
a,
we wish to make the angle between Df(x) u and
a,
D(dfb)(x) u
If x
a.
less than
But
a.
Now the angle between Df(x) u and Df(b) u
function of x and u,
so D(dfb)(x)
dfb(x) = Df(b) (x - b),
is a continuous
and its value is zero when x = b.
Since the do-
main of u is a compact set, there is some neighborhood U of b this angle is less than
Let
Lemma.
8.7
Given a > 0,
finite.
when x
a.
approximation to
there is a
on
f
U,
for any u.
Rn be a non-degenerate
f : K
g : K' - Rn which is a
map
is in U,
B > 0
in M(b,K);
let f
since
f
is non-degenerate,
for all b in a > 0.
Let
IKI and all b - ac/n;
5-approximation to
U.
Let
lie in
Cr
on U is an a angle-
f
Cr map which is a
g : K' - Rn be a non-degenerate
on
K
U.
Proof. Let e -min II dfb(x) II/Il x -b II x # b
Cr map;
such that any non-degenerate
B-approximation to for any
such that
a
be a simplex of K'; let b and x lie in a;
let
b
is the same as that between v = D(fla)(b) U and w - D(gIa)(b) u , where U = (x - b)/IIx-bII n/Ifv*ll, which is less The angle between v and w is less than 11 7 than en/a = a. (For if a is acute, then 11 7 - w II > IfII sin 8 > and if a is not 'acute, 11 7 - w II > IrII > IfII 9/,t.) IfV II 29 /n; U.
The angle
9
between dfb(x)
and dfb(x)
II
8.8
Theorem.
let K be finite.
Let
Rn be a
f : K
There is a
B > 0
Cr immersion, or imbedding;
such that any
B-approximation to
f
is an immersion, or imbedding, respectively. Proof.
We first make two preliminary definitions, and prove a
special case of the theorem. (1)
face of each.
Let
al
and
a2
The angle from
be two simplices, with al
to
a2
a = aln a.
is defined as follows:
a proper
Let
IMMERSIONS AND IMBEDDINGS OF COMPLEXES
§8.
be the barycenter of and
yd
ments
mum such angle Now intersect
and
a1
So we would get the same minimum
a2 - St(-0).
since y and
if
z
then range over compact sets,
z
a
and
z
The mini-
a2.
If we extend the ray from a to
is positive:
a
opposite
a1
This angle is positive.
a.
is called the angle from
9
85
between the line seg-
e(y,z)
lies in the face of
different from
a?
a2 - St(-a);
over
where y
za,
is any point of
Consider the angle
a.
it will
z,
ranged only a
is
positive.
Now the angle and
of
a1
a1
and
from
9
a1
is a function of the vertices
02
If we vary these vertices continuously (without-allowing
a2.
to become degenerate),
a2
to
then
varies continuously as well
9
(see Exercise (a)).
(2)
Let
projection of itself. a1
a1
be a proper face of the simplex
onto
a
Second, suppose
opposite
a;
fined points y px are parallel. into
a
let of
as follows:
p belongs to
and x
of
The projection of
We define the
First, it is the identity on a1 - a.
a be the barycenter of s1
a1.
Let a.
s1
be the face of
There are uniquely de-
such that the line segnents
a
a1
onto
a
a
yd
and
is defined to carry p
x.
Projection is a continuous map of
a1
onto
a;
furthermore, it is
natural in the sense that it is invariant under a linear isomorphism of
a1
with another simplex (see Exercise (b)). The fact about projection which is useful for us is the following:
Let
a = a n a2 1
be a proper face of both
a1
and
a2;
let p belong to
II. TRIANGUTATIONS OF DIFFERENTIABLE MANIFOIDS
86
and
at - a
into
q
a2 - a.
to
a2
(3)
to
a1
to
a1
xq,
G : L' -. Rn
is a non-degenerate
tion to
it follows that G
a.
z
and use the definition of the
et
and
e2
Let L be
a > 0
with each other.
Cr map which is an a/2 is
dGb(x)
has a non-zero projection on Z..
jection of Rn onto
.0,
in
If
angle-approxima-
1-1.
Let $ be the straight line in Now no vector
with vertex v
Let the images under F
For let (P be a 2-plane containing F(L). a . a.)
p
px and xq is no less
and let F : L -. Rn be a 1-1 linear map.
bisector of
carries
Now we prove a very special case of the theorem:
of the two line segments make an angle of
unless
a
a2.
a complex consisting of two line segments
F,
onto
To prove this, merely choose a point
a2.
az is parallel to
such that
angle from
common,
a1
then the angle between the line segments
x,
than the angle from in
If the projection of
((p
is in fact unique
(P perpendicular to the
is perpendicular to
If we let
it follows that
ctG
c
2,
so each
denote the orthogonal prois
1-1.
Hence G is
1-1.
$8. -LMMERSTONS AND IMBEDDING8 OF COMPTMS
87
This special case contains the crux of the entire argument. Proof of the theorem.
any
f be an immersion.
Let
(4)
First, choose
f. is a non-degenerate map.
s-approximation to
A second condition on
small enough that
a
This is easily done.
For each pair of simplices
6:
and
a
a2
1
of K such that
a - ain a2
is a proper face of each, consider
dfb: a1U a2-+Rn where b
is any point of
It is a
a.
tive angle from the image of
continuously with b, ranges over
Choose
a1,a2.
a/6
to the image of
al
a be the minimum of
small enough that any
8
This angle varies
a2.
angle-approximation to
f
It follows that any
a(a,,e2)
as b
a(a1,a2),
so there is a minimum such angle
Let
a.
t-t linear map, so there is a posi-
for all such pairs
8-approximation g to
is an
f
(using 8.7).
8-approximation g : K' - Rn to
is an im-
f
mersion, as we now prove.
Let b
be a point of
IKI;
we need to prove
dgb: is
Then
dgb(p) n dgb(q),
Suppose
1-1.
dgb
Rn for some points
maps the two rays beginning at b
respectively, onto a single ray of
on these rays as close to b
Rn.
such that
and such that
U
is an
dfb(x)
(by 8.6).
a/6
Then on the set
b
and
q,
q
dgb(p) - dgb(q).
such that
U
angle-approximation to
U,
in 'SE(b,K').
and passing through p and
lies in for
f(x)
is an a/6 angle-approximation to
dfb(x)
q
and
Hence there are points p
as we wish such that
There is a neighborhood U of
p
g(x)
St(b,K'),
x
in
U,
for x in
is an a/2 angle-approximation to
dfb(x)
dfb(x).
If p and enough to b
q
that the line segment pq
point of pq;
let
el - px and
linear on the complex with each other. L.
