Lectures on polyhedral topology

be

the set whose elements are nonempty sets of the following sort: Let

n}=

J .

Then if it is not empty the set Ρ Γ\Η.θ... ΠΗ. Π Ρ, Π .... Π Ρ, "^q^ ^ ^n-q)

•25

is -an element of P. The properties of Q!) follow from 1,4^.13.0 If the above, the union of elements of ^ constitute the boundary ^ A of A. Then using remarks preceding U5.2, we have, if

excluding A

1,4.9/ and the

(p, A. / A, then

dim A^ ^ dim A, We have seen that if A is a bounded open convex set of dim ^

1, then A = ^ ( ^ A), Hence by an obvious induction,·

we have 1.5.Proposition.

A closed convex cell is the convex hull of a

finite set of points. Q A partial converse of 1.5.5, is trivial: 1.5.6. The convex hull of a finite set is a finite union of open (closed) convex cells.' D The converse of 1.5.5 is also true. Μ

Ex. t.5.7· The convex hull of a finite set is a closed convex cell. Hints Let J^x^ >·'·> 1.4.16

^x^

^ finite set in vector space V. By ~ ^

x^)· It is enough to show

that 0(x^ }.·.) x^) is an open convex cell. Let Μ be the linear manifold generated by ^x^

· Let dim Μ = k . Write

A = 0(x^ J... J x^) -J A = K. ^ ^ ,..., x^^ . A is open in M. To prove tl:^e proposition it is enough to show that A is the intersection of half spaces in M. Step 1. A and

A are both union of open (hence closed) simplexes

with vertices in ^ ^ 1.2.7.

. The assertion for A follows from

26

Step :2. If Β is a (k-1)-simpiex in ^ Λ and Ν is the hyperplane in Μ .defined by B, then A cannot have points in both the half spaces defined by Ν in Mi Step 3.. It is enough to show that each point of closed

(k-1 )-simplex with vertices in

^A belongs to a

-^x^

x^^"· ·

Step 4. Each point χfcbA is contained in a closed vd-th verticer in

^x^

» To prove this let C^ )»··>

be the closed simplexes contained in x^ jii.j x^

which contain x^

(k-1simpleχ

c) A with vertices in

and Dj 3..., D^ ( C"^^) which do

not contain x. by Step I) U C. Ο D. - ^ A^ Consider any point i j , a ^ A and a point bfe0(a, x). Let C^ (resp. D^ ) denote the closed simplex whose vertices are those of C. (resp. D.) and 'a'.. ' J By the remark following 1.4.18. U C.' W D.' ^ A, Show that r i ^ j ^ then

C^

is a neighbourhood of b. If dim G^^k-1

for all i,

dim C^'^ k-1 for all i. Use 1.5.2 to show that in this

case [J C^

cannot be a neighbourhood of b.

Q

Since the linear image of convex hull of a finite set is also the convex hull of finite set, 1.5.?, immediately gives that the linear image of a closed convex cell is a closed convex cell. If A is an open convex cell in V and ψ

a linear map

from V to W, then φ(Α) = <φ(Α), by 1.4.16, hence by 1.4.15 C|)(A) is an open convex cell. Therefore 1.5.5. Proposition. The linear image of an open (resp. closed) convex cell is an open (resp. closed) convex cell. Q

27

It6. Presentations of polyhedra If ^ is a set of sets and A is setj we shall write .

AV^iP

when A is a union of elements of (p . For exaijiple (d) of U5.4 can be expressed as "If A ^ (p, then

'bAV"(p

We make the

obvious convention, when 0 is the empty set> that φ matter what (p U6iU

no

is.

Definition. A polyhedral presentation is a finite set (p

whose elements are open convex cells, such that A ζ (p Implies bAVp. 1.6.2. Definition. sentation ^

A regular presentation is a polyhedral pre-

such that any two distinct elements are disjoint, that

is,

implies A Π B = 0.

Ex, The (p of proposition 1,5.4 is a regular presentation. t.6.3· Definition.

A simplicial presentation is a regular pre-

sentation whose elements are simplicies and such that if A then every face of A also belongs to (p . If

(ps·^® polyhedral presentationsj we call Q j

subpresentation of ^

, If ^

a

is regular (resp. simplicial) then(^

is necessarily regular (resp. simplicial). The points of the 0-cells of a simplicial presentation will be called the vertices of the simplicial presentation. The dimension of a polyhedral presenΡ tation^is defined to be the maximum of the dimensions of the open cells of (p . 1.6.4. Definition. li" (p is a polyhedral presentation

|(pl will

28

be used to denote the union of all elements of

» We say that(J)

is a presentation of ^(J·^ or that ^(p^ has a presentation

,

Recall that in 1.1, we have defined a polyhedron as a subset of a real vector space, which is a finite union of convex hulls of finite sets. It is clear consequence of 1.5.4, 1.5.5 and 1.5.6 that t,6.5. Proposition. Every polyhedron has a polyhedral presentation. If (p is a polyhedral presentation, then

is a polyhedron. D

Thus, if we define a polyhedron as a subset of a real vector space which has a polyhedral presentation, then this definition coincides with the earlier definition. 1.6.6. Proposition. The union or intersection.of a finite number of polyhedra is again a polyhedron. Proof; It is enough to prove for two polyhedra say Ρ and Q. Let φ and Q) be polyhedral presentations of Ρ and Q respectively. Then

is a polyhedral presentation of Ρ \JQj hence Ρ U Q

is a polyhedron. To prove that Ρ Π Q is a polyhedron, consider the set

consisting of all nonempty sets of the form A Π B,

for

and Β t©^·

It follows from 1.4.13 that

polyhedral presentation. Clearly \ Ρ Π Q is a polyhedron. If X C Y

is a

| = Ρ Π Q. Hence by 1,6.5

Q polyhedra, we will call X a subpolyhedron'

of Y. Thus in 1.6.6, Ρ Γ\ Q is a subpolyhedron of both Ρ and Q. 1.6.7. li* (p

0"» sre two polyhedral presentations consider

the sets of the form A K B ,

A ^(p ,

Clearly A X Β is

29

an open convex cell, and by 1.4.20 b (AXB) is the disjoint union of

bA X

the fom if both ρ

B, A / S 5 b

and ό A > 0 B. Thus the set of cells of

A X B , A^(p., Β ^ (3i is a polyhedral presentation, regular and

are. This we will denote by (p y.

.. As above,

we have, as a consequence that Ρ K Q "is a polyhedron, with · presentation ^ X

.

Ex. 1.6.8, The linear image of a polyhedron is a polyhedron (follows from the definition of polyhedron and the definition of linear map).Q Recall that we have defined a polyhedral map between two polyhedra as a map whose graph is a polyhedron. 1.6.9· Proposition.

The composition of two polyhedral maps is a

polyhedral map. Prooft

Let X, Y and

Ζ be three polyhedra in the vector spaces

U, V and W respectively, and let f ί X polyhedral maps. Then ~P(f)(^ U polyhedra.

By 1.6.7,

^ Y, g : Y

V and P ( g ) C _ V

and X A p ( g )

are

are also polyhedra

ζ)Γ\( 'ί )
in U>tsV><.W. By 1.6.6,

> Ζ be

' is a polyhedron.

This intersection is the set S : ^(x, y, z) ( X ^ X, y = f(x), ζ = g(y)j in U A V X W .

By 1.6.8 the projection of U y ^ V X W

to

UKW

takes S into a polyhedron, which is none other than the graph of the map g ο f : X .

^ Z. Hence g ο f is polyhedral. Q

If a polyhedral map

f :Ρ

^ Q, is one-to-one and onto

we term it a polyhedral equivalence. Ex. 1.6.10. If, f 5 Ρ

^ Q is a polyhedral map, then the map

30

f': Ρ

^P(f) defined by f'(x) = (x, f(x)) is a polyhedral

equivalence. 1.6.11. Dimension of a polyhedron. The dimension of a polyhedron Ρ is defined to be Max. dim C

j C ^(p » where (p is any polyhedral presentation of P. Of course we have to check that this is independent of the

presentation chosen. This follows from 1.5.2. Let Ρ and Q be two polyhedra and f : Ρ a polyhedral map. Let

^ Q be

: Ρ X. Q — ^ , Ρ and jJu : Ρ ^ Q

be the first and second projections. If "PCf), then the open cells of the form

> Q

is any presentation of (C), CT

presentation of P, regular if and only if ^

is. a

is regular. If f

is a polyhedral equivalence, then the cells of the form |Jl(C), C ^

is a presentation of Q. This shows that

1.6.12. Proposition. invariant.

The dimension of a polyhedron is a polyhedral

α

1.7. Refinement by bisection. 1.7. 1. Definition. If say that (p refines

and

are polyhedral presentations, we

, or (p is a refinement of

provided

(a) (b) If

Aei(p, and

In otherwords, ^

and

B ^ C ^ , then AAB = 0 or

ACb.

are presentations of the same

polyhedron and each element (an open convex cell) of (p is contained in each element of

which it intersects. Hereafter, when there

31

is no confusion, we vd.ll refer to open convex cells and closed convex cells as open cells and closed cells. A polyhedral presentation is regular if and only if it refines itself. Let (3 = (P ; H^, H~) be a bisection of the ambient vector space V (1.3.9); G^ a polyhedral presentation offi.polyhedron in V, and let A ξ (2.. We say 'that

admits a bisection by

(B at A, provided; Whenever an open cell A, ζ. Q. intersects B a (i.e. A, n d A i Φ), and dim A^ < dim A, then either A^

Ρ or

or

(in particular this should be true

for any cell in the boundary of A). If (j^ admits a bisection by (J^ at A, then we define a presentation Q

as follows, and call it the result of bisecting

by (3 at A; consists of

with the element

the nonempty sets of the form, Α Π Ρ or

A removed, and with

E* or A n h"^

that is d

=U(I-iA})

A O H ^ A π η"}

^

I

By 1.4.13j and the definition of admitting a bisection, Q , a polyhedral presentation.

Clearly

refines

is

, if ^ ^ is

regular. We remark that it may well be the case that A is contained in Ρ or h"^ or H~. In this event, bisecting at A changes nothing at all, that is

= CS- . If this is the case

we call the bisection trivial. It is also possible, in the case of

m

32

irregular presentations, that some or all of the sets ΑΛΡ, A Λη·*·, A Π Η

may already be contained in Q . - ^A] in

this event, bisection will not change :§,s much as we might expect. Ex. 1.7.2». Let A and Β be two open cells,, with dim A.^dim Β and A / B. Let

(B

: fp J h!

hTI

be bisections of

space such that A is the intersection of precisely one element from some of the

(β .'a.. J

that Β Π Ρ

BOH

If A Π Β / 0, then 3 .and Β Λ H~

an S.

1 ^

m, such

are all non vacuous. Q

What we are aiming at is to show that every polyhedral presentation ^

has a regular refinement,, which moreover is obtained

from (p by a particular process (bisections).

The proof is by an

obvious double induction; we sketch the proof below leaving some of the details to the reader. 1.7.3. Proposition

REFI ( (p ,. (P' j {s^)

There is a piOcedure) whidh) applied to a polyhedral presentation (p , gives a finite sequences cells by bisections of space), which start on and

of bisections (at , give end result (p ,

is a regular presentation which refines ^ ..

Proof: Step 1. First, we find a finite set

(B . ;·("ρ 5 H'^j H ] , j = 1,.,.n

J

j

3 jJ

of bisections of the ambient space, such that every element of (p is an intersection one element each from some of the (3 j's». This is possible because every-element of ^

is a finite intersection of

hyperplanes and half spaces,, and there are only a finite number of elements in (p

33

Step 2. Write (p = (p^, index the cells of (p^ in such a way that the dimension is a non decreasing function. That is define p^ to be the cardinality of fp , arrange the elements of Μ 0 such that dim D^^ dim for all - 0 < k ^ κ >«5t

fp as D^,..., D*^ , vJ 0 0 , Pq ρ - 1. Ο

Step 3. SQ ^ denotes the process of bisecting ^ ^ Inductively, we define S^ tPojk: ^^ i+1

to be the process of bisecting

G^i 5

(Po k+1

defined since the elements of nondecreasing dimension.

Step 4. Write

at D^ by

result. This is well

^ are arranged in the order of

This can be done until we get S„

i = (PQ ρ

and

^ repeat step (2) and then the step (3)

with bisection

(j^j instead of 2 And so on until we get

is clearly a refinement of ^

=

is regular. Each element by

φ

. 1 (O , when the process stops, (p ^ η y η ; it remains to show that belongs to some

fp

^

and

each element of (p -is a finite intersection of exactly one element each from a subfamily of the (p) s. It is easily shown by double J induction that if A ^ Ο , then for any j, i ^ 0 $ either α Γ P. ^ η J or A(2h. or H^. That is 0 admits a bisection at A by · J j V η (Β. for any j, but the bisection is trivial. Let C , D

e (P

,C

C ο D ^ 0, dim C ^ dim D, Then since C is an intersection of one element each from a subfamily of the (j^ .'s, by 1.7.3, there exists an

such that D Π Pj|_ , D Π H^ and D O, Η £

But this is a contradiction.

Hence (p

are all nonempty.

is regular. Write

D

34

+ Ρ^

(Pn-

j· This gives the

Ό

"REFI ((P . p', {s.p".

Q

We can now draw a number of corollaries; 1.10.4. Corollary.

Any polyhedron has s. regiolar presentation. Q

U10.5.

Any two polyhedral' presentations (p »

Corollary.

of the same polyhedron X have a common refinement obtained from

^

, which is

and from G^ by a finite sequence of bisections.

To see thisJ note that ^ of X. The application REFI

is a polyhedral presentation ^

provides (p. .

Let ^T^^ and ^U^ | denote the subsequences applying to respectively; observe that they both result in (j^ .

and

D

1.10.6. Corollary. Given any finite number

(P

polyhedral presentations, there is a regular presentation ^ with

^l(prl ' (p j_| =

^

subpresentations

of of

>·.·>

>

for all i and φ) . is obtained from

_

by a finite sequence of bisections. This is an application of R Ε F I ( φ and an analysis of the situation.

Q

y .. ^ ^

,

35

Chapter II Triangulation As we have seen, every polyhedral presentation φ) regular refinement.

has a

This implies that any two polyhedral presentations

of X have a common regular refinement, that if Χζ] Y are polyhedra there are regular presentations of Y

containing subpresentations

covering X, etc.. In this chapter we will see that in fact every polyhedral presentation has a simplicial refinement, and that given a polyhedral map Ρ

f : Ρ ——^ Q, there exist simplicial presentations of

and Q with respect to which f is "simplicial",

2.1. Triangulation of polyhedra.. A simplicial presentation

of a polyhedron X is also

known as a linear triangulation of X, We shall construct simplicial presentations from regular ones by "barycentric subdivision". 2.1. 1. Definition.

Let ^

be a regular presentation.

of (p is a function Τ| : (p

> such that

A centering G, for

every C ζ; p . In other words, a centering is a way bo choose a point each from each element (an open convex cell) of (p . 2.1.2. Proposition.

"^q

^]

^k ^^^ elements of (p ,

ordered with respect to boundary relationship, then

(C^),,..,

is an independent set for any centering T^ of ^ » Proof; Immediate from 2.1.3. Proposition.

1.4.9?

by induction.

D

Suppose that φ) is a regular presentation and

36

C^^p. C is the disjoint union of all open simplexes of the form

where A^ ^ ^ Proof;

, ^o < ^^ <

By induction.

of dimension < form 0

< \

and

Assume the proposition to be true for cells

dim C.

^ C is the union of all simplexes of the

η(Α^), ... ,η(Α^)) where A.

and AQ<_ ..

A^ (since <

, A. <

C

is transitive). Since C is abounded

open convex cell C is the union of 0( Tl(C), x), χ ^^C and

(C)

(see the.remark following 1.4.18). Now 2.1,2 completes the rest. D It follows from 2.1.2 and 2.1.3, that if (p is any regular presentation, then the set of all open simplexes of the form 0( η(θ^),

^(Cj^)), for C.

a simplicial presentation of

, with CQ< ..

, is

. This leads to the following

definition and proposition. 2.t.4. Definition. of ^

If |p is any regular presentation, Y| a centering

; the derived subdivision of ^

open simplexes of the form 0( CQ<1 ...

^(Qq),

relative to Y| is the set of

^(^j^))

. It is a simplicial presentation

, "^i ^

(of ^(p

(P >

) and is

denoted by d ((p , rj ). The vertices of d ((P , "Γ| ) are precisely the points (0-cells) 'Tj (C) , C

When Vj is understood, or if the particular

choice of T| is not so important, we refer to d( derived subdivision of ^

,

) as a

and denote it by d(p .

2.1.5· Proposition. Every polyhedral presentation admits of a simplicial refinement. Q

37

Hence every polyhedron can be triangulated. 2.2. Triangulation of maps. Now, we return, to polyhedral maps. If f : Ρ •—^ Q is a polyhedral map, we have seen that the map f': P - 9 T ( f )

given by f'(x) = Qx , f(x)) is a polyhedral

equivalence and that any presentation of ~P(f) gives a presentation of Ρ by linear projection. Also, we saw in 1.3, that if A is a convex subset of vector space V and ^ : A — ^

W a map of A

into a vector space W,

ί Ql — ^

A,, AgeCJL, A, 4

is called combinatorial if for all

A^ implies

φ(Α^) ^ φ(Α2).