Since
is
1-1
a < s,
on
lie in the same simplex
L,
L,
Further,
and
dgb
of
lies in U.
e2 - xq;
dfb(e1)
a
and
take them close
K,
Let
x be the mid-
let L - e1U e2. dfb(e2)
Now
dfb
make an angle of
is an a/2 angle-approximation to
dfb
the special case of our theorem applies to show that contrary to hypothesis.
is x
on dgb
Otherwise, let
p and
p
image of enough to
b
from x
and
dgb
and
a1
be the simplices of K containing
a2
in their interiors.
respectively,
q,
under the projection of
q
is an
lie in
and
p
is linear on the complex L = e1U e2,
dfb
a/2 angle-approximation to
on
dfb
L.
and
We claim that the angle between the line segments dfb(el) is at least
dfb(e2)
dgb
is
1-1
on
it will then follow from the special case that
a;
dfb(p) = dgb(q).
contradicting the assumption that
L,
close
q
and the line segment
x
to
x be the
Let
Take
a.
from p
e,
Now
U.
onto
a1
that the line segment to
neither simplex is a
Since
is a proper face of each.
a = aln a2
face of the other,
e2
MANIFOLDS
II. TRIANGULATIONS OF
88
To prove this, note that
is a linear isomorphism on ay because pro-
dfb
jection is natural with respect to linear isomorphisme, the projection of the simplex
onto the simplex
dfb(al)
dfb(a)
into
must carry dfb(p)
This means, as noted in (2) above, that the angle between dfb(el) dfb(e2)
is at least as large as the angle from dfb(al)
angle is, in turn, at least
(5)
K' - Rn to
f,
g is
angle-approximation to
1-1
a
Vn a1
al,a
is a proper face of
into Now
simplex
a
el . px
and
this
s
as above.
Given b
in
such that for any s-approximation
b
The proof is as follows:
on V.
on which dfb of
lying in
b,
such
of simplices of K containing b the projection of
al,
onto
al
a
a/3
is an
carries
U.
g is an
a/6 angle-approximation to
angle-approximation to dfb For suppose
dfb(a2);
Then choose a neighborhood V
f.
such that for any pair
that
to
E-neighborhood U of b
First choose an
U,
choose
be an immersion;
f
and
a.
there is a neighborhood V of
IKI,
g :
Let
dfb(x).
K,
let
e2 a qx.
so L - e1U e2 with each other.
It follows that
g is
If p and
q
1-1
Since U is an Now
a < x,
E-neighborhood,
and
dfb(el)
g is
1-1
on
dfb(e2) L,
is an
a/2
on V.
lie in the same
x be the midpoint of the line segment
lies in U. Since
U.
p and q in V.
g(p) a g(q), of
on
g
hence
f;
U n a
pq;
let
is convex,
make an angle of
by the special case of
our theorem, contrary to hypothesis.
The argument in the case where
simplex of K is similar.
Let
p
p and
q
lie in Int al
do not lie in the same and
q
in
Int a2,
n
48. IMMERSIONS AND IMBEDDINGS OF COMPLEXES where
a = aln a2
line segment
px;
Let x be the image of p
is a proper face of each.
under the projection of let
onto
a1
e2
a;
then x
is in
be the line segment
other.
The special case of the theorem applies to show
g
lies
a with each
make an angle of at least
dfb(e2)
be the
e1
Then L - e1U e?
qx.
and
and
Let
U.
in U,
dfb(el)
89
is
on
1-1
L,
contrary to hypothesis.
(6)
Let
f be an imbedding;
let
be chosen as in (4).
8
ing (5), choose a finite number of compact sets IKI such that any
Exercise (a) of 3.10, there is a C0
80-approximation to Hence if
morphism.
g
8-approximation
whose interiors cover
Ci
is necessary 1-1 on
such that if
so > 0
g is 1-1 on each
and
f
f
to
g is both a
80
and a
Apply-
g :
1K! -+ Rn
Ci, then
By
Ci.
is a
is a homeo-
g
6-approximation to
f,
g
will be an imbedding.
(a)
Exercise.
Let
pact subset of Y and define is continuous.
f : X x Y -+ R
fA(x) = min f(x,y)
for
R
to
is a continuous function of the vertices of (b)
Exercise.
y in
A.
Conclude from this that the angle
fA: X a2
Let A be a com-
be continuous.
a1
from
a
and
Prove that a1
a2.
Derive a formula for the projection of
a1 onto
a.
Conclude that projection is continuous, and natural in the sense previously described. (c) 6
Exercise.
Generalize the theorem to the case
will have now to be a positive continuous function on (d)
Exercise.
Show that given a
any sufficiently close approximation
providing g
carries Bd 1K!
g
to
into Bd M.
of course.
1K!,
Cr triangulation f
K infinite;
f:
K - M,
is also a triangulation,
The Secant Map Induced by
S9.
Given a complex
K into euclidean space g
of some subdivision
a finite subcanplex
K,
of K into Rn such that
K'
the secant map g induced by C1 approximation to
f,
9.1
Definition.
f
g is linear on each
The second step is to
in such a way that the ex-
SKI
f
(9.7 - 9.8).
Let
f : K -' Rn be a Cr map.
simplex which is contained in a simplex of which equals
f by a Cr map
is subdivided very carefully,
K.
C1 approximation to
Let
of
f
K1.
(9.1 - 9.6).
K1
g may be extended to all of
tension will be a strong
Cr map
which is automatically linear, will be a
on
f
of
K;
The first step is to show that if
prove that
and a
K1,
We would. like to approximate
Rn.
simplex of the corresponding subdivision
strong
f.
on the vertices of
s
The linear map
K.
be any
Rn
Ls: a
is called the secant map induced
f
f.
Similarly, if K'
is a subdivision of
then the secant map
K,
induced a f, Ii,: K' -, Rn, is the linear map which equals ces of
Note that if
K'.
the secant map induced by
9.2
Definition.
f
carries a simplex into e or
The radius
positive numbers.
tained in
a
Let
f
:
There is an
r(a)
so does
Rn-1,
t(a)
a - Rn be a Cr map; E > o
of
let
c
of
d(a) a
s
is the mini-
is
be
to s
con-
and thickness greater than fps.
a
r(a)/d(a)
and
such that for any simplex
is a a-approximation to 90
a
The diameter
The thickness
having diameter less than
the secant map Ls
of the simplex
a to Bd a.
is the length of the longest edge.
Lemma.
on the verti-
f.
mum distance from the barycenter
9.3
f
to,
THE SECANT MAP INDUCED BY f
§9.
Given
Proof.
in
b
where
f(x) - Fb(x) = h(x, b),
Then
approximation to less than
f
is a 8/2
Fb
such that
b
E1 neighborhood of
on the
this implies that
E1;
Df(x) - DFb(x) = Df(x) - Df(b).
E1 independent of
It follows that there is an
(x - b).
'
uniformly as
IIh(x, b)II/Iix - bli - 0
Also,
(See (2) of 1.10.)
Iix - bli--0.
Fb(x) = f(b) + Df(b)
let
a,
91
We require
b.
for
iih(x, b)Ii < 8/2
to be
e
We
Iix - bli < E.
further require that ilh(x, b) II/Iix - bli < 5t 0/4 for lix - bil < E. We prove that if than
then Ls
e,
center of
has thickness at least
s
s/2 approximation to
is a
where
Fbis,
This suffices to prove our theorem, since
s.
approximation to
of a simplex
into Rn
s
is the bary-
b
Fbis
is a
8/P
fis.
If L and
We need a preliminary result:
(1)
and diameter less
to
and
11
F
are linear maps
for all
L(x) - F(x)II < 9
then
x,
11 <28/r(s)
IIDL(x) for all unit vectors U in the plane of a. For let
b be the barycenter of
L(x) - L(b) + DL(b) (x-b),
and
s.
DL(x) = DL(b).
Similarly for
[DL(x) - DF(x)) u - [DL(b) - DF(b) J for some x in Bd s.
L is linear,
Since
DF.
Then
(x-b) /ll x - b II
This equals
([L(x) - L(b) I - [F(x) - F(b) J) /II x - b , II DL(x) - u - DF(x) u II < 29/11 x - b < 29/r(s) II
so
11
Proof of the lemma.
(a)
let
x - E aivi
Let
v0,...,vp
be the general point of
s.
Since
be the vertices of L.
Ls(x)
=
ai L8(Vi) - E ai f(vi)
Fb(x)
=
ai Fb(vi)
and
Fb
s;
are linear,
and
so that II
L8(x) - Fb(x) II
< 8/2 whenever
=
ii E ai h(vi, b) II < maxllh(vi, b) II . lix - bll < E,
so that II L8(x) - Fb(x) II
Now
Ilh(x,b) II
8/2,
which is the first part of our desired conclusion.
Furthermore,
II h(x, b)Ii/ll x - b II < 6to/4 whenever II x - b < E, so that ii La(x) - Fb(x)11 < maxlivi - b11bto/4 < d(s) sto/4 < 8 r(s)/4 11
since
Ls
to < r(s)/d(s).
From the preliminary result (1) we conclude that
is a 8/2 approximation to Fbis.
<
II. TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS
92 (a)
Exercise*.
Show by means of an example that the theorem fails
without the hypothesis that the thickness of
9.L
that
Let K be a finite complex.
Lemma.
There is a
to.
We first remark that if the theorem is true for the complex
Proof.
it is true for any complex which is isomorphic to K.
K,
such
to > 0
K has arbitrarily fine subdivisions for which the minimal simplex
thickness is at least
be a linear isomorphism. 11
is bounded away from zero.
s
For each simplex of
For let
f : K - K1
the ratio
K.
ro.
f(x) - f(y) II/II x-y II is bounded, and bounded above
Choose a and
so that
0
and
y belong to the same simplex of
be a subdivision of K of diameter less than
a
(i.e., the maximal diameter of a simplex of
to
Given
K.
is less than
K'
than
at0/9,
has diameter less than
K1
0, let
K'
and thickness grater than
the minimal thickness of such a simplex is greater than ponding subdivision of
a
T
to).
e,
and
corres-
and thickness greater
ac
as the reader may verify, so our result follows.
Now the complex K is isomorphic to a subcomplex of some "standard simplex" - the "standard simplex" of dimension p being the one having eo,el,...lep
as vertices, where
natural basis vectors for R.
co
is the origin and the others are the
Hence it will suffice to prove the theorem
in the case where K is this standard simplex. Let
J
be the cell complex whose polytope is
obtained as follows:
RP
If
io,i1,...,ip
RP,
whose cells are
are integers, consider the cube in
defined by the equation C(i1,...,ip) _ (x
I
ii < xi < ij +
1
for
.1
= 1,...,p)
and intersect it with the region
R(i0) _ (x
I
i0 < x1+...+ xp < 10 + 1)
The resulting set will be a cell. tration shows J
C(o, o, o)
(To illustrate, the accompanying illus-
intersected with R(o), R(1), and R(2),) Take
as the collection of all such cells and their faces.
The cell complex
Sq. THE SECANT MAP INDUCED BY f
J
93
has two properties which are important for us: is the image of one of the cells contained in the
Any cell of J
(1)
C(o,...,o),
unit cube
The simplex
(2)
subcomplex of
J,
under a translation of R. Em
spanned by O,me1,...,mcp
for each positive integer
Now we subdivide
J
at each step we use the centroid
as the interior point of the cell
c
m.
into a simpliciel complex L by the step-by-
step process described in Lemma 7.8; of
is the polytope of a
c
which determines the subdivision.
It follows that conditions (1) and (2) hold for the complex
a result, the simplices of L have a minimal thickness diameter
c
L as well.
to > 0,
As
and maximal
d.
The similarity transformation on RP which carries
x
into x/m
does not change the thickness of any simplex, and it multiplies the diameter
by
Therefore the image of
1/m.
rectilinear triangulation of
of thickness at least
to
and diameter at
d/m. Since m is arbitrary, the theorem is proved.
most
Exercise*.
(a)
Let
E1
Em under this transformation will be a
Let K
consist of a 2-simplex and its faces.
be the barycentric subdivision of
K1
vision of
K1,
etc.
K,
Show that the thickness of
K2
the barycentric subdi-
Kn approaches zero as
n --w.
I
9.5
plexes
K.
Unsolved problem*.
One would expect
to
Generalize this theorem to non-finite com-
and
a
to be continuous functions on
but it is not entirely clear even what the proper conjecture should be.
fKJ,
II. TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS
94
Theorem.
9.6
f : K - Rn be a
Let
there is a subdivision
8 > 0,
induced by
Rn
LK': K'
Given
finite.
of K such that the secant map
K'
is a
f
K
Cr map;
a-approximation to
f.
We now obtain the following generalization of this theorem, which will follow immediately from the lemma which succeeds it:
subcomplex of to
Let
Theorem.
9.7
Given
K.
f : K -' Rn be a
be a finite
Rn
such that
f
(1)
g equals the secant map induced by
(2)
g
(3)
K'
Here
equals
f
outside
equals K outside
9.8
subcomplex of
mation
g :
K1'
equals
h
f.
Here
1K1I
and on
_
St(K1,K)
there is a
K';
and condition (3) means
appears in
Cr map.
6 > 0
K'.
Let
be a finite
K1
such that any
is the standard extension of
K'
outside
K1'.
5-approxi-
can be extended to an E-approximation
f1K1
K;
(see 7.12),
St(K1,K).
The subdivision
Proof.
to extend
to
f
is defined on
c > 0,
Rn
to
h : K' - Rn
h
Given
K.
induced by
K1
f : K - Rn be a
Let
Lemma.
on
St(K1,K).
is the subdivision of
Ki'
f
St(K1,K).
that every simplex of K outside
and
K1
8-approximation g : K'
there is a
a > o,
let
Cr map;
of K is specified, and the map h
K'
IK) - St(K1,K).
to each simplex of
The only problem remaining is
whose interior lies in
K'
and to see how good an approximation h
is.