But unfortunately there may be several distinct polyhedral tasips

— 1 ^ 1 inducing the same combinatorial map

and a map ^ (JL | — ^

( J L — ,

inducing some combinatorial map (JL —^(j?)

need not even be polyhedral (We will see more of the-se when we come to 'standard mistake'). a unique map

If turns out that a map

— ^ (12) induces

— — ^ \Φ>) ^^ ^ require that the induced map

to be linear on each cell of (JL · Bu.t in this case it is sufficient to know the map on 0-cells (vertices); one can extend by linearly. This naturally leads to simplicial maps. 2.2.2. Definition. Let X and Y be polyhedr-a,

and

^

3S

simplicial presentations of X and Y respectively. f ;X

A map

^ γ is said to be simplicial with respect to

and ^

,

iff 1) f maps vertices of each simplex in into the vertices of some simplex in

.

and 2) f is linear on the closure of each simplex in ^

. /

f is polyhedral, since its graph has a natural simplicial presentation isomorphic so ^ Let ^ (resp. ^ Q ) ^

·

of Λ

and ^

.

be two simplicial presentations.

be the set of vertices of Ji (resp. ^

—^^^

^^ ^

Sl simplicial map from

2.2.3. Ebcample. If φ centerings of ^

^

). If

carries the vertices of a simplex

into the vertices of some simplex of ^

say cii

Let

: ^D — ^

and

to ^

, then

(also) we vdll

.

is a combinatorial map

yj , Q

respectively, the map which carries Tj (c

to 0 ( (p(C)) is a simplicial map from d((p , η ) to d(

0).a

We now proceed to show that every polyhedral map is simplicial with respect to some triangulations. Let Ρ and Q be two polyhedra and f ; Ρ — ^ polyhedral map. Let (p , T^(f)(^P XQ.

and ^

be presentations of P, Q and

Let Ql be a regular presentation of Ρ X Q

refines

^-^d let

Q be a

which

be the subpresentation of

which covers "Πίί)· Let

and ^

be the projections of Ρ

Q onto Ρ and

39

Q respectively. tation

(^Λ

By the refinement process there is a regular present

of Q refining

If ·Α

such that:

,c 6

ΑΠρ-(Ο) / 0 , then

Then, if C ^ (2 , ^(C) is the union of elements of Now we look at the presentations of "yi (f). The cells of

C" = CO(AXB'),

C

^

On the other hand, since open cell Β

=

· ((pXO^')

are by definition of the fom

(2, Aejp , B' ^ (^λ

Clearly

is a refinement of

with B 3 b ' .

a refinement (3^ of (p (x, y) I: C

.

Since

C"CC 0 (j^^B').

, there is an

is a subpresentation of

, if , c" ^ 0, C C Α Χ B. Hence if

then -x T A, y Ζ, B', SO

Cn(uC\B')Ccr\(AXB') = C". Thus

(X, y) E A XB'.

C" =

Hence

Hence g"'

*

can be also described as

Now, clearly

(p ' = ^ ( g ' )

I D

regular presentation of Ρ ^ ^ TCi) is

1-1 and

a is lineai^with

reference to the ambient vector spaces . Now the claim is. that f induces a combinatorial map (

p(f))

(^D ' — ^ 0^'·

(A) is a cell of

Let A be any cell of (p '·

Ρ say some c Π jockB').

f(A) = fMC ^(c Π

= |A.(C)n B' = B' by (•). Thus is the

IS union

and "^C π the union of hence

5b') J (by 1.4.5) and so ^ C) r\ b', JAiC)

^ (C Λ f^C^s'))) , and (Λ ( 6 C)

Hence if A^^ A, f(A^) S^^B'. Thus f

Β',

40

induces a combinatorial map from the presentation

φ'

(p

to

comes from

to the closure of each cell of

^^

Moreover, since

the graph of f restricted is a closed cell, and hence f

is linear on the closure of each cell of

(p ' ·

The discussion so far can be summarized as: 2.2.4. Theorem. Let f : Ρ •—^ Q be a polyhedral map, and let polyhedral presentations of Ρ and Q respectively. Then there exist regular refinements

and

^

of φ and

such

that I 1) to

^

If A

I

t

( p , f(A) ^ ^ .

The. induced m a p from ' ( p

i s combinatorial. I 2) f is linear on the closure of each cell of

Qi

Furthermore, 2.2, 5.

If ^ϋ) and ^

tation

of 'p(f)

are regular and if there is a regular presensuch that

a) For each C £

, 7\(C) is contained in some element

b) For each C

,

of (p , is the union of elements

of ^ , then in the above theorem we can take

^

=

(in

other words, a combinatorial map can be found refining only ^ not

,

). To apply 2,2.4 to the problem of simplicial maps, we can

use 2.2.3 as follows: First we choose some centering 0 of I and then a centering Ύ| of ^ so that f( Ύ|(θ)) = θ (f (C)) for all C 6(p'.

41

ι

Since

f is linear an each element of I

, we have that f ί Ρ

>Q

I

is simplicial with respect-to d( φ) , η ) and d(

, Q). Hence,

2.2.6. Corollary. Given a polyhedral map f ; Ρ — ^

Q, there exist

triangulations ^

and

of Ρ and Q, with respect to which f

is simplicial. Moreover, J^ and ^ ^ can be chosen to refine any given presentations of Ρ and Q.

Q

'

Defining the source and target of a map f : Κ

^ L to

be Κ and L respectively. We may now state a more general result, details of the proof left as an exercise. 2.2.7. Theorem. Let ^K^ "^be a finite set of polyhedra, with 1 n; let f ί Κ ^ Κ be a finite set of polyhedral maps, the sources and targets being all in the given set of polyhedra. Suppose that for each Y

, ^ ^ < jiy^

^^

occurs as the

source of at most one of the maps f (i.e. Y ^ ^ implies Let

<ί. ξ )·

be a presentation of K y

of simplicial pre sent that for all Υ

, fy

for each Y . Then there is a set , with ^ y refining (py

, such

is simplicial with reference to

That is to say, the whole diagram

^ fy λ can be triangulated, Q

The condition on sources is not always necessary , for example; Ex. 2.2.8, A diagram of polyhedral maps

X

42

can be triangulated if f : X

^ Y is an imbedding.

However Ex. 2,2.9. The following diagram of polyhedral maps (each map is a linear projection)

cannot be triangiilated. Ex. 2.2.10. Let φ

U

be a presentation of a polyhedron Ρ in V,

Cp 5 V — > W be a linear map, then presentation of Ex. 2.2. 11. Let

φ ( (p) =

is a

φ (Ρ).Π f ίΡ

^ Q be a polyhedral map, and X a sub-

polyhedron of P. Then dim f(X) ^

dim X. D

Ex. 2.2.12. If f ; Ρ — ^ Q is a polyhedral map Y is a sub-1 polyhedron of Q, j. (γ) is a subpolyhedron of X. U Next, one can discuss abstract simplicial complexes, their geometric realizations etc. We do not need them until the last chapter. The reader is referred to Pontryagin's little book mentioned in the first chapter for these things.

43

Chapter III Topology and Approximation Since we know that intersection and vinion of two polyhedra is a polyhedron, we may define a topology on a polyhedron X, by describing sets of the form X - Y, for Y a subpolyhedron, as a basis of open sets. If, one the other hand, X is a polyhedron in a finite dimensional real vector space V, then V has various Euclidean metrics (all topologically equivalent) and X inherits a metric topology. Ex. These topologies on X are equal. The reason is that any point of V is contained in an arbitrary small open cell, of the same dime;ision as V. It is easy to see that a closed simplex with this topology is compact.

Hence every polyhedron, being a finite union of simplexes

is compact. The graph of a polyhedral map is then compact, and hence f is continuous.

Thus we have an embedding of the category of poly-

hedra and polyhedral maps into the category of compact metric spaces and continuous maps. It is with respect to any metric giving this topology that our approximation theorems are phrased. \

A polyhedron is an absolute neighbourhood retract, and the results that we have are simply obtained from a hard look at such results for A.N.R"s. It turns our that we obtain a version of the simplicial

44

approximation theorem, which was the starting point, one may say, of the algebraic topology of the higher dimensional objects. The theorem has been given a 'relative form' by Zeeman., and we shall explain a method which will give this as well as other related results. We must first say something about polyhedral neighbourhoods. 3. 1.» Neighbourhoods that retract. Let φ cells C of for such of Ο

regular presentations.

Consider the open

, with C Π \(p| / 0, together with A, A

C, Λ

The set of all these open cells is a subpresentationsN"

. \j\rlis a neighbourhood of |(p| in j ^ j . For, if Jsf'

is the set of cells c'^;(5^such that c'O ((p) = 0, then vj^^' is a subpresentation of

and

-

\

· If

is simplicial,

can be described as the subpresentation, consisting of open simplexes of

with some vertices in (p together with their faces.

^^(pCO^ is a subpresentation, we say that ^ in

J if for every C ζ·

either C

=0

is full

or there is a

A ζ: (p with C Py I (p) = A. In the case of simplicial presentations, this is the same as saying that if an open simplex

of

has all its vertices

in (p) , then g"· Itself is in ^ * An example of a nonfull subpresentation:

5

3.1.1.

If

regiilar presentations, then d (p

is full in

d Co . For, if Tj is any cehtering, then an element (an open simplex) of άφ^

is of the form 0( η

CQ<

If

... < C^.

that is in ip , then Ο(η(0ο),...,

3.. 1.2.

T](Ct)) e d(p

, are necessarily in (p

. Then

, and 0( η (CQ) ,...,

If ^ is full in

) Ο |d (p j

, the simplicial neighbourhood

ί i s t h e s u b p r e s e n t a t i o n o f d (3^ c o n s i s t i n g o f a l l

s i m p l e x e s o f d ^ w h o s e vertices vdth

C^ s

Q

Definition.

(P i s

, 0 ^ t ^ k, is the last element of the Cj, j ^

= 0( η(0ο)

^(Cj^)) ^Ο^^θ^

(CQ))

C Π |(p j

/ 0.

T | ( C ) are centers o f cells

I t i s denoted b y Ν

C of

(p ) (or N ^ ( ( p , T j )

w h e n w e want t o m a k e e x p l i c i t t h e centering). Clearly Ν

p ) is a full subpresentation of d

It can be also described as the set of elements cp of d which ψ Hence

r\ id(p| =

Ν ^ ( (p )

.

, for

plus the faces of such ^ . is a neighbourhood of

j ^ | in the topological

sense. Such a neighbo-urhood as

Ν

f) ) | of |(p( is usually

referred to as a 'second derived neighbourhood' of ^(p|in Ij0>| ' the following reasons

If Χζ^,Υ are polyhedra; to get such a

neighbourhood we first start with a regular presentation C?-of Y containing a subpresentation (jB covering X, derive once so that d(2) is full in d CTL , then derive again and take | ^dOX,^*^® ^ * Now we can define a simplicial map r ·. ^

(p ) — ^ d(p

46

using the property of fullness of (p in C Π C Π

/

. If C ^

we know that there is a A

, with

, such that

(pj = A, and this A is uniquely determined by C. We define

r ( η C) = η A. Ex. 3.1.3. The map r thus defined is a simplicial retraction of onto d(p

D

That is r is a simplicial map from which when restricted to d ^

to

d(P,

is identity, r defines therefore

a polyhedral map, which also we shall call r ; ^ ^ ((p ) — > |

·

We have proved 3.1.4.

If X is a subpolyhedron of Y, there is a polyhedron Ν

which is a neighbourhood of X in Y, and there is a polyhedral retraction r : Ν — ^

X. Ο

3.2. Approximation Theorem. We imagine our polyhedra to be embedded in real vector spaces(we have been dealing only with euclidean polyhedra) with euclidean metrics. • Let X, Y be two polyhedra, ρ ,

ρ' be

metrics on X and Y respectively coming from the-vector-spaces in which they are situated. If cjC , p) ·

^ ^

functions,

we define p(oC,i5 ) = Sup f (oC (x), ^(x)) xeY If A is a subset of X, we define diam A=-sup ρ (χ, y), X,y e A and if Β is a subset of Y, we define diam Β = Sup

f'(x,y) Β

47

We can consider X to be contained in a convex polyhedron Q. If X is situated in the vector space V, we can take Q to be large cube or the convex hull of X. Let Ν be a second derived neighbourhood of X in Q and r i Ν — ^ X be the retraction.' Now Q being convex and Ν a neighbourhood of X and Q/ for any sufficiently small subset S of X,

Ν

(recall that < (S)

denotes the convex hull of S),. This can be made precise in terms of the metric; and is a uniform property since X is compact. Next observe that we can obtain polyhedral presentations (p of X, such that diameter of each element of Φ

is less than a prescribed

positive number. This follows for example from refinement process. Now the theorem is 3.2,1. Theorem. Given a polyhedron X, for every ^ > 0, there exists a ί ^ 0 such that for any pair of polyhedra any pair of functions f : Y

^ X, g : Ζ

Υ, and

^ X with f continuous

and g polyhedral, if p(f| Z, g) < (S j then there exists a g polyhedral, i j ζ = g, and Proof;

fCf, g K ^ , .

We embed X in a convex polyhedron Q, in which there is

a polyhedral neighbourhood Ν and a polyhedral retraction r ;Ν

^ X as above. It is clear from the earlier discussion,

that given 6 >

there is a "v^ ;> 0,. such that if a set A

diameter < Υ| , then Κ Define

=

X has

Ν and diameter r ( /<(Α))<ξ .

^ /y

Now because of the uniform continuity of f, (Y is compact), there is a θ > 0, such that if Β

Y and diam. (B) < 0,

4δ

then diam f (B) <

C

From this it follows that, still assuming B([]_Y, and diameter B < Q , and additionally that set

P(fjZ, g)< ·5 ; that the

f(B) (J g(B A Z) has diameter less than 3 ^ = rv . And hence

we know that

f /I (f (B) (J g (B Γ\ Z))

N, and

Cdiam r(K(f (B) Ug'(B nZ))^ < e Then we find a presentation

.

of Y, such that the closure

rr of every element of ^

has diameter less than 0. Also there is a

presentation 5s oi" ^ on the closure of every element of which g is linear. Refining ^

U

taking derived subdivisions (still

calling the presentations covei-ing Y

and Z, as Λ

anc^espectively),

WB have the following situation; , are simplicial presentations, of Z ^ Y , on each closed

-simplex g is linear, the diameter of each closed

Λ -simplex

0.

We now define h ; Y — > V of ^ Extend

Q as follows: On a 0-simplex

, h(v) = g(v).. On a 0-simplex w of

-^

, h(w) = f(w).

h linearly on each simplex, this is possible since Q is

convex. But now, if then h( (5- )

jj- = [V^ ,..., v ^

(fC ^ ) U g( ?

Z))

is the closure of a /^-simplex, N; this is a computation

made above (•>!·) since diam. (j- < Q. And so h(Y)([jN.

Also it is the case that h is polyhedral,

since h is linear on the closure of each simplex of J^

and on

= Z, clearly, h agrees with g. Define, g : Y — ^ X to be r ο h. Since r and h are

49

polyhedral so is g ; since hj Ζ = g and r is identity on X; it follows that gjz = g. To compute

ρ(g , f) we observe that any

y £ Y is contained in some closed simplex f(y) and h(y) are contained in l( ( Γ

f ) U s( ^

both )f

hence

both f(y) and g(y) are contained in

r

g(

c )υ

^ ^

z)))

This set by (*) has diameter< ^ . Hence

p (g, f) < ^

Ο

We now remark a ntimber of corollaries: 3.2.2, Corollary. Let X, Y, Ζ be polyhedra, , Z Q y , and f : Y a continuous map such that

f Ζ is a polyhedral.

Then f can be

approximated arbitrarily closely by polyhedral maps g ; Y that g Ζ = f

^ X such.

z. a

The next is not a corollaiy of 3.2.1, (it could be) but follows from the discussion there. 3.2.3.

Any two continuous maps f^, f^ : γ · — ^ X, if they are .

sufficiently close are homotopic. not Y or the maps involved).

(Also how close depends only,on X,

If f^ and f^ are polyhedral, we

can assume the homotopy also to be polyhedral,, and fixed on any subpolyhedron on which f^ and

f^ agree.

Proofί Let Ν and X be as before. Let if A Q l , diam A < η

,, then

K(A)Cn.

be a n-umber such that If-f(f^,

F(y,t) = t f^(y) + (1-t) f2(y) 6 N, for and r . F, where r : Ν homotopy. If f^, f^

1 and all

y^Y

^ X is the retraction, gives the required

are polyhedral, we can apply 3.2.1 to obtain

a polyhedral homotopy with the desired properties.

^X

D

50

Remark; The above homotopies'are small in the sense, that the image of X is not moved too far from f^(x) and f2(x)· 3-2.4. Homotopy·groups and singular homology groups of a polyhedron can be defined in terms of continuous functions or polyhedral maps from closed simplexes into X. The two definitions are naturally isomorphic.

The same is true for relative homotopy groups, triad

homotopy groups etc. The corollary 3.2.2 is Zemman's version of the relative simplicial approximation theorem. From-this (coupled with 4.2.13) one can deduce (see M. Hirsch, "A proof of the nonretractibility of a cell onto its boundary", Proc. of A.M.S., 1963, Vol. 14), Brouwer's theorems on the noncontractibility of the n-sphere,.fixed point property of the n-cell, etc. It should be remarked that the first major use of the idea of simplicial approximation was done by L.E.J. Brouwer himself; using this he defined degree of a map, proved its homotopy invariance, and incidentally derived the fixed point theorem. It should be remarked that relative versions of 3.2.1 are possible.