St(K1,K) - 1K1I,
We carry out this extension
step-by-step, first to the 1-simplices, then the 2-simplices, and so on.
For this reason, it will suffice to prove our lemma for the special case of a single simplex: Special case. its faces;
let
5-approximation tion.
Let
L1 - Bd a.
L be a complex consisting of a simplex Given
g: L' -+Rn to
Rn h: L' _, to fIL , where
c > 0,
fjL1
there is
8 > 0
a and
such that any
may be extended to an E-approxima -
L' is the standard extension of Li
Sq. THE SECANT MAP INDUCED BY Proof of special case.
is the barycenter of
a
linearly onto the line segment
g(y) f(a),
But there are difficulties here,
a.
for the resulting map will not be differentiable at definition, "tapering off" between
and
g(Bd a)
We modify this
o.
by a smooth function
f(a)
rather than linearly.-
a(t),
a(t),
Let for
95
g would be the
An obvious extension of
one mapping the line segment ya where
f
1/3 < t,
as usual, be a monotonic
and equals
1
for
ty + (1-t) a,
where y
mined, and
is unique if x / a.
y
t > 2/3.
is in Bd a
and
CO function which equals
If x is in t
0 < t < 1;
then
a,
o
x =
is uniquely deter-
Furthermore, if L is any subdivision
9 of
and
Bd a,
functions of
Let
L'
x
is the standard extension, then
on each simplex of
g be a
L',
y
and
except at the point
Cr map of a subdivision
of
L,
L1
are
t
Cm
x - a.
into
Rn.
Define h(x) = f(x) + a(t(x)) Since
a(.t(x)) = 0
each simplex of to
f
L'.
for
(g(y(x)) - f(y(x))I
in a neighborhood of
x
We show that
h
as good a
II
is of class
f
as
g
so
is.
a(h-f) /ax for t > 1/3. It equals
a'(t) (g(Y) - f(Y) I at/ax
+
Cr
fIBd a.
h(x) - f(x)II < II g(y) - f(y)II,
Co approximation to
Second, we compute
h
on
is automatically a good approximation
if g is a good approximation to First note that
a,
.
a(t) Iag/ay - of /ay) ay/
h will be
II. TRIANGUIATIONS OF DIFFERENTIABLE MANIFOLDS
96
where
f
are written as column vectors, as usual.
a(t) *Wax
and Ll'
and
g
and the map
Now
al(t)- at/ax
are bounded independent of the choice of the subdivision Hence we may make
g.
simply by requiring
and
Ig(y) - f(y)I
as small as desired,
a(h-f)/ax
Jag/ay - of/ayI
to be sufficiently
small.
Exercise.
(a)
need to be in order that
fag/ay - of/ayj
than
Compute exactly how small
Ig(y)
Ia(h - f)/axI
- f(y)I
and
should be less
s.
Complete the proof of the lemma using the result
Exercise.
(b)
for the special case proved above. (c)
Suppose that whenever then
g
carries
a simplex
a
(d)
al
plane
P
carries a simplex
into
Exercise.
, as well.
(P
,
then
h
at
of
K1
into the plane ( ,
Prove that whenever
carries
a
into (P
,
f
carries
as well.
k be a finite number of the
Let
Prove Theorem 9.7, with the additional conclusion that a
of K which
also carried by g (e)
f
of K into 9
planes in Rn. each simplex
Show that the following addendum to the lemma holds:
Exercise.
f
carries into one of the planes
6) i
is
into
Exercise.
Show that (c) does not hold if you replace the
throughout by a half-plane, for instance
Hm.
§1o.
Fitting Together Imbedded Complexes.
Suppose now we have entiable manifold
M.
Cr imbeddings of two complexes into the differ-
whose images in M
overlap.
We would like to prove
that by altering these imbeddings slightly, we may make their images fit together nicely, i.e., intersect in a subcomplex, after suitable subdivisions.
If M is euclidean space, this is not too difficult; the construcThe basic idea is to alter the imbeddings so
tion is carried out in 10.2.
that they are linear near the overlap (using the results of §9), for we know from §7 that two rectilinear complexes always intersect nicely.
If M is not euclidean space, we must use the coordinate systems on
which look like euclidean space, and carry out this alteration step-
M,
The process is not basically more difficult, but it is somewhat
by-step.
more complicated (10.3 - 10.4).
After this, however, the two main theorems
of Chapter II fall out almost trivially (10.5 - 10.6).
Definition.
10.1
phisms;
let
f1('K11)
(K1,f1)
and
(K2,f2)
images of of
with L2. of
f2(1K21)
K2,
be closed subsets of their union.
are polytopes of subcomplexes
respectively, and
fP1f1
and
L2
is a linear isanorphism of
L1
They intersect in a full subcamplex if
whose vertices belong to
K1
f2: K2 -. Rn be homeomor-
are said to intersect in a subcanplex if the inverse
f1(IK11) n f2(IK2I)
and
K1
and
f1: K1 - Rn and
Let
Li,
either for
L1
Li
contains every simplex
i .
1
such a case, there is a complex K and a homeanorphism
or for
i . 2.
f : K - Rn
In
such
that the following diagram is commutative:
Here
i1
and
i2
are linear isanorphisms of 97
K1
and K. with subcomplexes
98
TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS
II.
of K whose union equals
The pair
K.
it is called the union of
morphism;
the subcomplex
is unique up to a linear iso-
(K,f)
and
(Kl,f1)
Let L denote
(K2,f2).
i1 (K1) n 12(K2).
We will be interested in this situation in the case where
Rn and
f1: K1
f2: K2 -. Rn are, in addition,
The map
Cr imbeddings.
Cr non-degenerate haneomorphism in this case, but it
f : K-+ Rn will be a
may not be an imbedding, because the immersion condition may fail.
for some point b is.
of
the map dfb
ILI,
However, if
(See the example of 8.1.)
on 3t(LJ,Ki)
for
j
-
and
1
may not be
even though
1-1,
f
should happen to be linear
fi
then the union is necessarily an imbed-
2,
ding, for in this case dfb = ft3't b for b 1-1.
That is,
and
in ILI;
is given to be
f
Another case in
This fact will be of importance in what follows.
which the union is an imbedding is given in Exercise (c).
Let
Exercise.
(a)
fi :
Rn be a haneomorphism,
K1
j = 1,2.
Rn intersect in a subcomplex, prove that
f1: K1-+ Rn and
f2: K2
f1 : Kl' - Rn
f2: K2 -. Rn intersect in a full subccmplex, where
and
K2
and
are the barycentric subdivisions of Exercise.
(b)
union
K1
and
KI'
respectively.
K?,
Prove existence and essential uniqueness of the
(K,f).
Form an abstract complex K whose vertices are the points
Outline:
fi(0),
where
simplex of
vJ
is any vertex of
(j = 1,2).
If
(f1(vo),...,f1(vp))
let the collection
Kj,
K1
v0 ... vp
is a
be a simplex of
Then take a geometric realization of this abstract complex, and define To show
If
f
K.
f.
is a homeomorphism, you will need to consider the limit set of
f.
(c)
Exercise.
Let
fi :
K1 - Rn be
whose intersection is a full subccmplex. and let
ii
Int i1(IK1I)
be the inclusion of and
10.2
subcomplPxes of
Lemma. P
in K.
K1
then
Int
Let
f
:
P;
(j = 1,2)
f : K - Rn be their union,
If
IKI
is the union of
f.: K - Rn is a Cr imbedding.
Rm be a
P
whose union is
whose images are closed in Rm;
Let
Cr imbeddings
let
Cr map.
fIQ
let A be finite.
and
Let Q and A be fIA
Given
be
Cr imbeddings
§10.
s > 0,
and
FITTING TOGETHER IMBEDDED COMPLEXES
a-approximation g : P'-+ Rm to
there is a
such that
f
intersect in a full subcomplex, and their union is a
gfA'
g equals
Furthermore, St3(f-1f(A),P)
=
Proof.
f,
and
P0
equals
faces, so it contains f(P) - f(P0)
to
carry a point of
A.
P - P0
We assume
a
is less than half the distance from
a-approximation
into a point of
g(A).
g
to
f
cannot
We also assume that f,
gIQ'
is
a
and
Cr imbeddings.
are
Let
P1
be the subcomplex of
P whose polytope is Sf(PO,P).
Theorem 9.7, there is a a-approximation g : P' -. Rm to the secant map induced by f and
P.
along with their
f(A),
small enough that for any a-approximation g : P' -+ Rm to
f
Cr imbedding.
be the complex whose polytope is 3(ff(A)); it
so that any
f(A),
gjQ'
outside
P,
all stars being taken in
St(SE('g£(f-1f(A)))),
Let
P'
consists of all simplices whose images intersect
glA'
99
P'
equals
P
outside
on the subcomplex St(P1,P).
P,;
By
f which equals
further,
g equals
100
TRIANGULATIONS OF DIFFERENTLABLE MANIFOLDS
II.
Let Now
and
gIQ;
be the part of
Ql
that lies in
P1
so that
Q,
are linear imbeddings, so the images are finite recti-
gIA'
By Theorem 7.10, one may choose sub-
linear complexes in euclidean space.
divisions of these complexes whose union is a complex.
induce, via g 1,
a subdivision
of
P1
intersect in a subcomplex. gIQ." and
10.1).
Pl, 11
gIA"`intersect in a full subcomplex (see Exercise (a) of
We now claim that
in A"'.
of
PiT
in the standard way to a subdivision
P,
(see 7.12).
P'
For suppose
gIA?
and
gIQ1
1
We take the barycentric subdivision
Extend this subdivision of of
P "'
These subdivisions
such that
P1
1
so that
P1 - QQU A.
intersects
g(al)
Then
g(a2),
must lie in
al
so that in particular, sect in a subcomplex.
and
gIQ"'
Hence
intersect in a subcomplex.
where
a1
and
gJQ"'
But
IQ1I.
gIQ"'
is in Qf"
and
a2
by our original assumption on
IP0I,
lies in
al
gIA "'
and
is 8,
inter-
gIA "'
intersect in a subcomplex.
gIA"'
A similar remark shows this subcomplex is full. It follows that the union of ding;
and
gIQ "'
we apply the remark at the end of 10.1.
b1
in
B.
But
and b2
then
g is linear on
P1",
which contains YE(IPOI,P "').
gIQ"'
and
Corollary.
Let
M be a non-bounded
f(A)
Cr submanifold of some
is replaced by M
is contained in some coordinate
P which
and subdivide them finely enough that each has diameter
less than one-fourth the distance from P'
of
P
8E4(f-lf(A),P')
and
£-1#'(A)
in the standard way. into
to
Extend to
f-1 (M - U).
The result is that
f
U.
We then take the subcomplex
lemma applies.
Hence the
First take the collection of those simplices of
f-1f(A)
(f-1f(A),P')
by choice of
IP0I,
M.
Proof.
a subdivision
for
Cr imbedding.
providing we assume that
(U,h) on
intersect
is a
must lie in
The preceding lemma holds if Rm
euclidean space. throughout,
gIA"'
b1
Cr imbed-
g(b1) = g(b2),
IAI,
10.3
carries
For if
in
IQI
union of
system
is a
gIA "'
consider the
J
of
Cr map
P'
whose polytope is
hf : 3-+ Rm.
The preceding
There is an approximation G : J' - Rm to
hf;
G equals
§10. FITTING TOGETHER INEEDDED COMPLEXES
hf and J' equals
J
on 3t4(f 1f(A),P'),
g - h-1G
St3(f-1f(A),P').
outside
obtain the desired map;
and
g = f
no simplex of
101
Hence we may define
outside 3E3(f-1f(A),P') outside
P'
J
to
needs to be sub-
divided at all.
The preceding lemma guarantees that G is a homeomorphism.
g
sure
is, we need to have .g and
g(3M3(f-if(A),P'))
approximate
g(P - St4(f-1f(A),P'))
the lemma guarantees that G
are disjoint.
is a Cr imbedding;
10.4
Let M.
f
Given
g': L'
g : L - M be there are
M to
and
f
neighborhood on L : A1,A2,...
and
g'
to
and
g
(s
is to be continuous on the
ILI.)
carries each simplex of
L
into a coordinate
Order the simplices of
Choose a neighborhood Vi
of
g(Ai)
in M,
for each
is a locally-finite collection of compact subsets of
(Vi) (*)
f': K' -+ M
in such a way that each simplex is preceded in the ordering
by all its faces. so that
s-approximations
(See Exercise (a) of 7.13.)
M.
Cr submanifold of R.
Cr imbeddings whose images are closed in
Cr imbedding.
and
(KI
Assume
Proof.
f(A).
g respectively, which intersect in a full subcomplex,
such that their union is a
disjoint union of
lies outside the coor-
f(x)
Let M be a non-bounded
Theorem.
s(x) > 0,
if
chosen containing
(U,h)
: K- M and
Finally, one uses Exer-
is an imbedding.
g(x) a f(x)
in the preceding corollary:
dinate neighborhood
h-1 G
Note that the following additional conclusion holds
Exercise.
(a)
I
g
Similarly,
the fact that
is an imbedding follows from Exercise (a) of 8.3.
cise (c) of 10.1 to prove that
To be
closely enough that
f
Assume and
f
s
g,
M.
is small enough that for any B-approximations respectively,
g'(A1)
is contained in Vi
1,
f'
and is
disjoint from f'f-1(M - vi). Let
f0 a f
and
go m g.
Induction hypothesis.