For example define a pair

(X^, X2) to be a space (or a

polyhedron) and a subspace (or a subpolyhedron) and continuous (or polyhedral) maps f : (X^, X^) sort of function X^ Theorem 3.2.1

^ (Y^, Y^) to be the appropriate

^ X2 which maps

into Y2 . Then

can be stated in terms of pairs and the proof of this

exactly the same utilising modifications of 3.1.4 and the remarks at the beginning of 3.2 which are valid for pairs. Another relative version of interset is the notion of

51

polyhedron over A, that is, a polyhedral map f ! (^

S X — ^ A)

such that

cL. =

:Χ

^ A. A map

^ ( <55 ί Y - — ^ A) is a function f : X

> Y

· fj we can consider either polyhedral or continuous

maps. The reader should state and prove 3.2.1 in this context (if possible). 3.3. Mazur's criterion. We shall mention another result (see B. Mazur "The definition of equivalence of combinatorial irabeddings" Publications Mathematiques, No.3, I.H.E.S.j 1959) at this point, which shows that, in a certain sense, close approximations to enbeddings are embeddings (in an ambient vector space). Let

^ simplicial presentation of X, and let V be

a real vector space. Let

denote the set of vertices of ^ ^ .

Given a function C) :

V, we can define an extension

Φ

5 ^^ — b y

mapping each simplex linearly. Clearly if

is any polyhedron containing

We call φ

^ Y is

the linear extension of C) . Cp is called

an embedding if it Eiaps distinct points of X into distinct points in V. 3.3.1.

(Mazur's criterion for non-embeddings) If the linear extension φ

of Cp : o O ^ — ^ V is not

an embedding, then there are two open simplexes 5" and, with no vertic6'S in common, such that φ

of

(6~")Γ\

Proof; The proof i? in two stages. A>

V

If 6"= 0(vq

v^) and | 9 ( V Q )

is not indep.endent, then there are faces

ύ

and 6~2 of

φ(\)} 6~,

52

vdthout vertices in common, such that

(

) C\ φ(6~2) ^ 0

(This is just 1.2.6). B) Thus we can assume that for every (P .of ^

i-V ( , Cp

)

is also an open simplex of the same dimension. Consider pairs of distinct open simplexes^p , p 'j- such that Let

cy Τ be such a pair, which in addition has the property

dim g- + dim and

is minimal among such pairs. We can now^that

have no vertex in common. If

= 0(w^

w^), then if

0(vq

v^^) and

r-l

^

^

0""') ^ φ.

φ ( 6") Γ\ φ ( Τ )

0, there is an

equation

φ^ν ^ with r_+... + r 0 m

V = 0

=

··· ^

Ai,... + s . 1 η

strictly greater than 0, for otherwise dim

νHere r. and s. are X i dim ^

will not

be minimal. Now if v^ = w^, and r^ >

0

O'

and have a common vertex,, say, for example, s^. we can write

0-i^

0'

^

1

1

Multiplying by (1 - s^) , we see that some face of ^ intersects a proper face Cp^O (w^ that ^

( 6-)

of φ (

So

and fj' had not the minimal -dimension compatible- with the

properties Γ ^ ^ Τ

, φ(6~)Πφ rjJ

Now it easily follows, since to check CD is an embedding we need only

-check that finitely many compact pairs

Ιί^Φ ((f ) J $ ( τ )). > (Γ

τ

=

do not intersect;

53

3.3.2. Proposition. Let ^

be a simplicial presentation of X

contained in a vector space V, let ^ ^ Then there exists an £

such that if

be the set of vertices. Qp ; ^

- y V is any

function satisfying "^(v, φίv) )
:X

^V

is an embedding.

, then the D

This is a sort of stability theorem for embeddings, that is^ if we perturb a little the vertices of an embedded polyhedron, we still have an embedding.

5i

Chapter IV Link and Star Technique 4.1, Abstract Theory I 4.1.1. Definition Suppose

(Join of open simplexes) 3-nd ^

vector space. We say that

are two open simplexes in the same

^ ^

is defined, when

(a) the sets of vertices of ^

and

are disjoint

(b) the union of the set of vertices of

and IT

is independent. In such a case we define

to be the open simplex

whose set of vertices is the union of those of

and of «Γ . If

β" is a 0-simplex, we will denote 6~ Τ X is the unique point in

^

or

{xj where

.

We also,, by convention, where g" (or Τ ) is taken to be the empty set 0, make the definition 0 6"- 6 - 0 = 5 " Clearly dim ^

= dim

+ dim Τ

+1, even when one

or both of them are empty. Ex. 4. t.2.

Τ

^^ defined if and only if (F

Τ

~ 0> and any

two open intervals 0(x, y) , 0(x', y') are disjoint, where Xj

0- J

y' ζ: Τ » ^ ^

the union of open

1-simplexes 0(x, y), xt C

In this case> yt

·

is Q

This is easy. Actually it is enough to assume

0(x, y) r\0(x', y') = 0 for x, x' e 6" , y, y ' 1

x / x' or y ^ y'

55

That it is true for points of (j" and

and

^

r\

= 0 follow

from this. Ex. 4. U3.

i'ftien ζ- cf

same as

is defined, the faces of

' , where

respectively.

ff ' and

If either

' ^

is a proper face of Ex. k. 1.4» In V ^

are the

faces of

and·^

Γγ'

, then

qf '

. 13

Let ^

and c^ be in ambient vector spaces V

W % iK , let

^

or

S" Τ

= Γ X 0 )< 0 and ^

=0

and W.

«f X ' · Then

is defined. Q

4. t.5. Definition. Let an element of

be a simplicial presentation, and «5~

, Then the link of ζ" in

denoted by ' Lk (

,

is defined as =

is defined, i

Lk (G- , ^ ) - ^

if r* =

Obviously Lk (

,

) is a subpresentation of

In case ζΤ is O-dimensional, we write Lk (x, Lk(

ί

. ) for

) where χ is the unique element in ^ .

Ex. 4.1.6. If Τ ζ Lk(

/Ϊ ), then

LK(CR , 4.1.7. Notation. If

)) = LK( R T , 4). D

is an open simplex, then ·|ςΓ~ j- and ^^

denote the simplicial present'ations of n- and

made up of faces

of (Γ . Ex. 4.1.8. If

= p ^ Lk (

, and dim p ^ 0, then

p , [d^} ) f^ L r ] ) =

will

r]

α

)

56

A..t.9. Definition. Let QL and (j^ be simplicial presentations such that for all if

QL , q;

6"'

or

£) / Γ Τ

defined, andQ-TOC

ff '. Then we say that the join of

is defined, and define the join of QLand (J})

0 and;^

, denoted by QL·^*· (R)

to be the set ί 5" ^ I Γ 6 O t J ft t (β ^ „·

By 4.1.3

(Tor cr may be empty"! but not both. j

^ simplicial presentation.

If 0

is empty, we define Q l * 0 = 0 * 0 ^ = (JL. In case 0 L and (j^ are presentations of polyhedra in V -V/ and W, then we construct, by 4.1.4, QX

and (p which are isomorphic

to 0Land(/6 5 and for which we can define 0 L ·ϊ<· (Jj . depends only on

It clearly

and (J^ upto simplicial isomorphismj in this way

we can construct abstractly any joins we desire. Ex. 4.1.-10.

OV.* (ii - β

* Ot

That is whenever one side is defined, the other also is defined and both are equal.

D

Ex. 4.1.11. If

,

Lk(oC(i, In particular, when

. then ) ^ Lk

*Lk ( (i

).

0,

Lk ( ο 6 , σ ι . * ( β ) = Lk CoC , 0 1 ) and when dC = 0 ,

Lk {β , If

ψ ,(B).n

is the presentation of a single point ^ vj , and is

joinable to (j^ , then we call

(JL*

J the cone on (Q with^vertex

57

ν, and denote it by CCijb). (κ) is called the base of the cone. If we make the convention^ that the unique regular presentation of a one point polyhedron v, is to be written {{v]j , then C((B) ) = jjvj] 4.1.12. Definition. Let

be a simplicial presentation, andg"^ J^ ,

Then the star of . cf in ^ be

, denoted by St{ ^ , Ji ), is defined to

Lk( r , i ). Clearly St( r , 4 ) is a subpresentation of

equal to

(J

Τ f

,

and is

·

In case tP contains only a single point x, we write St(x, Ex^ 4,1.13. Let / ξ be a simplicial presentation, (5~ an element of ^

. If ^

is a face of

with dim ^

Lk( ^ , Λ ) = 4. 1.14. Definition. k-skeleton of

* Lk( ^ , Λ ) ) . D

If /C is a simplicial presentation, the

, denoted by

/i

k is defined to be

= U {[^j j rf 4

Clearly Ex. 4.1.15. If

,

dim

is a subpresentation of

Γ ^ δ" ;

= dim Γ" - 1> then

k|.

/I.

and dim 5- = 1 , ( X ^ k), then j^) - Lk( 6", Λ

Ex. 4.1.16. Let f ; Ρ — respect to presentations

Q

_ ^ __

£3

be a polyhedral map, -simplicial with and

of Ρ and Q respectively.

Then

2) « «"ίτΛ .

C st(f
5δ

3) For every

^

if and only if f maps every a 1-simplex of

C Lk(f(6" ), Λ')

, f(Lk( Γ,Λ))

1-simplex of

onto

'.

(Strictly speaking, these are the maps induced by f). 4.2, Abstract Theory II 4.2.1.

Definition.

Let φ

centering of φ) , Let

be a regular presentation and Yj a

A t jp . Then the dual of A and the link

of A, vjith respect to

, denoted by

Q A and -Λ α

are defined

to be

CQ where

η ο ^ ) I A
k

0 j

t (p for all i. Clearly ^ A and

A are subpresentations of d ^ = d((p j'rj ).

When there are several regular presentations to be considered, we will denote these by

^(p·^ ^nd

^ A. Ύ^ will be usually omitted from

the terminology, and these will be simply called dual of A and link of A. 4.2.2. Every simplex of d ^ belongs to some 4.2.3.

A is the cone on

^ A with vertex

T| A. -O

Ex. 4.2.4. Let dim A = p, and consider any p-simplex ^ of d (p contained in A i.e. ^ = 0( B ^

BQ

... < B P = A, Then /\ A = Lk(

.., "Γ|ΒΡ), for some , d ( p ). •

4.2.5. Suppose (p is, in fact, simplicial. both

A and

Then we have defined

Lk(A,(p ). These are related thus:.

59

A vertex of There is a unique

^ A is of the form

Β of (p such that

Y| C where A <. C.

C = A B . T|B

is a typical

vertex of d(Lk(A, (f> )). The'correspondence Vj C

Β defines

a simplicial isomorphisni:

AAO

d (Lk {A,

(p )). Q

Ex. 4.2.6. With the notation of 4.2. 1, A < Β if and only if (5b C ΛΑ.

For any Α ^ ψ , /\A. is the union of all ζ Β

Ex. 4.2.7. If (p is simplicial, A, 3 £(p , then

for A <.B.D

5αγ\ ί Β / 0

if and only if A and Β are faces of a simplex of Q!) . If C is the minimal simplex of (p containing both A and Β (that C is the open simplex generated by the union, of the vertices of A and B), then

<5\AnS Β

c^.C. Ο

4.2.β. Definition. centering of ^

If (Ρ is a regular presentation and

, the dual k-skeleton of

defined to be k

a

, denoted by ,φ ^ is

• Γ

(P =

( Η CQ

CQ<

...
dim

CQ^

k,

k Clearly

^ - is a subpresentation of d(p , and is, in

fact the union of all all

A for dim A ^ k. It is even the union of

A for dim A = k.

C

Λ

^

Thus ο A, /Λ A, (p E^· 4.2.9.-

d(p= (P° 3 (p'

are all simplicial presentations. ...

3 (p""' = 0, where k

η

is the dimension of

. "Dim (p

= η - k. D

We shall be content with the computation of links of vertices of (p

60

Ex. 4.2.

If A ξ ρ

, dim A

Lk (V^ A, φ

k, then'

. [d A^^'^ λ k.

Ο

Next, we consider the behaviour of polyhedral maps vdth respect to duals. Let f : Ρ — 7 Q be a polyhedral map; let ^ and

be

two simplicial presentations of Ρ and Q respectively with respect to which f is simplicial. If /{p is a centering of ^

, it can

to a centering V^ of (p , that is f( Yj A) =-{r> f(A) for all A

. (see 2.2).

f is simplicial with respect to d((p ,'η )

and d(0) -ζη ) als». Now, 4.2.11. If Α ς ρ , f( 4.2.12. If Β t

A) C

Ο

then f"^ (<S Φ^Β) = U ^

A

J f( A) = Β ]. D

Remark; All these should be read as maps induced by f, etc. Since each such A must have dimension ^ dim B, we have 4.2.13. Proposition. With the above notation, for each

This property is dual to the property with respect to the usual skeleta "f ( (p , ) Κ

O^i'" · κ

4.2.14. Corollary. If dim Ρ = n, then dim f~\

^ η - k. Π

In particular, if dim Q = m, and q is a point of an (open) vn-dimensional simplex of

, f~^(q) is a4(n-m)-diinensional

subpolyhedron of P.

Ο

Ex. 4.2.15.

= (p ' J if and only if every 1~simplex off

f

is mapped onto a 1-simplex of (^N . (i.e. no 1-simplex of collapsed to a single point).

0

(p is

.

61

4.3. Geometric Theory. 4.3.1. Definition. Let Ρ and ' Q be polyhedra in the same vector space V. We say that the .ioin,.,of Ρ and Q is defined

(or Ρ » Q

is defined, or Ρ and Q are joinable)» if; (a) F η Q = 0 (b) If X, X' J 4

P, y, y' ζ. Q, and either x / x' or

then 0(x, y)Π 0(x', y') = 0.

If the join of Ρ and Q is defined, we define the join of Ρ and Q , denoted by Ρ * Q to be P^Q-

yl(xi:P, y

By definition, P. * 0 = 0 ^ Ρ = P. Every'point

ζ ^ Ρ * Q can be written as s

z=(l»t)

x + ty,xtP,y?:Q,

O^t^l.

The number t is uniquely determined by z; y is uniquely determined if ζ (: Ρ (i.e. if t ^ 0), χ

is uniquely determined if ζ

Q

(i.e. if t ^ 1). 4.3.2. Let φ

and

be simplicial presentations of Ρ and Q;·

and suppose the (geometric) join Ρ * Q is defined. Then by 4.1.2, the (simplicial) join (p ^^

defined, and we have jf

Ρ *

This shows that Ρ * Q is a polyhedron. • 4.3.3. Definition. If P^, Q^, P^, Q2 P^ * Q^ and P^ * Q2 are defined, and are maps, then the join of f and .

are polyhedra such that f : P^

^ P^, g : Q ^ — ^ Qg

denoted by f

g, is the map

from P^ * Q^ to P^ * Q^ given by, (f * g) ((1 - t) X + t y) = (1-..- t) f(x) + t g (y) y e Q i , 0 < t 41.

62

4.3.4. In the above if f ; P^ — ^ with, respect to (p ^, (p^;

P^, g ! Q^

.

> Q^ are simplicial

f ^^ g '· P,

is simplicial with respect to φ ^ ·ί<·

Pg ^ %

and (p ^ * Og · Thus the join

of polyhedral maps is polyhedrei,!. , D 4.3.5. If Ρ * Q is defined, (id)^) ^ (id.) = Id ρ „ . . If, P^ * Q^, P^ ^^ Q^, P^ * Q^ are defined and f^ : P^ ·' ^2

^

' §1 '

^

>

P2 ,

^ S

'

(f2 c f,) ^ (g^ ο g^) = (f^ * g^) 0 (f^ * g^) α This says that the join is a functor of two variables from pairs of polyhedra for which join is defined and pairs of polyhedral maps, to poljrhedra and polyhedral maps. The join of a polyhedron Ρ

and a single point ν is

called the cone on P, (sometimes denoted by C(P)) with base Ρ and vertex v, Ex. 4.3.6. C(P) is contractible.

Ο

Ex. 4.3.7. Ρ * Q - Q contains Ρ as a defomation retract. Hint;

Use the map given by (*) below · Q Let us suppose that Ρ * Q and the cone C(Q) with vertex

V are both defined.

The interval

lj is 0

1 ,

and so two

maps can be defined: oC ί Ρ ·«• Q — ^ [ 0 , OJ

the join of Ρ

> 0, Q

—1,

p5 : C (Q) — ^ [ 0 , j], the join of ν — > 0, Q

) 1.

Simply speaking, p((l~t)x+ty) = t oC( (1 - t) V + t y) = t , for χ

y e <2.

63

The correspondence : . (1 - t) X + t

s (x, (1 " t)

V

+ t y)

is a well defined fiinction between '

([p, 1))· and Ρ χ

([0, l)).

It is a homeomorphism, in fact. But it fails to be in any sense polyhedral, since it maps, in general, line segments into curved lines. Example: Taking Ρ to be an interval, Q to be a point.

cca)^ The horizontal line segment corresponds to the part of a hyperbola under the above correspondence,. We can however find a polyhedral substitute for this homeomorphism. 4.3. β. Proposition. Let P> Q, oC J ^

be as above, let 0 <

< 1,

Then there is a polyhedral equivalence. -1

( CoA]

ρ χ pr' (

)

which is consistant with the projection onto the interval [^0, Tj « Proof: Let ^

and

be simplicial presentations of Ρ

and take the simplicial presentation ^ of

Lo. t] ·

='

ί·£Τ] f

and Q )

64

Gonsider the set of all sets of the forai A( ^ > ^ , i)> where Ρ ζ(ρ , Γ ζ Ο

J i £^

and Γ = 0 iff 1 = [O] , defined -thus:

A( p , 0 , 0 ) = f Mp

i ) = f ^ O d f ^ (i)

The set of all these A( f , 6Γ , i), call it 01- . It is claimed that C5l is a regular presentation of that A( ρ , er", i)^ A( p',

iO

)> and

if and only if

i^i'.

Secondly, consider the set of all sets of the form B( Ρ ,

. i) where ρ ^ (p

, (f e 6 ^ ^ ^ ί ^ and 6" = 0 if i =

defined thus:

B( F , 0 , 0) = Ρ B(f

£Y}

i) = f X^( trfvjrxp"' (i))

It is claimed -chat (j2) of all such B( Ρ ,

i) . is a

regular presentation of Ρ /<s 12)"'([o, ^J), and that B(f,r,i)4

B( p', r', i') if and only if (Γ-4 β-

f'

and i 4 i'. Hence the corrsspondence is a combinatorial equivalence Tj and-^ of 0Land(j3

Β( ,·ρ ,

, i)

"'^(jh . If we choose the centerings

respectively such that

oC ( η(Α( f and

A( p , (f, i)

p)(2 nd coordinate of

(c, τ ))) = (B( -p , βΤ , (0, Τ ))) =

The induced simplicial isomorphism d((jU,^rj ) a polyhedral equivalence consistent vd.th the projection onto

)

Ρ X

) gives fj )ν

,Γο^ 'Τ J .

It should perhaps be remarked that by choosing ^

and

65

fine enough, our equivalence is arbitrarily close to the correspondence (*) on page 6?. Ο 4.3« 9. Corollary. Let C(F) be the cone on Ρ with vertex v, and : C(P) any

^ [0, lj be the join of Ρ

ζ· (0,· l),

Ρ >· [ρ,

) is polyhedrally equivalent to

by an eqiiivalence consistent with the projection For, take Q = ν in 4·3-8.

4.3.10. Corollary. Let J^ P-^

^ 0," ν — ) ί. Then for

0, Q — ^

^

? *

[o,. l] be the join of

1 ; let 0 < Y <
1. Then

is polyhedrally equivalent to P·^ Q

, S3 by an equivalence

consistent with the projection to Ijy^ , For, by 4.3.8, where p> : C(Q) By 4.3.9, interchanging

β

)

j J ^ Ο

)

· Ρ X

^^ ^^e join of Q

) >1

0 and 1, we see that

and vertex

> 0,,

(i^V [Y ,

combining these and noting the preservation of projection on ^"V^ ,

0; ,

we have the desired result. [~\ 4.3. t1. Definition. Let Κ be a polyhedron and χ ζ; K. Then a subpolyhedron L of Κ is a said to be a (polyhedral) link of X

in K, if L * X is defined, is contained in K, and is a

neighbourhood of χ in K. A (polyhedral) star of χ in Κ is the cone with vertex X

on any link of χ in K. Clearly, if a ζ. K^ C

^^^^

is a, neighbourhood of * a'

in K, then L<;3 ^^ is a link of ' a ' in K,^., if and only if it is a link of' a ' in K.

66

To show that links and stars exist, we triangulate Κ by a simplicial presentation

with χ as a vertex. Then |

is a link of χ in K, and

St(x,/i

In this case

)\

is a star of χ in K.

St(x, Λ )1- Lk(x,/J )\ is open in K; this need not

be true for general links and stars. Ex. 4.3.12. If ^ G 6" 6 4

is any simplicial presentation of K, and

J then

and

* Lk(

{b 6"} ,Λ )

Ex. 4.3. ^2'.

(a) Let

as in 4.2.1, if χ ^ A, ^ A ·«· | Λ D

f :Κ

κ' be a one-to-one polyhedral map,

simplicial with reference to presentations K'. Then 'for any ^ ^ /ζ , χ f f f

is a link of χ in K,

is a star of χ in K.

(b) With £a, is a link of χ in K.

Lk( tr , Λ )

and

' of Κ and

^

\β·] * Lk(

Λ )1

Lk(

)|

is the join of and

X—>f(x).

Formulate and prove a more general statement using 4.1.16 (b) With the hypothesis of 4.2.15, if A^ is a 0-cell

of ^ and f

,

) C I^AQI

A(fAQ)

is the join of A^

> fCA^)

and f

•

If χ and a are two distinct points in a vector space, the set of points (l - t ) x + t a , t ^ O

will be called 'the ray

from X through a'. Let L^

and L2 be two links of χ in k, then for each

point a t Lp the ray through a from χ intersects L2 in a unique point h(a) (and every point In L2 is such a image). It

67

intersects L2 in atmost one point, since the cone on L^ with vertex χ exists. It intersects L2 in at least one point since the cone on L2 must contain a neighbourhood of the vertex of the cone on L^.

^

The function h : L^

^ L2 thus defined is a homeomor-

phism. But, perhaps contrary to intution, it is not polyhedral. The graph of the hiap h in this simple case is a segement of a h3φerbola. The fallacy of believing h is polyhedral is old (See, Alexander "The combinatorial theory of complexes", Annals of Mathematics, 31, 1930); for this reason we shall call h the standard mistake after Zeeman (see Chapter I of "Seminar on Combinatorial Topology"). We shall show how to approximate it very well by polyhedral equivalences. It migh be remarked that the standard mistake is "piecewise projective", the category of such maps has been studies by N.H. Kuiper /"see "on the Smoothings of Triangulated and combinatorial Manifolds" in "Differential and combinatorial Topology"^

A symposium in Honor

of Marston Morse,Edited by S,S. Cairns_7. 4.3·

Definition. Let A and Β be two convex sets.

one function from A onto B, linear; if for each a^, a^ f A,

cL i ^ ••-

A one-to-

Β is said to be quasi-

(["a^, a^ ) ^ j^(a^),

In other words, J^ preserves line segments. It is easy

68

to see that cL

is also quasi-linear.

Example i Any homeomorphism of an interval is quasi-linear.

In

,

the map;

as a map from A = ^ (r^, r^) Β = -^(Λ^,

0 <

< ,

^^y

to

quasi-linear. Q

4.3.14. Proposition. Let cC · ^ — ^

Β be quasi-linear. Let

a^l be an independent set of .points in A, defining an open simplex ^

. Then • cLi^o) }···>

and the simplex they define is of 6~ if and only if

is independent,

( C" )· Consequently

eC ( Τ ) is a face of

The proof is by induction. definition.

(Sj^) \

is a face

dL ( ΙΤ)·

For η = 1, this is the

The inductive step follows- by writing

and noting that quasi-linear map preserves joins,

=

1

Π

4.3.15. Theorem. Let L^ and L^ be two links of χ in Κ with h : L^—^

Lg, the standard mistake. Suppose

polyhedral presentations of L^ and L^. Then there exist simplicial refinements

1

and

h(^) ^ f :L — ^

L

2

of ^ ^

^ and h

^

1

and ^ ^

2

such that for each

is quasi-linear. If

is defined as the linear extension of h restricted

to the vertices of /ή ^, then f is a polyhedral equivalence simplicial with respect to =

^ and

^ and such that

for

Proof; We can suppose that ^ ^ ^ and

simplicial, and find

69

a simplicial presentation ^D of (L^"'^ x (Si^^fe]]

)· Define Λ

that every simplex

^

= Lk(x,(p ). It is clear

is contained in tT

and hence the standard mistake h^ :

)C J

—^

>

L^ takes

to

· • The restriction of h^ to

let

Lg* x ) refining

ζ; ^

55-

; the three points

is quasi-linear. For,

a^, a^, χ determine a plane

and in that plane an angi^ar region ^

, which is the union of all

rays from χ through the points of • by definition, takes Pa , a Γ ^ • 1 2-J geometrically obvious, is just

· to L.

standard mistake, X

, which, it is

h^(a^), h^Cag)^.

This, "together with 4.3. 14, enables us to define I=

tr )

C" t A '·

presentation refining ^

, and to see that this is' a simplicial

.

Similarly, via the standard mistake h^ ί | Λ

—^

L,

we have

Λ

2

Since, clearly, h 5 L^

L^ is h^ · h^

and the

composition and inverse of quasi-linear maps are again quasi-linear, the major part of the theorem is proved. The last remark about f is obvious.

Q

4.3. t6. If in 4.3.15, for a subpolyhedron K* of L^, h^K' is polyhedral, then we can arrange for f : L ^

y Lg of the theorem

.1 = h to be such that f | K'

I K. For, all we need to do is to assure that

has a

70

subpresentation covering Κ; then because h is linear on each simplex in K, the resultant 4.3.17. Corollary.

f is identical with h there. D

Links (resp. stars) of χ in Κ exist and

all are all polyhedrally equivalent. 4.3-1 β. Proposition. If

D

f : Ρ — ^ Q is a polyhedral equivalence, ,

then any link of χ in F is polyhedrally equivalent to any link of X in Q. For'triangulate are obviously isomorphic.

f, and look at the simplicial links; they Q

4.3.1?· allows to define the local dimension of polyhedron Κ• at x. This is defined to be the dimension of any star of χ in K. By c

4.3.17 this is well defined. It can be easily seen that (by 4.3.12) the closure of the set of points where the local dimension is ρ is a subpolyhedron of K, for any integer p. We will next consider links and stars in products and joins. Ex. 4.3.19. Let C(P) and C(Q) be cones with vertices ν and w. Let Ζ = (P

(Q)) U (C(P)% Q). Then

(a) C(P)

C(Q) = C(Z), the cone on Ζ with

vertex (v, w) (b) Ρ y. w and ν X. Q are joinable, and (P A w) * (v KQ) is a linkf of (v, w) in C(Z). Hence by straightening our the standard mistake^ we get a polyhedral equivalence Ρ -^ί· Q C^i Z, which extends the canonical maps Ρ

^

71

and Q — ^

νQ.

Hint; It is enough to look at the following 2-dimensional picture for arbitrary ρ t P> q G- QJ

ν

(υ.ν) ih^)

Ex, 4.3.20. Prove that Ρ * Q 4.3.8. If

φ : Ρ * Q — ^ [^0, Q

(C(p) >. Q O P ^ C(q)) utilising is the join of Ρ — >

the equivalence can be chosen so that P><.C(Q) and Ex. 4.3.21.

C() ( [o,

[^/z ,· \] ) goes to C(P)

)

0, Q — ^ ^o

Q. U

(Links in products). If x ^ P, y t Q, then a link of

(x, y) in Ρ X Q is the join of a link of χ in Ρ and of y in Q.

a link

Q

The join of X to a polyhedron

consisting of

two points is called the suspension of X with vertices Xj and X2 and is denoted by S(X). Similarly K^^ order suspensions are defined. Ex. 4.3.22.

(Links in joins).

In Ρ * Q.

1,

72

1) Let J.L·Ρ * Q - (Ρ Ο

and let

X = (1-t) Ρ + t q, ρ ^ P, q e Qj 0 <; t < 1. If L^ is a link of ρ in P, L^ a link of q in % then p, q) is a link of χ in Ρ

L^) (with vertices

Q.

2) If ρ G P, and L is a link of ρ in P, then L * Q is a link of ρ in Ρ * Q. , Hint; for 1. Consider simplicial presentations (p and and Q having ρ and q as vertices. is a link of()(p, q) in (p in Ρ

Then Lk(p, ^p ) * Lk(q,

follows from this.

)

, by 4. 1. 11. Hence a link of χ

Q = • ^p, q^ * Lk(p,(p ) * Lk(q,(^)

Lk(p, (p) •»• Lk(q, (3^)

of Ρ

or the suspension of

with vertices ρ and q. The general case

Q

4.4. Polyhedral cells, spheres and Manifolds. In this section, we utilize links and stars to define polyhedral cells, spheres and manifolds and discuss their elementary properties. Let us go back to the open and closed (convex) cells discussed in 1.5. If A is an open cell, then the closed cell A is the cone over

A with vertex a, for any a ^ A,

4.4.1. Proposition! If A and Β are two open cells of the same dimension, then

A and ^ Β are polyhedrally equivalent. More-

over the equivalence can be chosen to map any given point χ of δ A 'onto any given point y of

B.

Proof; Let dim A = η = dim Β = η. Via, a linear isomorphism of the linear manifolds containing A and B, we can assume that A

73

and Β are in the same n-dimensional linear manifold, and moreover that A Λ Β / 0. Then

"^-A and

point of Α Λ Β

Hence 'b A and

in 'AljB.

Β are both links of any Β are polyhedrally

equivalent. A rotation of A will arrange for the standard mistake to map X to y. And 4.3.15, we can clearly arrange for χ and y to be vertices in

^ and

A

^

By joining the above map with a map of point of A to a point of B, we can extend it to a polyhedral equivalence of A and B. Thus any two closed cells are polyhedrally equivalent. 4.4.2. Definition»

A polyhedral n-sphere (or briefly an n-sphere)

is any polyhedron, polyhedrally equivalent to the boundary of an open cell of dimension • (n + l). By 4.4. 1 this is well defined. 4.4.3. Definition. A polyhedral n-cell (or briefly an n~cell) is any polyhedron, polyhedrally equivalent to a closed convex cell of dimension n. By the remark after 4.4.1, this is well defined. All the cells and spheres except the 0-sphere are connected. Consider the "standard n-cell", the closed n-simplex, and the "standard

(n-1)-sphere", the boundary of a n-simplex. By 4.1.8,

4.3. 1δ and 4.4.1, we have 4.4.4. Proposition. (n-1)-sphere.

The link of any point in an n-sphere is an

^

4.4.5. Corollary. An n-sphere is not polyhedrally equivalent to an (m)-sphere, if m ^ n.

74

Proof,: By looking at the links'iising 4.4.4·, and induction. 4.4.6. If f n-simplex of

^ C

D

is an equivalence of an n—cell with a closed

, we see that for points of C corresponding to points , the link in C is an (n-l)-cell; and for points of C

corresponding to points of ζΤ , the link in C ' is an (n-1)-sphere. 4.4.7. Proposition.

An n-sphere is not polyhedrally equivalent to

an n-cell. Proof; Again by induction.

For η = 0, a sphere has two points and

a cell has only one point. For η ^ 0, an n-cell has points which have (n-l)-cells as links, where as in a sphere all points have (n-1)-spheres as links. And so, by induction on η they are different.

D

This allows us to define boundary for arbitrary n-cells, namely the boundary of an n-cell C, is the set of all points of C whose links are (n-l)-cells.

We will denote this also by

^ C.

This coincides with the earlier definition for the boundary of a closed convex cell, and the boundary of a n-cell is an (n-1)-sphere. And as in 4.4.5, an n-cell and an (m)-cell are not polyhedrally equivalent if m / n. By taking a particularly convenient pairs of cells and sphere, the following proposition is easily proved: Ex. 4.4.β. Whenever they are defined, 1) The join a m-cell and an n-cell is a (m + η + l)-cell. 2) The join of a nir-cell and an n-sphere is a (m + η + 1)-cell.

75

3/ The join o? a m-sphero and an n-sphere is a (τη + η + O-sphere. If in (1) of 4,4. δ, C, '.^nd C^ are the cells, then Cg) =

CC^

of

the cell, and S the sphere 4.4.9. Definition.

if C is

υ (C ^t· S) - 5 C * S.

Q

A PL-manifold of dimension η (or a .

PL n-manifold) is a polyhedron Μ

such that for all points x ^ M,

the link of χ in Μ is either an (n-l)-cell or an (n-1)-sphere. 4.4.10. Definition.

If Μ is a PL n-manifold, then the boundary

of Μ denoted by ^ M, is defined to be

Μ =^χςΜ link of X in

Μ is a cell^ . Notation; If A is any subset of M, the interior of A and the boundary of A in the topology of M, vdll be denoted by int and bd ^ A respectively, Μ - ^ Μ

A

Μ is also usually called the 0

interior of M, this we vdll denote by int Μ

or M. Note that

int^ Μ = Μ, where as int M. = Μ - ^ M. It is clear fx-om the propositions above, the manifolds of different dimensions cannot be polyhedrally equivalent, of course, from Brouwer's theorem on the "Invariance of domain", it follows that they cannot even be homeomorphic. 4.4.11. Proposition.- If Μ is a PL n-manifold, then b Μ is a PL (n-1)-manifold, and

^ ( ^ M) = 0.

Proof: We first observe that Μ - ^ Μ is open in M, For if XfeΜ -

Μ, let L be a link of χ in M, S the corresponding

star, such that S - L is open in M.® S is a cell and

S = L.