Cr imbeddings, where K1 of
L;
and
respectively.
fi
and
gi
Suppose
fi: K1 - M and
is a subdivision of K and Li are
(1 - 1/21)
Furthermore, if J1
gi: Li - M are is a subdivision
s(x) approximations to
denotes the subcomplex of
Li
f
and
whose
g,
TRIANGULATIOIQS OF DIFFERENTIABLE MANIFOLDS
II.
102
suppose that
polytope is A1U...U Ai,
f : Ki - M and
intersect in
gilJi
i
a full subcomplex, and their union is an imbedding.
Let h : Q - M be the and
KLU Ji.
as subcomplexes of
in order that
h to
Extend
nate system on M.
to h such that
Given h'IQ'
h'jAi+1
h{iKj
and
h': P' ^ M
Cr imbeddings which intersect in
are
This union is the
a full subcomplex such that their union is an imbedding. same as the union of
Cr map;
is contained in a coordi-
there is an e-approximation
e > o,
and
on Ai+1
gi
The map h : P - M is a h(Ai+1)
Cr imbeddings, and
are
hIAi+1
we assume this done without
P by letting it equal
Now apply the preceding corollary.
hIQ and
Li,
We may have to subdivide
Ji.
should be a complex;
P
change of notation.
Q
which we also
P be the complex obtained by identifying each point
let
with the corresponding point of
of Bd Ai+1
(Ki,fi)
so that
Q,
is the polytope of a subcomplex of
Now Ai+1
denote by Ai+1;
Ai+1
and Ji
Consider Ki
giIJi.
Cr imbedding which is the union of
so the union of the latter
h'IJi U Ai+1,
pair is also an imbedding. If
mation to to
is small enough,
a
(1
s(x)/2i+1 approxi-
1/2i+1)S(x) approximation
-
as desired.
f,
Further, if Cr map
to a
is an
fi+1 - h'lKi
so that
h,
h': Ki - M will be a
gi: Li
is small enough,
a
gi+1: Li+1 - M which is a
s(x)/21+1 approximation to
Here we use Lemma 9.8 again.
M.
satisfied, so that
fi
and
We would like to define similarly for
g'
and
L'.
Thus the induction hypothesis is
are defined for all
gi
may be extended
h': Ji U Ai+1 -, M
f' a lima - - fi
i.
and
and
K' = lim Ki;
The induction hypothesis does not contain enough
information to enable us to do this, so let us examine the definitions of fi+1
and
gi+1
more closely.
Corollary 10.3 tells us that outside
St3(h-1h(Ai+1),P).
for simplices outside (*)
on
5,
carried
x
equals
Kip
fi
Hence
St3(fi1
Q'
fi+1
equals Q and equals
of K into
into V. Thus it is certain that outside
lection of subsets of
St3(f-1(Vi),K). K,
so that
and
equals
Ki+1
h,
equals
Ki,
Now by the original assumption
gi(Ai+1),Ki).
carries a point x
fi,
h'
gi(Ai+1) fi+1
equals
only if f1,
f
and
Ki+1
These sets are a locally-finite col-
l m fi
and
lim Ki
do exist (see 7.13).
§10.
equals
g1+t
Similarly, St3(gi-1
FITTING TOGETHER IMBEDDED COMPLEXES
gi(Ai+1),Li)
St3(Ai+1, L1)
-
f
St3(Ai+1, L).
C
outside
L1,
Hence the limits 8(x)-
That their union is a Cr im-
g, respectively.
and
equals
The limiting maps will automatically be
exist in this case as well.
approximations to
and Li+1
gi,
103
bedding is easily checked. Exercise.
(a)
By examining the proof of this theorem more closely,
draw the following additional conclusion: g(ILI),
equals
then we may so choose f,
outside
L,
and
f'
that
f'
Assume
equals
and
K,
Cr aubmanifold N of
may be chosen so as to carry these subcomplexes into
10.5
Theorem.
g : L - M be
f'
f and g carry subcomplexes of K
respectively, into the non-bounded g'
K'
Apply Exercise (a) of 10.3 and Exercise (d) of 9.8
to strengthen 10.4 as follows: and
and
St3(f-1(V),K).
Exercise.
(b)
K'
If V is any neighborhood of
Let M be a Cr manifold;
Cr triangulations of
M.
let
M.
Then
N.
f : K - M and
There are subdivisions of K and
L which are linearly isomorphic. In fact, given
8(x) > 0,
g': L' - M which are
and
is a linear isomorphism of Proof.
there are
L'
f
f'(IKI) - M.
10.6
triangulation.
so that
Similarly, if
Theorem.
(f')-1 g'
8(x) > 0
so that any
In the other case, we consider
and apply Exercise (b) of 10.4 to assure that
carries Bd IKJ
f'(Bd SKI) - Bd M,
such that
and g are necessarily onto maps, using Lemma 3.11.
M as a submanifold of D(M), K' - D(M)
f': K' -+ M
with K'.
Our result then follows from Theorem 10.4.
:
M,
In the non-bounded case, we choose
8-approximations to
f'
s-approximations
Cr triangulations of
into Bd M.
f'(IKI) C M. a
If
8
Again, if
is =all enough,
If M is a non-bounded
is small enough, 8
is small enough,
g'(ILI) - M.
Cr manifold, M has a Cr
If M is a manifold having a boundary, any Cr triangulation
of the boundary may be extended to a
Cr triangulation of M.
(If
104
II.
f : J - Bd M
is a
TRIANGULATIONS OF DIFFERENTIABLE MANIFOLDS
M
triangulation g : L of
Proof.
systems
Let
countable collection Let
covers
M be non-bounded.
Let
M.
Kii
M by m +
chosen so that
coordinate
1
of disjoint bounded open sets in Rm (see 2.8).
hi(Vij)
such that the collection
(Cii)
be a finite rectilinear complex in Rm contained in
Ki be the com-
let
U3a1 K. Then hi1: Ki - M is a Cr imbedding whose image contains
plex
Ui C. Let gi: Ki - M be a
a neighborhood of
chosen so that
I = o,...,m,
for
hit
si(x)-approximation to
(Ko,g0),...,(K,gm)
triangulation of M if contains
Ui Cij
lies in Rn.
is chosen small enough that
M has a boundary;
We triangulate the space For each simplex
Rn+1.
whose polytope is
of
a
a x(1 - 1/n,
consider the cells contained in
1
as follows:
JJI x[o,1) J,
To
Cr tri-
Suppose
and each positive integer
- 1/n+tl
and
a x(1 - 1/n),
J
n,
which are
The collection of all such cells is a cell complex IJ?x [0,1);
this cell complex may be subdivided into a - 1/n)
a x (1
We denote the resulting simplicial complex by Let
gi(JKi1)
f : J - Bd M be a
let
simplicial complex without subdividing any cell kind.
Cr
(see Exercise (b) of 3.11).
Now suppose angulation.
ei(x) > 0
intersect in
This union will be a
a full aubcomplex and their union is an imbedding.
let
Cr
is the union of a
hi(Ui)
and containing a neighborhood of hi(Ci1);
hi(Vij),
is a
is a linear isomorphism
f
Cover
be a closed set contained in Vii
Cij
f
L.)
i - o,...,m,
(Ui,hi),
g-1
such that
M
of
with a subcomplex of
J
an extension of
0 r triangulation of Bd M,
of the second
K.
be the subcomplex of K whose polytope is
IJJx (0,5/61;
P : Bd M x[o,1) - M be a product neighborhood of the boundary.
The
composite
Jx[0,1) fxi> Bd Mx[0,1) P> M when restricted to
K0,
is a
closed in M and contains
,
Cr imbedding g : K0 - M whose image is in its interior.
P(Bd Mx [0,4/5))
Let h : L - M be a Cr triangulation of the non-bounded manifold Int M.
By subdividing L if necessary, we may assume that any simplex of
L which intersects the set P(Bd Mx [0,3/4)),
P(Bd M x(4/5))
is disjoint from the set
using Exercise (a) of 7.13.
Let
L0
be the subcomplex
of L consisting of simplices whose images under h intersect
§10.
FITTING TOGETHER IMBEDDED COMPLEXES
M - P(Bd M x[0,4/5)), and their faces.
whose image is closed in M,
Then h : Lo - M is a
contains M - P(Bd M x[0,4/5])
h': Lo -' M to g and
g': Ko -' M and
Cr imbedding in its interior,
By Theorem 10.4, there are
and is contained in M - P(Bd M x[0,3/4]).
approximations
105
respectively,
h,
whose intersection is a full subcomplex such that the union is an imbedding.
According to Exercise (a) of equals
on
K0,
IJI x(0,1/2].
thus be the required h'(ILoI)
cover
Now h(ILOI)
10.4, we may assume
The union of
Cr triangulation of
M,
contains
contains
providing
M - P(Bd M x[0,3/4])
P(Bd MX [0,1/21).
K
interior. then
and
in its interior, and
g'
and
will contain these two
h'
and
Thus in this case
Exercise (b) of
g'(IKOI)
g'(IKOI)
automatiand
h'(ILOI)
M.
10.7 Problem*. f:
KO
will
g'(IKOI)
in its interior.
closed sets, if the approximation is good enough;
Let
and
and (LL,h')
(KO,g')
P(Bd M x[1/2,4/5))
3.11 applies to show that the images of
do cover
g,
M.
g(IKOI)
cally contains
equals
g'
fIKO
Let A be a closed subset of the manifold
M be a Cr imbedding such that
If KO
contains A in its
f(IKI)
is a subcomplex of K such that
may be extended to a triangulation of
M.
M.
fIK0
triangulates
A,
106
NTIABIE MANIFOLDS
II. TRIANGULATIONS OF DI 10.8
Let M be a non-bounded
Problem*.
W -+ M be the retraction of a neighborhood W of M in
of
Rn.
Rn
onto M which was defined in 5.5.
Let
r:
plex K in Rn
Prove there is a rectilinear com-
lying in W such that
a suitably chosen smooth triangulation this will not suffice. Ki
with
is linear on
IKKI C Int IK1+1I,
Ki,
If M
is not compact,
f is isotopic to there exists
K' - M to
f.
on K.
g
Prove that any map
Specifically, prove the fol-
5(x) > 0
there is a map
f,
g: K' - Rn
and use the fact that if
f: K - M be a Cr map.
Let
Given E(x) > o, g:
M.
then the secant map induced by g equals
sufficiently close to
approximation
f: K
Instead write K as the union of finite subcom-
10.9 Problem*.
lowing:
r1K is a Cr triangulation of M.
If M is compact, one may take the secant map induced by
Hint:
plexes
Cr submanifold
such that for any 5ft:
IKI x I - M such
that
is of class
(1)
ft
(2)
ft: K'
(3)
f0 - f and
on the cell
Cr
for each simplex
a x I,
a
of
K',
Hint:
M is an e-approximation to
t,
and
f1 - g.
Use the techniques of Problem 5.15.
Let
10.10 Definition.
of the non-bounded n-manifold
if for each x
for each
f,
M.
N1, ..., Nk be non-bounded submanifolds They are said to intersect transversally
in the intersection of any collection Ni ,...,Ni
of these
1
submanifolds, there is a coordinate system such that
h(U n Ni )
that of Nik,
for each
10.11 Problem.
K,
U
Rn about x in M
is contained in a plane in Rn whose dimension is k.
Generalize Theorem 10.4 as follows:
is a subcomplex of L and complex of
h:
f-1g
(1)
is a linear isomorphism of H
then we may require that
(f')-1g'
-
f-tg on
If H onto a subIHI.
be non-bounded submanifolds of M which intersect
N1, ..., Np
Let
(2)
and
L,
respectively, into the submanifolds
and
g'
to carry these subcomplexes into
then we may require
Ni,
Assume
H is a full subcom-
denote the simpiices of L not in
A1, A2, ...
gi: Li - M are Cr imbeddings which are
to
and
f
f1:
M
K1
and
fi: Ki - M
approximations
1/21) 8(x)
-
on
IHI.
Let
denote
Ji
suppose that
whose polytope is H U Al U ... U A1;
Li
the subcomplex of
fi gi = f-1g
and
respectively,
g,
(1
Alter the
H.
Suppose
induction hypothesis in the proof of 10.4 as follows: and
f'
N1.
To obtain condition (1), proceed as follows: Let
of K
L(1)
Condition (2) will follow readily from Exercise (d) of 9.8.
Hint:
plex.
and
K(i)
carry subcomplexes
g
and
f
If
transversally.
107
FITTING TOGETHER IMBEDDED COMPLEXES
§10.
intersect in a full subcomplex and their union is
giIJi
a Cr imbedding.
Let
non-bounded manifold M. Given
closed in M. g
such that (1)
f-1g':
g
If
g(L)
M
g': L'
to
L' -+ K is linear.
is linear on the subcomplex H of
require that (2)
L -+ M be a Cr imbedding, with
g:
8(x) > 0, there is a s-approximation
f-tg
If
f: K - M be a smooth triangulation of the
Let
10.12 Theorem.
on
g' - g
L.
we may
IHI.
L1, ..., Lk
carries subcomplexes
L into
of
the transversally intersecting non-bounded submanifolds N1, ..., Nk
of
a neighborhood of may require that Proof.
respectively, and if g(LL)
K,
in Ni
(f')-1g":
f(f')-1g"i
where
a
1,
then we
is a linear isomorphism of
f-1g
H
since this may be obtained by a preliminary subdivi-
We apply 10.11 to obtain e-approximations
L' - M,
for each
triangulates
f
g'(L1) C N1.
We may assume that
with a subcomplex sion.
M,
is small enough that
L' - K is linear, and equals
f'
f': K'
M and
is a triangulation.
f-1g on
IHI.