76

If y t S _ L, then a link of y in S is a link of y in M, since S is a neighbo\irhood of y. y

Since S is a cell and

S - O S , the link of y in S is a (ίβϋ. Hence y £ Μ - Ο M,

for all y ζ· S - L or Μ

Μ is open in M. Hence

Μ is closed

in M. If then

is any simplicial presentation of Μ and

^J. * Lk( Γ ί

by 4.3. 12. Hence Γ C 0

) J· is a link of χ in Μ Μ

or

also is contained in ^ M, since ^ Μ uion of all such ^

L

X ^

for all χ ^

C Μ - c)M.

If

C ^

is-closed.

^ Μ being the

^

is a subpolyhedron of M.

Let X be a point of S-= L

ζ; ,

M, L a link of χ in Μ and

, the corresponding star such that S - L is open.

is an (n-l)-cell.

And by 4-4.8, S is an n-cell with

δ S ^ L O x ^^ ^ L. If y f X

^ L - 5 L C 3 - L, then a link of

y in S is a link of y in Μ as above. But a link of y in 3

is a cell, since y ζ. ^ S. Hence x-ii-^L-^ L C b M. Since

^ Μ is closed, χ * ^ L C ^ M, and since χ -if- L is a neighbourhood of X in b M, ^ L is a link of χ in ^ M. Hence a PL (n-1)-manifold without boundary. Remark;

Μ is

D

Thus, if χ f ^M, there exist links L of χ in Μ

(for example, the links obtained using simplicial presentations), such that

L C

and "b L a link of χ in ^ M. This need

notbe true for arbitrary links.

Also there exist links L of

X ^

^ Μ = ^ L. For example, take a

Μ

in M, such that L

regular presentation (p of Μ

in which χ is a vertex and take

77

4.4.12. Proposition. Let Μ be a PL n-manifold, and

a

simplicial presentation of M. If

CTC^^

or Μ -

ζ

, then either

M, and 0

I Lk( (i , Λ )1 is a (n-k-l)-cell if 6" C δ Μ

2)

Lk( ^ ,

is a (n-k-1 )^sphere if

Μ - £) Μ

where k is the dimension of (f Proof?

That

C ^ ^ or Μ

b Μ

is proved in 4.4.11.· The

proof of (1) and (2) is by induction on k. It k from definition. Lk(

If k> 0, let

0, this follows

be a (k-l)-face of

. Then

) = Lk(6·}^ Lk( C , Λ )), and

being the link of a point in fi" is either (n-l)-sphere or a (n-l)-cell. Hence, by induction, or sphere of dimension C δ

Lk( ^ ,

)

is either a cell

(n-l) - (k-l) - 1 = (n-k-l). If

Lk( 6~ , Λ )

cannot be a sphere, since then

is a cell. Hence | Lk( • ^ S'J ^i· Lk( 6"", Λ ) = b

would be a sphere.. Thus if ^ C 0

| L k ( ) j

Λ )! *

ζ"

is a (n-k-1 )-cell.

Similarly if 6" Q Μ - ^ M, | Lk( eT , Λ ) is a (n-k-1 )-sphere. Q Ex. 4.4.13.

(1) Let Μ be a PL m-manifold, and Ν a PL n-manifold.

Then My. Ν is a PI (m+n)-manifold and ^ (M V.N) is the union of ^MXNandMK^M. Hint! Use, 4.3.21

and 4.4.8.

(2) If Μ

Ν is defined, it is not a manifold except

for the three cases of 4.4.8. Hint; Use 4.3.22.

D.

^

78

k:k, 14. Proposition; (a) If f 5 S — ^ S' is a one-to-one polyhedral map of an n-sphere S into another n-sphere S', then f is onto. (b) If f ί C — ^ C

is a one-to-one polyhedral map of an

n-cell C into another n-cell C

such that f(b C)

C', then f

is onto. Proof of (a); By induction.

If η = 0, S has two points and the

proposition is trivial. Let η ^ 0. Let f be simplicial with respect to presentations

and

of S, X ζ. ζ- for some L, = L2 =

\[b

of S and S'. If χ is any point ^ /S

δ",

s^

(f 6Γ )] ^^ Lk(f

Since f is infective

,

Λ 2 ) 1 . and S^ = ||f

f maps L^

f L^ and χ — ^ f(x). L^ induction f L^ is biJactive.

Lk(f 0-, Λ

^ L^, and f S^ is the join of

and L^ are (n-1 )-spheres, and by Therefore

open in S'. Since S is compact is connected^

Consider

Hence f(S) is

f(S) is closed in S'. Since S

f(S) - S'. (b) is proved similarly.

^

By the same method, it .can be shovm Ex. 4.4.15. There is no one-to-one polyhedral map of an n-sphere into an n-cell.

D

Ex. 4.4.16. a) A PL-manifold cannot be imbedded in another PL-manifold of lower dimension. . b) If Μ and Ν are two connected manifolds of the same dimension, "b Ν If

0, and Β Μ = 0, then Μ

Ν is also empty, and if Μ

cannot be embedded in N.

can be embedded in Ν then Μ jai» N.

79

Ex. 4.4.17. a) If Μ ς Ν are two PL n-manifolds, then Μ - ^ Μ C Ν - ^ N, and Μ - ^ Μ is open in N. Hint; Use 4.4. 14 and 4.4, 15. In particular any polyhedral equivalence of Ν has to taken Ν - "δ Ν onto Ν - "δ Ν and h Ν onto ^ Ν. b) If M C N - ' ^ N , both Μ and N. PL (n)-manifolds, and X any point of ^ M, show that there exist links L of χ in M, such that a link of χ in Μ is an (n-l)-cell Dd L, and D α bM = 'b D. D Ex. 4.4. ie. In 4.2.14, show that if Ρ is a PL n-manifold is a PL (n-m)-manifold and ^ (f~\q))C^P.

Q

4.5. Recalling Homotopy Facts. Here we discuss some of the homotopy facts needed later. The reader is referred to any standard book on homotopy theory for the proof of these. 4: 5. 1. We define a space Ρ to be polyhedra Y C

with dim X

(k-1)-connected iff, for any

k, every. continuous map Y — ^ Ρ has

an extension to X. Thus, a ( -1)-connected polyhedron must just be non-empty. A k-connected polyhedron for k

can be anjrthing. For k

0,

it is necessary and sufficient that Ρ be non-empty and that ίΓ^(Ρ) = 0 for i ^ k. 4.5-2. A pair of spaces (A, B) where Β Q λ, is k-connected if for any polyhedra Y Q X with dim X ^ k, and f : X — ^ A such that i"(Y)C

"t^'^en f is homotopic to a map g, leaving Y fixed, sucW

that g(X)Q B.

80

This is just the same as requiring that

B) = 0

for i 4 k. If A is contractible (or just (k-1)-connected) and (Aj B) is k-connected, then Β is (k-1)-connected. We shall have occasion to look at pairs of the form A - B), which we denoted briefly as (A, - B). The following discussion is designed to suggest how to prove a result on the connectivity of joins, which is well known from homotopy theory. U. 5.3. Let A^Q A, B^

B, and suppose

(A, - A^ ) is a-connected,

(B, - B^) is b-connected. Then (A X B, - A^

B^) is (a + b + 1)-

connected. Let Y C X, dim X$.a + b + 1, and

f:X — ^

A χ B, with

f(Y) Π A ^XB, = 0. We must now triangulate X finely by say /4 . Look at

so the two coordinates of f are homotopic, using homotopy extension, to get a map, still called

Because X - X^ has X^ f^ (Χο)Γ\Β, = 0

Β

2

1

f^ such that

as a deformation retract, we can first get

and then f~\A X Β ) is contained in X - X .

1 1

2

By changing, homotopically, only the first coordinate, we get f"' (A^XB^) = 0. To go more deeply into this sort of argument, see Blakers and Massey, "Homotopy groups of Triads" I, II, III", Annals of Mathematics Vol, 53, 55, 58. 4.5.4. If Ρ is (a-1 )-connected, Q is (b-1 )-connected, then Ρ

Q

81

is (a + b)-connected. For, let C(p), C(Q) be cones with vertices v, w. Then (C(P), - v) is a-connected, (C(Q), - w) is b-connected. Hence by 4.5.3, (C(P)

C(Q), - (v, w)) is (a + b + 1)-connected. By 4.3.19,

this pair is equivalent to (C (P * Q), - (v, w)). Hence Ρ (a + b)-connected.

Q is

For a direct proof of 4.5.4, see Milnor's "Con-

struction of Universal Bundles II (Annals of Mathematics, 1956, Vol.63). 4,5.5. The join of k non-empty polyhedra is (k-2)-connected. In particular

(k-l)-sphere is (k-2)-connected. The join of a (k-l)-

sphere and a a—connected polyhedron is (a+k)—connected. k

suspension

Thus a

(same as the join with a (k-1)-sphere; of a

connected polyhedron is at least k-connected.

82

Chapter V General Position We intend to study PL-manifolds is some detail. There are certain basic techniques which have been developed for this purpose, one of which is called "general position".

i\n example is the assertion

that "if Κ is a complex of dimension k, Μ a PL-manifold of dimension > 2 k, and f : Κ

^ Μ

is any map, then f can be appro-

ximated by imbeddings". More generally we start with some notions "a map f ,: Κ

^M

being generic" and "a map f : Κ — ^^ Μ being in

"generic position" with respect to some Y (_ M**. This "generic" will be usually with reference to some minimum possible dimensionality of "intersections", "self intersections" and "nicety of intersections". The problem of general position is to define useful generic things, and then try to approximate nongeneric maps by generic ones for as large a class of X's, Y's and M's as possible (even in the case of PL-manifolds, one finds it necessary to prove general position theorems for arbitrary K). It seems that the first step in approximating a map by such nice maps is to approximate by a so called nondegenerat'e map, that is a map

f ! Κ — ^ Μ which preserves dimensions of subpolyhedra. Now it happens that a good deal of 'general position' can

be obtained from just this nondegeneracy, that is if Y is the sort of polyhedron in which maps from polyhedra of dimension ^

some η

can be approximate by nondegenerate maps, then they can be approximated

83

by nicer maps also. And the class of these Y's is much larger than that of PL-manifolds. We call such spaces Non Degenerate

(n)-spaces or

ND(n) -spaces. The aim of this chapter is to obtain a good description of such spaces and prove a few general position theorems for these spaces. 5.1. Nondegeneracy. 5.1.1. Proposition. The following conditions on a polyhedral map f JΡ— y

Q are equivalent! (a) For every subpolyhedron X of P, dim f(X) = dim X. (b) For every subpolyhedron Y dim f"^(Y)

of Q,

dim Y.

(c) For every point χ ^ Q, f ^(x) is finite. (d) For every line segment

, χ / j,

f([x, yj) contains more than one point. (e) For every φ ,

with respect to which f

is simplicial, f(
> ^

·

(f) There exists a presentation Q

of p, on

each cell of which f is linear, and one-to-one. Proof{

Clearly (a)

(d)

(b)

(c) ^

(e)

(f)

(d)

84

To see that (a) — ^ (b) : Consider a subpolyhedron Y of Q; then, f Dim (Γ\Υ)) = dim f (f"^(Y)) by (a) and as f (f"VY))CY, dim f (rkl)) ^dim Y. Hence dim (Γ'(Υ) 4 dim Y. To see the (d) Let CP 6 (p · li" that of

(c): ίΤ) has not the same dimension as

, two different vertices of Q"" say v^ and v^ are mapped

onto the same vertex of f( ίΤ") say v. Then ^v^, v^^ is mapped onto a single point v, contradicting (d), Finally (f)

(a):

To see this^ first observe that if f is linear and oneto-one- on a cell C, then it is linear one one-to-one on C also. Thus if A is a polyhedron in G, dim f(A) = dim A. But, X = U

(X Π C), and dim X = Max (dim X Λ c). It follows that

dim f(X) = dim X. Thus we have

t

and therefore all the conditions are equivalent. 5>t.2. Definition. We shall call a polyhedral map f which satisfies any of the six equivalent conditions of proposition 5.1.1. a nondegenerate map.

85

Note that a nondegenerate map may have various "foldings"; in other words it need not be a local embedding. Ex. 5.1.3. ( 0 Ρ

If f : Ρ

·•· yPj^j

Q is a polyhedral map, and

is a subpolyhedron of P, 1 4 i ^ k, and if

f /P^ is nondegenerate, then f is nondegenerate. (2) If f : Ρ — y

Q is nondegenerate, and X C Ρ a

subpolyhedron, then f / X is also nondegenerate. \ /"Hint : Use U C j .

0

Ex. 5.1.^. Proposition. The composition of two nondegenerate maps is a nondegenerate map.

Q

Ex. 5.1.5. Proposition. If f ; P^ — ^ nondegenerate, then f * g

P^

P^

Q^, and g : P^ — Q ^

are

^ Q * Q^ is nondegenerate.

In particular conical extensions of nondegenerate maps are again nondegenerate. /~Hint.5 Consider presentations with respect to which f, g are simplicial and use Let f : Ρ — ^

1, f_J7,

Ο

Q be a polyhedral map,

and

tri-

angulations of Ρ and Q with reference to which f is simplicial. and ^

^

as usual denote the k^^ skeletons of J^ a n d ^ ..

Let 0 , T| be centerings of

,

f(e ^ ) - ^(f δ") for

- Let

respectively such that /J^

and ^^^ denote

the dual skeletons with respect to these centerings. Then Ex. 5.1.6,.

f is nondegenerate if and only if f

^

C '

86

(b) f is nondegenerate if and only if

'(c) Formulate and prove the analogues of (a) and (b) for regular presentations. 5.2. ND (n)-spaces. Definition and Elementary properties. 5'.2.1. Definition. (read Non-Degenerate

A polyhedron Μ is said to be an ND (n)-space (n)-space) if and only if}

for every polyhedron X of dimension ^ n, and any map f t X — ^ Μ and any ^ ^

0, there is an ^ -approximation to f

which is nondegenerate. This property is a polyhedral invariant: •5.2.2. Proposition.

If Μ is and ND (n)-space, and

oC ' Μ — ^ M'

a polyhedral equivalence, then m' is also ND (n). Proof; Obvious.

pj

Before we proceed further, it would be nice to know such spaces exist. Here is an exampl; 5.2.3. Proposition. ' An n-cell is an ND (n)-space. Proof;

By 5.2.2, it is enough to prove for A, where A is an open

convex n-csll in

. Let f ;X — ^

A be any map from a poly-

hedron X of dimension ^ n. First choose a triangulation X, such that f is linear on each simplex of be the vertices of so that f'(v )

of

, Let

ν

. First we ' alter the map f a little to a f' f'(v )

are all in A = Interior of I.

This

7

is clearly possible? We just have to choose points near f(vj_)'s in the interior and extend linearly. Next, by 1.2.12 of Chapter I, we can choose' y

y 1

so that y r

is near f'(v.) and y 's are 1 1

1

in general position, that is any (n + 1) or less number of points of y's is Independent.

If we choose y's near enough f'(v^)'s, the

y's will be still in A, that is why we shifted interior. Now we define of

= y^

to a get a map X

f(v^)'s into the

and extend linearly on simplexes

^ M, which is non-degenerate by 5.1.1 (f).

And surely if f(v_) and y. are near enough, g will be good approximation to f.

Ο

The next proposition says, roughly, that an ND (n)-space is locally ND (n). 5.2.4. Proposition.

If

is any simplicial presentation of an

ND (n)-space, and 6 Λ ^ then j[St(6r,y^ ) is an ND (n)-space. Proof; If χ is a point of β' , then ( St( ) is a cone with vertex χ and base

lLk(

M; and

St(6",A

f :X

>\st( 6", Λ )

^

η and ^ >

so that f'(X) Q

f

which is a link of χ in

Lk(€~,A))

is open in M. If

is any map from a polyhedron X of dimension

0, we first shink it towards χ by a map f St(

(f, f') <

Now Ν

-

^ 6~ ^jlkC cr.

Μ

is a subpolyhedron of M, and Ρ (f'(X), N) >

)

say

so that

)| - ό 6Γ * (Lk( ζ-, Λ )] ί'(Χ)Γ\Ν = 0. Therefore

^ 0. Let T| = min( ^ , € /2). Since Μ is ND (n),

we can obtain an ^

-approximation to

nonde gene rate, g is an ^

f', say g which is

-approximation to f and g(X) Γ\ Ν = 0,

88

g(X)r Μ, Therefore is

an

N D

g(X) C

)j . Hence I St(r ,Λ )|

(n)-spaGe.

Next we establish a sort of "general position" theorem for ND (n)-space. 5.2·. 5. Theorem. Let Μ be an ND (n)-space, Κ a subpolyhedron of Μ of dimension ^ k. Let

f :X

^M

be a map from a polyhedron X

of dimension. ^ η ^ k — 1. Then f csn be approxiniated. by a nicLp Λ

g !X

^ Μ such that g(X) ΓΝ Κ = 0.

Proof! Let D be a (k + 1 )-cell. Let

f' s D

^ Μ be the

composition of the projection D X X — > X and f, that is f'(a, x) = f(x) for a 'r D, X e X. By hypothesis, dim (D ><X) ^ n. I

Hence f

can be approximated by a map g

The dimension of g'-^(K) ^ k. Consider

which is nondegenerate. ΓΓ (g'~^(K)); (where Π" i®

the projection D A X — > D), this has dimension ^ k; hence it cannot be all of the (k + 1 )-dimensional D. Choose some a ^ D Then g' (aV.X)nK - 0 . X ζ; X. Since

iV(g'~^(K)).

We define g by, g(x) = g'(a, x), for

f(x) = f'(a, x), and g' can be chosen to be as close

to f' as we like, we can get a g as close to f as we like.

Ο

We can draw a few corollaries, by applsdng the earlier approximation theorems. Ex. 5.2.6.

If Μ is ND (n), Κ a subpolyhedron of Μ of dimension

^ k, then the pair (M, Μ - Κ) is (η - k - 1)-connected. /"Hint! It is enou^ to consider maps

f s (D, ^ D)

^ (M, Μ - Κ),

and show that such an f is homotopic to a map g by a homotopy which is fixed on ^ D, and with g(D)

Μ ~ K. First, by 5.2.5, one can

89

get a very close approximation g^ to f with since g^ ^^D and f j^D

Μ - K. Then

are very close, there will be a small

homotopy h (3.2.3) in a compact polyhedron in M - Κ with hQ = f

D, h^ = g^ 6 D, Expressing D as the identification space

of b D X. I and D, (a cell with

^ D^ = b D X l) at ^ D X 1 t and patching up h and the equivalent of g on D , we get a map

J g : D — ^ M, with g j^D = f

^ D, g(D)(2 Μ - Κ and g close

to f. Then there will a homotopy of f and g fixed on

DJ7.

As an application this and 5.2.4 we have: 5.2.7. Proposition. If ND (n)-spaGe and Σ- Ζ.

is a simplicial presentation of an , then |LK( ζ" , Λ )

IS (η - dim ^ -2)~

connected. Proof: For by 5.2.4,

IS

|pt(5-,/i)| is |st( ^ , Λ )| - ? j i

ND (η), and by 5.2.6,

(n - dim ΰ" - 1 )-connected, thus giving that St( t5" , ) - ^ (n - dim C - 2)-connected, But Lk( ) is a deformation retract of

st(

A) -

C

is

π

5.3. Characterisations of ND (n)-5paces. We now introduce two more properties: the first an inductively defined local property called A(n) and the second a property of simplicial presentations called B(n) and which is satisfied by the simplicial presentations of ND (n)-spaces.

It turns out

that if Μ is a polyhedron and / 4 a simplicial presentation of M, then Μ is A(n) if and ohly if

is B(n).

Finally, we complete

the circle by showing that A(n)-space have an approxima,tion property

90

which is somewhat stronger than that assumed for ND (n)-spaces. A(n) shows that ND (n) is a local property. B(n) is useful in checking whether a given polyhedron is ND (n) or not. Using these, some more descriptions and properties of ND (n)-spaces can be given.. 5.3.1. Definition

(The property A(n) for polyhedra).

Any polyhedron is A(0). If η ^ 1 , a polyhedron Μ

is A(n) if and only if the

link of every point in Μ is a (n - 2)-connected 5.3.2. Definition.

(The property B(n) for simplicial presentations),

A simplicial presentation every fT

Ji ,

A(n - l),

Lk( f ,

)

is B(n), if and only if for

is (n - dim tT - 2)-connected.

By 5.2.7i we have 5.3.3. Proposition. If Μ is ND (n), then every simplicial presentation

of Μ is B(n).

Q

The next to propositions show that A(n) and B(n) are equivalent (ignoring logical difficulties), /

5.3.4. Proposition. If Μ is A(n,),' then every simplicial presentation of Μ is B(n). Proof! The proof is by induction on n. For η = 0, the B(n) condition says that certain sets are (^ - 2)-conhected,.i.e, any X is B(0), agreeing with the fact that any Μ is A(0). Let n ^ 0, and assume the proposition for m Let|A|c)M,rt4and dim C

n. = k. f

If k = 0, then by the condition A(n), the link of the element of Q"" , which can be taken to be | Lk(

, A)

IS

91

(η - 2)-connected. If k ^ Ο, let X be any point of if . Then a link of χ in Μ is

^ (Γ ^|Lk(

)j / which is A(n-l) by hypothesis.

Hence by inductive hypothesis, it^ simplicial presentation ^ ^ L k ( ^ , y^j ) satisfies B(n-l). If face of ^

is any (k-1 )-dimensional

, = (LkCT , [ ό ψ Lk(

which is ((n-l) - (k-l) - 2)-connected ) is B(n-l). 5.3»

Proposition.

))

i.e. (n-k-2)-connected since

•

If a polyhedron Μ

has a simplicial

presen-

tation which is B(n), then Μ is A(n). Λ Proof: The proof is again by induction.

For η = 0, it is the same

as in the previous case. And assume the proposition to be true for all m < η > 0. Let X ξ Μ.

^hen χ belongs to some simplex ίΤ" of /f ,

and a link of χ in Μ is ^ ^ ^ Lk( ίΤ , Λ ) ^ . We must show that this is an (n - 2)-connected A(n-l), As per connectivity, we note (setting k = dim ^

is a (k - l)-sphere; and by B(n), Lk( tT"» Λ )

connected.

) that is (n-k-2)-

As the join with a (k-l)-sphere rises connectivity by

k, ό Γ * Lk( 4 ) is (n-2)-connected. To prove that ^ ^ * Lk( )1 I enough to show that

Lk(, Λ )=

is A n-l

it is

' say is B(n-1); for

then by induction it would follow that

' = ^

* Lk( €Γ ,Λ ) |

is A(n-l). Take a typical simplex oC

/!'· It is of the form

92

p>V

, β C-

,/

Lk( Γ Λ

possible. Now Lk( ^ ,

), with

Pi or V

^ Lk( p, , {5-Γ] ) ^ Lk( V , Lk(

Let a, b, c be the dimensions of pectively.

a

Lk( p, ,

= 0 being

>.

/J )).

, V' res-

b + c + 1. Remember that dim g" = k. Therefore )

is a (k - b - 2)-sphere.

Lk(V , Lk(tr, /5))| =(Lk( V ^

,

Now^

) I ; and

by B(n)

assumption, this is (n - (c + k + 1) - 2) connected. Hence the join of

Lk(p>,

)

Lk( Y' , Lk( (Γ , Λ )) 1 which is

and

Lk(X , Λ') IS

(n - (c + k + l) - 2) + (k - b -

2) +1

-connected

that is ((n - 1) - a - 2)-connected. Thus

is B(n-l), and therefore by induction )l

a link of χ in Μ is a (n-2)-connected

A(n-1). Hence Μ is A(n).

0

We need the follovdng proposition for the next theorem. 5.3.6. Proposition. Let ^

be a regular presentation of an

A(n)-space Μ and V| be any centering of ^ . Let' A_ be any element of Ip , and dim A ^ k. Then ^ A

is an (n - k - 2)-connected

is a contractible A h-k' Proof; We know that ^ A is the link of a k-simplex in and

Since d ^

satisfies B(n), If k = 0,

Xk

^A

^ ·

is (n-k-2)-connected.

is the link of a point and therefore

A(n-l), since Μ is A(n). k > 0, then ^ A * \/\ A

is a link of a point in M,

93

and so is A(n-1). Take a (k-1 )-simplex ^ of d ^ ^A

is Lk( ^ , d ^δ·^^·^

in

^ A; then

which (by induction on k), we know

to be' a presentation of an A((n-1) - (k-l) - l)-space. To prove that

| ί A ( is A(n-k), we prove that ο A is

B(n-k). Consider its vertex 7|A, then Lk( ^ A

is (n-k-2)-connected.

Lk( CTJ cf A)| simplex <Γΐ; = If

C (Lk(

For a simplex ^ A))

^ [ η a] ,

Lk(

,

A)

^ ^ A, we have

which is contractible. For a

^

Cf , ff A) = Lk(

has dimension t, ^has dimension

A(n-k-l), Lk( Γ > Ά

A, ^A) =' ^^ A, and

(t-l); and so

A)( is ((n-k-l) - (t-l) - 2)-connected, i.e.

Q

5.3»7. Theorem. Let Μ be an A(n)-3pace, Y

degenerate. f

k\ being

is ((n-k) - t - 2)-connected. This shows that ζ k

satisfies B(n-k).

dimension . ^

Λ A).

n, and f t X —— ^ Μ Given any

ξ

X polyhedra of

a map such that f Y is non-

0, there is an

^ -approximation g to

such that g is nondegenerate and g | Y = f Y.

Proof; The proof will be by induction on n. If η =

we take

g = f, since any map on a 0-dim.ensional polyhedron is nondegenerate. So assume η η to be true for all m

0, that the proposition with m instead of n.

Without loss of generality we can assume that hedral. Choose simplicial presentations ^ and Μ

Q

f is poly-

j'^TfX of Y?· X

such that f is simplicial with respect to

a n d j

such that the diameter of the star of each simplex in '•'llYC. is less than ζ

. Let β , 'Tj be centerings of

and^A^ with

and

94

f(Q

IT ) - η

εε

C ^ ^

every ^

(f-6") for all

6Λ

. Then clearly is less than ζ· for

and the diameter of \ 0 f .

Consider an arrangement A^

A^ of simplexes of '^Υΐΐ^

so that dim k^ - > dim A. Ι-ίΊ,, for 1 i < k. The crucial fact about such an arrangement is, for each i, (*) }\A, is the union of S A. for some j's less than i. We construct an inductive situation

such that

-1

(1)

...

U|iA.\)

(2) Y. . x.n Y (3) g, : X, - —^ M, a nondegenerate map (4) g. (f-'lcTAj) C l ^ A . j (5) g.

i-1

(6) g. Y. = f Y. 1

η

, , is the union of certain A^ s in the beginning, say A^ s

with i ^ i .

f-^^"") ch""

and

1 Λ " is 0-dimensional.

Hence f f (

···

we take this to be g

all the above properties are satisfied and we

have more than started the induction* that g.

) is already nondegenerate. If

Now let i > i, and suppose

is defined, that is we already have the situation

1-1

It follows from (4) and (5), that for j < i,

aJ ) into

I^A. .

^

.

95

Also this shows that if x ^ f

(^A.l) then both JI

and f(x) are in

, which has diameter <. ξ- , and so X is an ^ -approximation to f i-Γ There are now two cases. Case 1. —

dim A. ^ k 1, 1 7 Look at A.

a n d Y i ( Y n x ' ) U f ·-1 ^ ( \'},A.\).

x' - r^

Y Π X' and g. ^ on ^

This is a contractible

,

( (a

(

A(n-k). Let The m a p s

f on

agree where both are defined by

and are nondegenerate by h3φothesis and induction.

Hence patching them up we get a nondegenerate map f' ; γ' — Since S \ \ is contractible

f' can be extended to a map (still

denoted by f') of X' to

aJ . Since x' ς

and

dim x'^n-k,

A^ is A(n-k), there is a nondegenerate map f" : x' - — ^ |<S

such that f

Y' ^ f' Y' , by using the theorem for (n-k) < n~1.

We now define g^ to be g^ ^ on X^ ^ and f these two maps agree where both are defined, namely f \

on X ; )·

Thus g^ is well defined and is nondegenerate as both f" and g, ^ are nondegenerate.

And clearly all the six conditions of

are

satisfied. Case 2.

Dim A^ = 0. Let B. 1

onto A^.

Β be the vertices of s

which are mapped

Then

... Let X' =

0ΐ>3\

\Sb

96

χ' is of dimension ^ n-1, and contains

f ( ^A^^ ). Let

Y' = f ( ^A^ ). Here the important point to notice is, that Y H x ' C y'. This is because so f(Y Γ\Χ') C

f(Ynx')

f Y is nondegenerate: Y Π x' C

It is also in

I al5A.j. We first extend

g.

\ξ A^

SC^,

and therefore

|;\aJ. Y

to X

and then by conical

extension to Ϊ \ ^ A^ ). g^ ^ maps γ' into ^ A^ and is nondegenerate on Y . Since

is (n-2)-connected, and dim X^n-1,

can be extended to a map f' of x' into \ is also A(n-l). Hence by the inductive hypothesis we can approximate f' by a nondegenerate map f" such that f" γ· . f' Y' =.g • Y'. 1-1 Hence f is nondegenerate and maps , 1 4 j 4s ^ B^ into I ^ · We extend this to a map h^ : ^B^ | — ^ ^ A^ by mapping B. to A. and taking the join . h. 3 1 J is clearly nondegenerate. Since lis I α \<ί Β.λζ^ l^B ι ^ n s . Ιζ^ χ', J J . j j if j / j'j h^s agree whereever their domains of definition overlap. J

Similarly h. and g agree where both are defined. We now define J i-1 g· to be g on X and h on cfB Thus g. is defined ^ i-1 i-T j on

0

^~

i® nondegenerate

since g^ ^

and

h.'s are nondegenerate. It obviously satisfies conditions 1-5 of ^ J to see that it satisfies (6) also: Let ij" is any simplex of' d ^ in

, i

S Β , if Β is not a vertex of Q" there is nothing to prove; j J if B. is a vertex of, ^ , write ξΤ = -Γβ I -ς-' . Both h. and f J JJ J agree on and B. and on Τ both are joins, hence both are equal J

97

on

. Then (6) is also satisfied and we have the situation

· D i

This theorem shows in particular that ND (n) is a local property; and that ND(n)-spaces have stronger approximation property than is assumed for them. The following propositions, which depend on the computations of links are left as exercises, Ex. 5.3.8. Proposition. X is an (n-2)-connected

C(X) and S(X) are ND (n) if and only if ND (n-l). Ο

Thus the k^^ suspension of X Is ND (n) if and only if X is an (n-k-1)-connected Ex. 5.3·9. Proposition.

Let

Then X is ND (n) if and only ND (n-l) for each vertex ν of Ex. 5.3.10. Proposition. 1ΐ ND (n)-space, and the dual skeleton

k ^

ND (n-k)-space. be a simplicial presentation of X, Lk(v, Λ ) .

is (n-2)-connected

Ο

is a simplicial presentation of an

n, then the skeleton is ND (n-k),

^ is ND (k), and

0

Thus the class of ND (n)-spaces is much larger than the class of PL n-manifolds, which Incidentally are ND (n) by the B(n)-property. The resiiLts of this section can be summarised in the following proposition: 5.3.It, Proposition.

The following conditions on a polyhedron Μ are

equivalent; 1) Μ is ND (n) 2) Μ is A (n)

3) a simplicial presentation of Μ is B(n) 4) every simplicial presentation of Μ is B(n) 5) there exists a simplicial presentation Μ such that | L K ( V , i s for all r ^

of

(n-2)-connected

and dim ν = 0

6) Μ satisfies the approximation property of theorem 5.3.7. 7) M)( I is ND (n+1).

Q

5.4. Sinfflilarity Dimension. 5.4.1. Definitions and Remarks. Let Ρ and Μ be two polyhedra, dim Ρ = p, dim Μ = m, p.^ m, and f ί Ρ

Μ a nondegenerate map.

Ee define the singiilarity of f (or the 2-fold singularity of f) to be set^x £ P^f"' f(x) contains at least 2 points • , and denote it by S(f) or S^ (f). By triangulating f, it can be seen easily that S(f) is a finite union of open cells, so that S(f) is a subpolyhedron of p. Similarly, we define the r-fold singularity of f for to be the set ^ x It· Ρ |

f(x) contains at least r points ι This vdll

be denoted by S^ (f). As above S^ (f) is a finite union of open cells, so that S^ (f) is a subpolyhedron of P. Clearly S^ (f)^S^ (f))

•

and S^ (f) are empty after a certain stage; since f is nondegenerate. The number (m - p) is usually referred to as the codimension; and the number r(p) - (r-l)m, for

2 is called

the r-fold point dimension and is denoted by d^ (see e.g. Zeeman "Seminar on combinatorial Topology", Chapter Vl). Clearly d - d -(ήν-ρ), r r—i It will be convenient to use the notions of dimension and

99

imbedding in the follovdng cases: (l) dimension of k, where A is a union of open cells. In this case the dim.· A denotes the maximum of the dimensions of the open cells comprising

A and is the same as the

dimension of the polyhedron A. (2) Imbedding C is an open cell and Μ f

a polyhedron.

f of G ——^ M, when

This will be used only when

comes from a polyhedral embedding of C. In such a case f(C) will

be the union of a finite member of open cells.

And if A

Μ

is some

finite union of open cells, then f ^(A) will be finite union of open cells, and one can talk of its dimension etc., A nondegenerate map

f : Ρ ——^ Μ will be said to be In

general position if dim (S^ (f))^ d^

, {dL cJX ^V·

If ρ = m, this means nothing more than that

f is nondegenerate, so

usually ρ <_ m. 5.4.2. Proposition. Let ^ Ρ

such that for every C

be a regular presentation of a polyhedron j f C is an embedding. Let the cells

of ^p tie C^

C^ , arranged so that

dim C^ ^ dim

1 ^ i ^t,

and let P^, i ^ t be the sub-

polyhedron of Ρ whose presentation is , G^ ,·'·} G^^· Then i) SgifJPi) =

(f,|p._^)g|c. Af~Vf(P.Jj)]

r\f(c.))] ii) S^(f|Pi)

UiVi

(f|Pi_i)nf-Hf(c.))]