We let
g":
Then g'
II. TRIANGUTATIONS OF DIFFERENTIABLE MANIFOLDS
108
To obtain (2), choose
a neighborhood in Ni tain
of
If
f(Ki)
Corollary.
Let
angulations of the manifold
M
g': L'
is small enough,
a
choose
equal to
g'
Proof.
K - M and
f:
Given
s(x) > o,
g
A restriction of
(M C Int N).
a triangulation of Hint:
f-1g': L'- K
L, then we may
f
g'
Assuming the subcomplexes
and
g,
obtaining triangulations
and
f
to be the submanifold Bd M
will be the desired triangulation of
of
M.
Let M be a Cr submanifold of N which is closed
°..cw there is a cr triangulation of N which induces M.
Suppc
the hardest case.)
that both M and N have boundaries.
Extend the imbedding
a neighborhood V of M in D(M) which
there is a 6-approxima-
Bd M to be full subcomplexes, we may
We now apply 10.12, taking N
10.14 Problem.
of M.
Then
Ni).
If M is non-bounded, this follows at once from 10.12.
double the triangulations
of D(N)
will con-
on IHI.
of K and L which triangulate
in N
f'(KI)
such that
M,
In the other case, we form the manifold D(M).
D(M).
and contains
Ni
g: L - M be smooth tri-
is linear on the subcomplex H of
f-1g
If
M.
g which triangulates
to
is linear.
of D(M).
lies in
Ni.
C
10.13
tion
so that
(see Exercise (b) of 3.11; by 10.11, both lie in
g' I (I,)
g'(L1)
Ki
g(Li).
is
M - N to an imbedding
into Int N.
luces a triangulation of Bd N;
Apply 10.11, letting
(This is
Let let
N1 = i(V), N2 - Bd M, and
f
i of
be a triangulation
g be a triangulation N3 = Bd N.
REFERENCES (1)
(a_]
Locally tame sets are tame,
R. H. Bing, 145-158.
J. G. Hocking and G. S. Young,
Ann. of Math., 59 (1954),
Topology, Addison-Wesley, Reading,
Mass., 1961. (3]
W. Hurewicz and H. Wallman,
Dimension Theory, Princeton Univ. Press,
Princeton, N. J., 1948. (4]
J. Milnor,
Differential Topology (mimeograp)ed), Princeton Univ., 1958.
(5)
J. Milnor,
[6]
J. Milnor,
17I
E. E. Moise, Affine,structures in 3-manifolds, theorem and Hauptvermutung, Ann. of Math., 56 (19
On manifolds homeomorphic to the 7-sphere, Ann. of Math., 64 (1956), 399-405. Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math., 74 (1961), 575-590.
(8)
D. A. Moran, R8,7.sing the differentiability clash euclidean space (thesis), Univ. of Ill., 1962.
(9]
M. Morse,
.triangulation 96-114.
a manifold in
Oii elevating manifold differentiability, Jour. Indian Math.
Soc., XX1V (1960), 379-400. (10]
J. R. Munkres,
(t structions to the smoothing
able homeomorphisms, Ann. of Math., 72 (1960),j', (11)
C. D. Papakyriakopoulos,
1121
N. Steenrod,
'iecewise-differenti.1-554.
A new prpof of the i -'iance of the homology groups of a coWlex, Bull. Soc. Math. Grace, et 1-154. The topology of fibre bundles, pr..
Univ. Press,
Princeton, N. J., 1951. (13)
J. H. C.
Manifolds with transverse f:I
f-a euclidean
apace, Ann. of "Math., 73 (1961), 154-212. (14]
J. H. C. Whitehehd, On C1-complexes, Ann. of
(15]
H. Whitney,
Differentiable manifolds, Ann. of
It (1940), 809-824.
(1935), 645-
68o. (16)
J. Stallings, The piecewise-linear structure of euclidean space, Proo. Cambridge Philos. Soc., 58 (1962)0 481-488.
(17]
M. Kervaire, A manifold which does not admit any differentiable structure, Comment. Math. Helv., 34 (1960), 257-270.
log
INDEC OF TERMS angle-approximation
differentiable map,
8.5
approximation of a map,
for manifolds for complexes ball,
Bm
7.1
barycentric coordinates
7.1
barycentric subdivision
7.1
of a complex,
boundary, Bd M
1.1
8-approximation
df dfb
euclidean space
cell complex
7.6
euclidean half-space
3.5, 3.8
combinatorial manifold
8.4
complex
7.1
coordinate neighborhood
1.1
fine
coordinate system
1.6
full subcomplex
1.0
Grassmann manifold,
3.1
Df(x)
1.3
7.2 7.6
1.1,5.10 1.0
EP
1.0
3.5
F1(M,M)
7.1
curve
1.1
Rm
combinatorial equivalence
Cm
8.2
double of a manifold, D(M)
3.5 7.2
cube,
3.3
of a manifold of a cell of a cell complex
cell coarse topology
1.6
dimension,
basis for a differentiable 1.6
8.1
differential of a map, of a manifold,
structure
1.7
differentiable structure
1.0
barycenter
Co
1.3
in euclidean space of a manifold of a complex
face,
of a simplex of a cell
7.1
7.4
Cr topology,
3.5,3.8 10.1
5.1.
Cap,n
1.0
Am imbedding, immersion, of manifolds
1.8
of complexes
8.3
8-approximation, for manifolds
3.5
for complexes
8.5
interior of a manifold, Int M
diameter of a simplex
9.2
"intersect in a subcomplex"
Diff M
6.3
lifting of a map
diffec*norphism
1.8
differentiable homotopy
3.9
differentiable isotopy
3.9
differentiable manifold
1.6
limit set of a function,
111
1.1
10.1
5.3 L(f)
2.9
linear map of a complex
7.1
linear isomorphism of a complex
7.1
112
INDEX OF TERMS
locally contractible
Riemannian metric
3.0
secant map
9.1
5.1
simplex
7.1
manifold
1.1
skeleton
7.6
manifold-with-boundary
1.1
sphere,
non-bounded manifold
1.1
non-degenerate map
8.1
normal bundle
5.5
normal map
5.4
locally-finite M(p,q),
M(p,q;r)
5.16 1.1,1.2
Sm-1
1.0
standard extension of a subdivision star,
St(x, K)
strong
C1 approximation
strong
C0 approximation
3.5
open simplex
7.1
subcomplex
7.1
orientable manifold
5.9
paracompact
7.1
1.1
subdivision, of a complex
of a cell complex partition of unity
7.6
2.5,2.6
submanifold
piecewise-linear
7.1
polytope
7.1
product neighborhood
5.9
radius of a simplex
9.2
rank
1.7
rectilinear triangulation
7.10
refinement
1.1
regular homotopy
3.9
retraction
5.5
tangent bundle,
2.1
T(M)
3.2
tangent map
5.4
tangent space
3.1
tangent vector
3.1
thickness of a simplex
9.2
triangulation
8.3
union of complexes velocity vector
10.1 3.1
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