100

This is obvious»

If we write Ρ =

(compatible with

the definition of S^'s, then S^(f P^) would be -^Mst P^, and only (ii) be written (with r ^ 2) instead of (i) and (ii). The proposition is useful in inductive proofs. For example, to check that a nondegenerate

f is in general position, it is enough

check for each little cell C^, that dim C^O f~Vf(S^

V

If we have already checked upto the previous stage; since f is nondegenerate

f"^ f(S

(flp. )) will of dimension d , and then I f 1— 1 ^ 1J we will, have to verify that f C. intersects f (S , (f P.))) in 1 r-1 1 codimension 'X (mr-p) or that (C.) intersects f(S (fjP )) in 1 r-1 ' i codimension ^ m-p, (We usually say that A intersects Β in codimension q if dim (APl B) = dim Β - q. Similarly the expression Ά intersects Β in codimension^ q' is used to denote dim (A Π B) ^ dim Β - q). The aim of the next few propositions is to obtain presentations on which it would be possible to inductively change the map, so that

f(C^) will intersect the images of the previous

singularities in codimension ^ (m-p). Proposition 5.4.7 and 5.4.9 are ones we needj the others are auxilary to these. Ex. 5.4.3. Let A, B, C, be three open convex cells, such that A Π Β is a single point and C

A Ο B. Then dim C ^ dim A + dim B.

/"Hint; First observe that if A' and B' are any two intersecting open cells then L^^t L^i = L^i p^ g' j where manifold spanned by X. Applying this to the dim C = dim L. V dim L, , = dim L + dim C (A U B ) A = dim L, + dim A

L^ denotes the linear above situation L - dim (L Γ\ L ) Β A Β L - dim (L„· _ Β AΛ Β

101

= dim L, + dim L . since Α.Γ\ Β

A

Β

is a point^J'' 5.4.4. Proposition. Let A be an open convex cell of dimension n, and QL a regular presentation of A with A tC?-· If L is anylinear manifold such that dim L H A = ky^O, then there is a Β of dimension^ η - k, vdth Β Π L / 0. Further, if A' is any cell of

contained in ^ A, we can require that A' Λ Β = 0.

Proof; If k = 0, we can choose A itself to be B. If k > 0, consider the regular presentation^ C

Π

C

C Π L Gα L

0,

have more than one 0-cell. Choose one of

these 0-cells of C. 'It must be the form Β Π L for some Β We >rould;iike to apply 5.4.2, for B, L A A L A A do not intersect.

and A. But Β and

Since we are interested in the dimension

of B, the situation can be remedied as follows! Let D be an n-cell, such that A Q Ό. L Γ\ D .is again k-dimensional. Since B<2 D, B O L C D Γ\ L, and as Β Ο ^ is nonempty, Β and D Π L intersect. Β

(D Π L) cannot be more than one point since

Bri(DnL)^]'BnL

which is just a point. Applying 5.4.2 to B, D O L and D we have η = dim D ^ dim Β + dim (D Λ L) = dim Β + k, or, dim Β ^ η - k. To see the additional remark, observe that all the vertices of

cannot be in A', for then L Π

A', _,contrary to the

hypothesis that L Π A is nonempty. Hence we can choose a 0-cell Β Π Lj Β ^ 0 \ of Β η A' = 0.

not in A'. Since OV is a regular presentation

Π

This just means that if. L does not intersect the cells of

102

QV. οί dim ^

, then dimension of the intersection is

η -i ,

or codimension of intersection is > !L . Using the second remark of 5.4.4. we have: 5.4.5.

Corollary. Let (p be a regialar presentation, containing a full

subpresentation ^ (which may be empty). Let

(p

=

dim C^ k

If L is any linear manifold which does not intersect ^ , then ^ Κ dim (L π ( (Ρ - O^) 4 η - k where η = dim ((J) - <3»). CX 5.4.6. Proposition. Let ^ A be a closed convex cell of dimension ^

k + q, let S be a (k-1)-sphere in ^ A s and B^ >··., B^ be a

finite number of open convex cells of dimension / q - 1 contained in the interior of A. Further, let

be a simplicial presentation

of S. Then there is an open dense set U of interior A if a ^ U, <5^ ^ ^ , then the linear manifold L, ( by
such that

. generated , a;

'a' does not intersect any of the B^'s.

Proof; For any {ρζ.

, consider the linear manifolds L

generated by CT and B., ^ Hence U = int A - {J ^ i int A. If a is any point ^ intersect any of the B.'sj

for

( (Γ > i < r. Dim L. ^ k+q-1. ^ V
L. C cr ί of U ^ , then L

does not . ( ^ , a) for if there were a B. with

3 L. OB. / 0, let b ^ L, ,ΠΒ·· L r L and (σ-, a) J a^ J («Γ, b ) ^ ((T , a) is of the same dimension, as L, ,, since b Is in the interior of A, Thus a t L vC L ( (Γ , a) contrary to the choice of (
t1 «ν β

» "U

satisfies our

103

5.4.7. Proposition. Let ^

be a closed convex cell of dimension

k + q, let S be a (k-l)-sphere contained in ^ A, and let

• Β W '

Β • be a finite number of open convex cells in int A. ΓJ

Then there is an open dense subset U , of int A such that if a ζ U, then S Proof; Let

a intersects each of the B/S

in codimension ^ q.

be some simplicial presentation of S. First let us

consider one B^. Let ^

^ be a regular presentation of B7

containing a full subpresentation ^ . covering B7 Γ)0^· Let (B - X J "iini q-l]. By 5.4.6, there is an open ^ i 1 J dense subset of int A say U^ such that if a ^ U^, 6" ζ; , then ^ intersect any of the elements of (β . B y 5.4.5, V tj J a; q-1 dim L. -Ύα ) ^ ~ where n. = dim Β . Hence \ D > a; ^i ' i i 1 i dim (S ·«· a Π B_) ^ n^ - q. Therefore df we take U = u^, where U J

constructed as above for each of Β 's, then U, satisfies the J

requirements of the proposition. 5.4.β. Proposition. Let q-cell;

D

(f be a k-simplex,

a regular presentation of

an open dense subset U of ^ manifold L.

Proof; Let C

C Γ\

and a, does not intersect any 0 and dim C ^ q-1.

, with C A "J- = 0 and dim C ^ q-1. The linear

manifold L, __ , c;

has dimension

dimension k + q + 1. Therefore L^ ^ and so Un =

Then there exists

, such that if a € U, the linear

. spanned by ^

cell C ζ (p satisfying

* Δ.

a closed convex

- L,

.

^^ Π Δ

is open and dense in A

. ) has

dimension^ q-1 · Define U

104

to be the intersection of all the U . If C C of (p of dimension ^ q-1, C Π ^ = 0, ^ choose bt C r>L : since b fli ( a) y = dim L and so L = L ^^ > ( r , a) ( or J a L contrary to the choice

a ^ U, and there were some with L, C ^ 0, V ί a; , dim L· =k + 1 = ( b) i.e. L L h) (
5.4.9. Proposition. Let S be a :(p-1)-sphere, ^ q-cell, Ip a regiilar presentation of S •«• ^ 1) a regular refinement

^

a closed convex

. Then there exists

of ^^

2) a point a € Λ 3) a regular presentation ^

of S * a

such that a) ^

contains a fulL subpresentation

b) Each C i.

covering S,

i

is the intersection of a linear manifold with a (unique) cell E^ ^ (p', if C

C', E_ / Ε

c

and if C< c', then E^ < Ε '

c c

c) dim C < dim Ε - q, for all C ^ ^ C

-Λ .

Proof; Let 01 , (j^ be simplicial presentations of 3, ^'

be a common simplicial refinement of

; and let

* (Ji and ^

. Since

Q Y is full in Ol-* (B , there is a subpresentation, say (p

covering S. If ζ" ζ: (Jl, S" * Δ

, of

is covered by a subpresentation

in , hence there is a subpresentation of (p \ covering

*

subset U ^

of A ·

for

. Applying 5.4.8 to ^ ^ ^

say

(P^j— t

, we get an open dense

intersection of the sets U ^

QX, . Let a ^ U. Obviously

'a' is in an (open) q-simplex

105 1

of ' (p

contained in l\ , Hence

(gj , call it

'a' belongs to a q-simplex of

f .

We define ^

to be union of J^ ,

intersections of the form L ^ ^ It is clear that L,

a) ^

<Γ t Φ- ' ^

\ 0 ® =

Moreover

\^ 3 (F may be empty) since that ^

t (p "/C ·

EHS^F,

Ρ^Λ

I

is full in

p

. This immediately gives

is a regular presentation^ using the fact that

is the disjoint union of cells C

, and all nonempty

^A

(AO B)

B, A Γ\ ^B, ^A Γ\. ^B, for open convex

A, Β with A η Β 0. Moreover ^ is full in Qp·. If is of the form C = L Π Ε, we write Ε as Ε . By

A

(er, a)

c

definition each C ^ (j> is the intersection of Ε with a linear Λ C manifold, and if c'
2

is-.r e o L , Ε , Ε t(p C 1 ^ Ο2 ^ Ε C,

, 0 ill ^ a 1 2

Ε . If C2

6Γ

case Ep C 5*· f, 1 '

If 5- - Τ , and C ^ C , clearly 1 2

, then C 1

C ^ Ρ' ^ f defined in the first paragraph of I»

the proof). But C ^ and

^

Remark; In the above proposition of S * A

refining

cannot be equal to C . In this 2

are disjoint, hence E^ ^ ^c ' ^

^

can be taken any presentation

and adjoin presentation of S * Λ .

5.4. to. Proposition. Let Μ be an ND (n)-space. Let X

^

106

polyhedra such that Ρ

X U C, C a closed convex cell, and

Χ(Λ C = ^ C, and dim Ρ = ρ < n. Let f ; Ρ — ^ Μ be a map such that f / X is in general position.

Then there exists an arbitrary

close approximation g to f such /that g is in general position and g / X = f / X. Proof; If ρ = n, any nondegenerate approximation of f would do. So let ρ ^ n. In particular dim C ^ ρ ^ n. Step A. Let D be an

n)-dimr-cell containing ."^C in its

boundary, and such that 1) Ds'^C^^'A., /X a closed convex (n-p)-cell 2) Δ Π C is a single point of both C and

^

'd' in the interior

so that C = d * b C

3) Dr\ Ρ - C. This is clearly possible (upto polyhedral equivalence by considering Ρ

0 in V X. W, (where V is the vector space containing

P, W an (n-p)-dimensional vector space), and taking an (n-p)-cell Λ

through d

0 in d

of the identity on r : D —^

C. Thus

W, for some d ^ C ^ ^ C etc. . The join

C and the retraction Δ — ^ d gives a retraction (f/C) · r is an e^ctension of f/C. Since Μ is

an ND (n)-space, (f/C)» r can be approximated by a non-degenerage map, say h, such that h/

C = f/

C. Let us patch up f/X and h,

and let this be also called h; now h maps X U D = Ρ 0 D into Μ and is nondegenerate. Triangulate

h so that the triangulation of

X Ο D with reference to which h is simplicial contains a subpresentation ^

which refines a join presentation of

bc * A

. We

107

β. apply 5.4.9 now, ^

vdll be

(p ' there and we obtain, a point a ^ A >

a presentation (j^ (what vms called Q\ there) of ^ C * a. Each cell Β

of (3 not in

ζ) C, is the intersection of a unique Ε D a linear manifold, if b' < Β • then Ε / Ε and dim Β < Β Β Step Β. Let B^ ,,.., B^ be the elements of (p) not in ^ so that dim ^ dim for ' 1 i <; r. Let X^ = X U

of J^ vdth dim Ε - (n-p). Β C, arranged B^O ... UBj_ >

X. is a polyhedron. We define a sequence of embeddings J^ : X — ^ X UD, ^ i i such that 0

c£. X is the identity embedding of X in X U D

2) rL ^ is an extension of 3) 4) h X

is in general position. i

We shall construct the ^ ^ ' s vdth

one at a time begining

^ : X — } X yjD, the inclusion.

=

i®

general

position, and.»we can start the induction. Suppose JIj is already constructed. Then i-1 ^ dim S (h ) ^ d ; and by (2), (3), embeds 0 B. in Γ 1-1 r i_ t 1 C) Ε i Eg^.

; consider h""^ (h dC i-1

(S„ (h oG ))) intersected with ^ i-1

Since h is nondegenerate, these consists of a finite number

of open convex cells of dimension situation with q = n-p, A = Ε

d^. We apply 5.4.7 to this

.S=

( ^B.) and^B

B^

i_i

of 5.4.7 standing for the open cells of intersected with point in

say

for all r

1

^ ^

J

h~'(h «L (S (h dC )) i-1 ^ i-1

By 5.4.7, we can choose a

(a of 5.4.7) so that dG ^

^B^) * e^

108

intersects all these (i.e. The join of

|

for all r ^ 1) in codimension ^ (n-p). "^^P

^ point b^ of B^ to e^

gives the required extension on B^. Then dimj^^. B.r\h"^

'«^i-i

equivalently dim that is dim since h

is an extension of h ^^ i-1

1

Since Sr.i ( h ^ , )

- S^^^ (h

υ B.r\ (hc^.) u i s ^ (hoC.ix. L ^ 1' 1-1

1

(haC.) (B.)^ 1 1J

and since h

is already in general position, dim S ,(hdC )
can be chosen as close to f as we like is

£3

5.4.11. Theorem. Let Μ be an ND (n)-space, ^ C ^ polyhedra dim ρ ^ n

and

f jΡ—^M

a map such that f X is in general

position. Then there exists an arbitrary close approximation g to f such that g X = f X, and g is in general position. Proof: Let

be a regular presentation of Ρ with X

subpresentation

. Let

•="

covered by a

A"^ be arranged so

109

that dim A. 4 dim A.^, ,

^

Let P. =

Apply proposition 5.4.10 successively to (P , X ) 1 1

VJ A., X.= (P , X ). r, r

Thi^ requires the following comment: We must use our approximation theorem, which for Μ and 6 ^ 0

gives

) ^ 0, such

that for any Y 3 Z, h^ : Y ^ M,h2 : Ζ ^ M, if h2 is polyhedral, and h^ Ζ is a S { )-approximation to_ hg , then there is h^ : Y — ^ M, a polyhedral extension of h2 , which is an ^ -approximation to h^.' We want g to be an

t -approximation to f.

Denote f P. by f^. We start with g = f = f X. Suppose g 1 1 0 0 i-1 is defined on P. ^ such that g. ^ is in general position and is an f· . , approximation to f . Then we first extend g. to P, ^1-1 .1-1 1-1 i say f^' so that f^' is an ^ ^ approximation to f^ (this is possible since

, = 0 ( ^ i/2)) by the approximation theorem. Then we 1-1 I use 5.4.10 to get an ^ j_/2 s-PP^o^^^^tion g^ to . f. ' such that S. is in general position and g. 1 P. = g. g. is an ^ ~ 11-1 1-1 1 ^ ^-approximation to f^ and is in general position, g^ gives the required extension.

D

By the methods of 5.4.10, the following proposition can be provedJ $4.12. Proposition. Let Μ be ND (n); dim ρ ^ n, Ρ = X Ό C , where C is a closed p-cell, X Π C = δ C. Let f ! Ρ

Μ be

110

a map, such that f X is nondegenerate; and call dim X = x. Then J

there is a nondegenerate approximation g ί Ρ — ^ Mj arbitrarily close to f, such hat g I X = f X, and S(g) =

S(f|x)

plus (a finite number of open

• convex cells of dim ^Max (2p-n, p+x-n) Sketch of the proof! First we proceed as in Step k of 5.4.10. Now ρ = dim G. In Step B) instead of h) we write dim^B. Π

(h

(2ϊ>-η, p+x-n).

And in the proof instead of the mess before, we have only to bother about h h"^ (h =

(h cC (X )), intersected with Ε . i-1 i-1 Β i ^ (X. .)) = h"^ (h L . (X))Uh"^ (h dO (B U 1-1 1-1 1-1 ' i-1 1

(f(X))U

(h dC i-1

possibility of

..UB ) = i'l

(B \J ... U B )). Now the only 1 i-1

(f(X)) intersecting Eg^ is when

since h is simplicial. Since dim B^ ^ dim

E^Q

(f(X))

(r^-p)} it

already intersects in the right codimension. And the intersections with second set can be made minimal as before · Π 5.4.13. Theorem. Let Μ be ND ( n ) X C Ρ ; dim ρ ^ n, f ! Ρ — ^ a map such that f X is a map g ; Ρ

is an imbedding^

^ M, such that g

Then arbitrary close to f

X =f X

and calling

X = dim X, ρ = dim Ρ - X, dim S(g) ^ Max (2p - n, ρ + χ - n). Proof: This follows from 5-4.12, as 5.4.11 from 5.4.10.

Ο

This theorem is useful in proving the following embedding-

Μ

11

theorem for ND (n)-spaces. 5.4.14. Ρ

Theorem (Stated without proof). Let Μ be a ND (n)-space,

a polyhedron of dimension p ^ η - 3 and

connected map.

f :P—^ Μ

Then there is a polyhedron Q in Μ

homotopy equivalence

g :Ρ—

Q

a (2p-n+l)-

and a simple

such that the diagram

g Q inclusion N/ Μ is homotopy commutative.

•

The method of Step A in 5.4.10, gives; Proposition 5.4.15. Let Μ be an ND (n)-space, and Ρ a polyhedron of dimension ρ ^ η and presentation ^

of P,

f » Ρ — y Μ be any map. Then 3 a regular simplicial presentation^^TfC of Μ

arbitrary close approximation g to

and an

f such that, for each C

,

g (C) is a linear embedding, and g(C) is contained in a simplex of ^^tTCof dimension = dimension C + (n-p) and g( ^ C) = b Moreover -ΊΠΓζ. can be assumed to refine a given regular presentation of M. Also a relative version of 5.4.15 could be obtained, tS And from this and 5.4.13. 5.4.16. Theorem. Let

f :Ρ — ^

Μ be a map from a polyhedra Ρ of

dim - ρ into an ND (n)-space M, ρ ^ n, and let Y be a subpolyhedron of Μ

of dimension y. Then there exists an arbitrary close approximation

g to

f such that dim (g(P)a Y) ^ Ρ + y - n.

112

And a relative version of 5.4.16. X] 5.4.17. It should be remarked that the definition of 'general'position' in 5.4.1 is a definition of general position, and other definitions are possible^ and theorems, such as above can be proved. Here we formulate another definition and a theorem which can be proved by the methods of 5.4.10. ^^

A dimensional function d : P—^j^O, 1, ... J

^ function

defined on a polyhedron, with non-negative integer values, such that there is some regular presentation φ C t(p 5 X t

d (x)

dim C, and d is constant on C.

We say d^ If f 5 Ρ — function, and k^

of Ρ such that for all

if for all χ ζ-F, d, (x) ^ d2 (x). Μ

is a nondegenerate map, and d a dimensional

k^ non-negative integers, we define Sd (f; k^ ,.., kg) - 1

= f

m ςΜ

3 distinct points

X.1 ,..., Xs U P, such that d (x·) 1 ^^ k. X , and f(x.) 1 = m for all i,' ^ It is possible that such a set is a imion of open simplexes, and hence its dimension is easily defined. A map f ί Ρ

Μ is said to be n-regular with reference

to a dimensional function d on Ρ if it is nondegenerate and dim Sd (f; k^

k^)^ k^ + ... + k^ - (s-l)n.

for all s, and all s-tuples of non-negative integers. If dim Ρ ^ n, and since we have f nondegenerate then it

113

is possible to show that a map f is n-regular if it satisfies only a finite number of such inequalities, namely those for which all and

s

2n, The theorem that can be proved is this;

Theorem. Let X C Let

i" / Ρ

^^ where Μ is ND (n) and dim Ρ ^ η.

d^ and dp be dimensional functions on X and P, with

4

|

Suppose f X in n-regular with reference to d . Then f can be A

approximated arbitrarily closely by g : Ρ — ^

Μ with g X = f \ X and

g n-regular with reference to dp. The proof is along the lines of theorem 5.4.11. We find a regular presentation (p of Ρ with a subpresentation covering X, and such that d^ and dp are constant on elements of ^^ . We utilise theorem 5.4.10 to get g on the cells of ^

one at a time; in the

final atomic construction, analogous to part B) of 5.4.10, we will have SC^E where S is a (k-1)-sphere. Ε a cell of dimension ^ k + q, where q = n-p (the cell we are extending over is a p-cell, on which dp is constant^ p). We have to insert a k-cell that will intersect such things as ^s^^ in dimension

^

dim S^ (

5

l^s^ " ^

k^ + ... +

dp (p-cell) - s.n.

We can do this for our- situation; this inequality will imply

114

gj_ is n-regular. Ρ .Finally we can define on any polyhedron Ρ a canonical dimensional fiinction d : d (x) ! Min ^dim (Stary in fStar of χ in

fJ

y ζ: / Star of χ in p_7 J A function n-regular -with reference to this d -will be termed,· perhaps, in general position, it being understood that the target of the function is ND (n). Thus; Corollary; If X C. P> dim Ρ ^n, f ; Ρ — ^ M, Μ

a ND (n)-space, and

if f X is in general position then f X

can be extended to a map

g ;Ρ — Μ

g closely approximates f.

5.4.18.

in general position such that

Conclusion.

Finally, it should be remarked, that the above

'general position' theorems, interesting though they arej are not delicate enough for many applications in manifolds.

For example, one

needs : If f ; X — ^ Μ a map of a polyhedron X .into a manifold, and Y (^M, the approximation g should be such that not only dim (g(X) f^Y) is minimal, but also should have S^XgJ intersect Y »

minimally e.g. if 2x + y

2n, S(g) should not intersect Y at

all. The above procedure does not seem to give such results. If for example we know that Y can be moved by an isotopy of Μ to make its intersections minimal with some subpolyhedra of M, then these delicate theorems can be proved. This is true in the case of manifolds, and we refer to Zeeman's notes for all those theorems.

115

Chapter VI Regular Neipjhbourhoods The theory of regular neighbourhoods in due to J.H.C. Whitehead, and it has proved to be a very important tool in the study of piecewise linear manifolds.

Some of the important features of regular

neighbourhoods, which have proved to be useful in practice can be stated roughly as follows:

^

(1) a second derived neighbourhood is regular

(2) equivalence of two

regular neighbourhoods of the same polyhedron

(3) a regular neighbour-

hood collapses to the polyhedron to which it is regular neighbo\irhood .(4) a regular neighbourhood can be characterised in terms of collapsing. Whitehead's theory as well as its improvement by Zeeman are stated only for manifolds.

Here we try to obtain a workable theory of regular

neighbourhoods in arbitrary polyhedra; our point of view was suggested by M. Cohen, If X is a subpolyhedron of a polyhedron K, we define a regular neighbourhood of X in Ρ to be any subpolyhedron of Κ which is the image of a second derived neighbourhood of X, under a polyhedral equivalence of Κ which is fixed on X. It turns out that this is a polyhedral invariant, and any two regi£Lar neighbourhoods of X in Κ are equivalent by an isotopy which fixed both X and the complement of a common neighbourhood of the two regular neighbourhoods.

To secure

(4) above, we introduce "homogeneous collapsing". Applications to manifolds are scattered over the chapter. These and similar theorems

116

are due especially to Newman, Alexander, Whitehead and Zeeman. 6.1. Isotopy. Let X be a polyhedron and I the standard 1-cell. 6.1.1. Definition.

An isotopy of X in itself is a polyhedral self-

equivalence of X ^ I , which preserves the I-coordinate. That is, if h is the polyhedral equivalence of X 'XI, writing h(x,t) = ^h^ (x,t), h^ (x,t)), we have h2 (x,t) = t. The map of X into itself which takes χ to h^ (x,t) is a polyhedral equivalence of X and we denote this by h . Thus we can write h as h(x,t) = (h^ (x), t). We usually say that

'h is an isotopy between hQ and h^ ' , or 'h^

is isotopic to h^' or

'h is an isotopy from h^ to h^ '. The

composition (as functions) of two isotopics is again an isotopy, and the composition of two functions isotopic to identity is again isotopic to identity, Now we describe a way of constructing isotopies, which is particularly useful in the theory of regiilar neighbourhoods. 6.1.2. Proposition. Let X be the cone on A. Let f : X — ^ X be a polyhedral equivalence, such that f A = id^. Then there is an isotopy h : X X I and

> X .)ς I, such that h \ ( X X O ) y A X I = identity

h^ = f.

Proof; Let X be the cone on A with vertex v, the interval I =[l,o3 is the cone on 1 with vertex 0. Therefore by 4.3.19, X X I is the cone on X X 1 U A Λ I with vertex (v,0). Define

117

by h (x, 1) = (f{x), 1) for X ζ: X h (a, t) = (a, t)

for a t A^ t ^ I. ^

•

Since h | A = id^, h is well defined and is clearly a polyhedral equivalence. . We have h defined on the base of the cone; we extend it radially, by mapping (v, O) to and Identity on (v, 0),

(v, O),, that is we take the join of h

Calling this extension also h, we see that

h is a polyhedral equivalence and is the identity on (A χ I)

(v, O).

Since X X O U A K i C ( A X I ) ^^ (v, O) , h is identity on X y. 0 U A X I, To show that h preserves the I-coordinate, it is enough to check on (X Κ l)

(v., O), and this can be seen for example'

by observing that the t(x, 1) + (l-t) (v,0) of X )( I with reference to the conical representation is the same as the point (t χ + (l-t) v,t) of X X I with reference to the product representation, and writing down the maps,

Q

If h is an isotopy of X in itself j A h A

X, and if

= Id(A Κ l) as in the above case, we say that h leaves A

fixed. And some times, if h is an isotopy between Id^ and h., Λ

we will just say that

I

'h is an isotopy of X', and then an arbitrary

isotopy will be referred to as 'an isotopy of X in itself.

Probably

this is not strictly adhered to in what follows; perhaps.lt will be clear from the context, which is which. From the above proposition, the following well known theorem of Alexander can be deduced!

118

6.1.3. Corollary.

A polyhedral automorphism of an n-cell which is the

identity on the boundary, is isotopic to the identity by an isotopy leaving the boundary fixed,

D

It should be remarked that we are dealing with I-isotopics and these can be generalised as follows: 6. l./t. Definition.

Let J be the cone on Κ with vertex 0, A

J-isotopy of X is a polyhedral equivalence of X X J

which preserves

the J-coordinate. The isotopy is said to be between the map X ><. 0 - — ^ X ^ 0 and the map Χ Υ Κ — ^ X >> Κ, both induced by the equivalence of

XXJ.

And we can prove as above: 6.1.5. Proposition. Let X be the cone on A, and let f : X )<> Κ — ^ X

Κ be a polyhedral equivalence preserving the K-co-

ordinate and such that

f A X Κ = Id. . Then f is isotopic to AK Κ ^

the identity map of X by a J-isotopy h ; X >C J — ^

X

that on A "A J and X X 0, h is the identity map.

•

J, such

This is in particular applicable when J is an n-cell. 6.2.

Centerings, Isotopies and Neighbourhoods of Subpolyhedra. Let ^

be a regular presentation of a polyhedron P, and

let Y , 0 be two centerings of ^

Vjc

Gc

. Then obviously the correspondence

,

c f(p

gives a simplicial isomorphism of d( φ , V^ ) and d(

, θ), which

gives a polyhedral equivalence of P. We denote this by ^QJ» (coming from the map and f Yj

Tk C ' Q 0 fQ

0 C). Clearly f ^ = (f ) • η, 0 θ , rj ΐ ^ where V-j , 0, are three

119

centerings of 6.2.1. Proposition.

The map f

described above is isotopic to

the identity through an isotopy h : P ) < . I — ^ ^ ^(p >

Ρ χ ΐ ί such that if for

and θ are the same on C and all D ^ (p with D < C

then h C X I is identity.^ Proof; First, let us consider the case when Y| and β a single cell A. Then ^ Q St(ηA, d

((p,R]

aj^d f ^^ η

differ only on

is identity except on

ΘΑ, d ((p, θ

))| = |st(

This is a cone,

is identity on its base; then by 6.1.2.· We obtain an

isotopy of 1 St( V^A, d ((p , V] )) , which fixes the base. Hence it will patch up with the identity isotopy of Κ - |(St(V|A, d( φ , The general ^

^

))|.

is the composition of finitely many

of these special cases, and we just compose the isotopies obtained as above in the special cases. For isotopies constructed this way, the second assertion is obvious. . D Let X be a subpolyhedron of a polyhedron P, and let ^ be a simplicial presentation of Ρ

containing a full subpresentation

TO) covering X. We have defined Νφ( Χ») (in 3.1·) as the full subpresentation of d ^

, whose vertices are Y^ C for C

C Π X / 0. This of course depends on a centering Y\ of ^

with , and to

make this explicit we denote it by is usually called a 'second derived neighbourhood of X'. We know that (XJ , V^ )

is a neighbourhood of X, and that X is a deformation

retract of ^ Ν ^ ( Xj , ^ ) | (see 3.1). Our next aim is to show that any two second derived neighbourhoods of X in Ρ are equivalent by

120

an isotopy of Ρ leaving X, and a complement of a neighbourhood of both fixed. We go through a few preliminaries first. Ex. 6.2.2. With the same notation as above. Let V^ and ^ centerings of ^ C m

such that for every C

be two

> with

^ 0, η C = θ C. Then ί Ν φ ( ? ς , η )( =

(Λ.

) .

/"Hint! This can be seen for example by taking subdivisions of which are almost the same as d((p ,

) and d( (p , θ ), but leave

unalteredJ7. 6.2.3. Proposition. two centerings of φ

With X, P,}Cj , (p as above, let , and XJ

and 6 be

the union of all elements of ^

, whose

closure intersects X. Then there is an isotopy h of Ρ fixed on X and P - U

, such that h^ ( | Ν φ ( X,, Vj ) |) = j N(p ( Χ , 6 ) |

Proof; We first observe that P - U is a full subpresentation (J\. of ^

is a subpolyhedron of Ρ and there which covers P - U , namely, C

if and only if G Π X = 0. By 6.2.2 we can change Yj and and Q l without altering

Νρ (

may assume that V| and θ h

; Y| )

and

are the same on

of proposition 6.2.1 with the new f

desired properties,

.

on

Ql. ^

© ) . So we and (FL . The isotopy

^^ in the hypothesis has the

θ,η

D

With X, P, ")(Q Ί (P

as above, let CD t Ρ — ^

"ο, ϊ"

be

map given by: if ν is a ^-vertex φ ί ν ) = 0, if ν is a (φ _ X, )vertex- Cip(v) = 1, and CD is linear on the closures of ^ Then

cp~\o) = X, since ^

( p - %^ , then

X

is full in (p i If

0, if and only if

-simplexes.

is a simplex of

φ ( 1Γ) = (0, 1). If

121

^ s a simplex of (Jt ( CFl- as in the proof of proposition 2.3) then φ ( 6") = I· Roughly, the map <) ignores the parts of Ρ away from X

and focusses its attention on a neighbourhood of X. We vd.ll use this

map often. 6.2. U. Proposition.

With the above hypotheses, if 0
then there is an isotopy h of P, taking

φ

0, then Cp (

and O V . Let

) ^ "y , and choose

both in and out of

Ρ )

Let

with

arbitrarily on be the sub-

(To' I = \ )(>|= X. Obviously ( N^j ( X

[,Ο,"/^ ), and if

centerings

is a simplex of

(p' denote d((p,^).

presentation with φ

fixed.

Qo, Γ] described above. Choose a

centering• of (pas follows: if C Π X

^ is any simplex of

> then

^ and

of

(p

then

arbitrarily otherwise.

'CyC U

^ ( [_0, (i ) onto

) and leaving X and Ρ Proof: Let (D be the map: Ρ

<

φ

φ

)|=

with vertices

i γ ) = {0,y ). Now choose two

such that if

η C) = p> and

ρ

(p

and

Cf) ( β C) = c^ and

Then clearly

Ν

(ρ·

=


and Ν

ρ'

)(=

cp~uco,
We apply 6.2.3 now, and 'AjL of 6.2.3 in this case happens to be

9 ( Co, 7)) . D 6.2.5. Proposition. Let ^ subpresentation of Q

,

be a simplicial presentation, ^

φ = P,

and Ν = Ν φ ( ^ , Vj ) . Let ^

a full

^ = X, Vj a centering of (p ; be a siniplicial refinement of

122

(p ί

) T^ith

t e r i n g of (3^ and

"the s u b p r e s e n t a t i o n covering n'

θ

Ν'; and l e a v i n g

Β

a

cen-

.. Finally., let Λ λ be a .

n e i g h b o u r h o o d o f N . Then there i s a n i s o t o p y onto

X ; let

h o f P , takifig Ν

Ρ - λ Λ . a n d X fixed.

Remark' Note, that if

were somewhat lai;'ge, or if there were no

u

in the statement, then the proposition is an immediate consequence of 6.2.3 and 6.2.4. •t Proof; Ive first replace the centering T| by a centering followsi Let ^ : Ρ Ο

η

as

Qo, 1J be the usual function given by,

(^-vertex) - 0, Cp ( ^ _ ^ )-vertex = 1, and φ is linear on

the closures of

-simplexes. Choose

Γ|' such that if p is a

simplex of (J) with φ ( f ) = (0, l), then (|) ( Vj' ^ ) = h, ' is a polyhedral equivalence carrjring N,(P t. -1 onto Ν (p( % » η ' )| = φ ' ( a ] )· A c t u a l l y f η '"^η i s isotopic to the identity, but we will need only that it is a polyhedral equivalence. Let f I r| ("U.) = XL . As XiL' is a neighbour-1 hood of [_0, s3 we can find a Ύ > i such that

<) " ^ ( [ Ο , Ύ ^ ) C \Λ!·

^ince f ·γ| ι ^ γ^

to d( (p, V| ) and d( (p, rj') f y^ ι ^ V^ this

is refinement of

carries (3^onto a refinement of d( and similarly f η , ^ η (Λ^ ) by

the centering of

and

and

is simplicial with reference,

induced ^ from Q be

d( φ, η

T^ ').

Let us call

'. |Λ^· ( = X. Let We have,

123

Now choose another centering Let dC

< dC <

in X, then

^^

as follows!

k) be such that if ν is a vertex of

φ (v) >

. 0 ^ is chosen so that if

vertices in and out of X, then

Ν

of

Cp ( θ^

=

h

)=

taking

6" ζ:

. Then clearly

Ύ )) fixed, with

s}) onto

Ji] ). By 6.2.3 there is

an isotopy h' leaving X and complement of •with h /

taking

has

By 6.2.4, there is an isotopy

of P, leaving X and complement of

h^

(5^' not

φ ~ ((O, cJCJ ) -

fixed. onto

Ν Let

f

isotopy of Ρ in itself given by

i'Yj'^if] (p. t) = (f-Yj is the required isotopy. carries Ν

(p), t) ρ First

are alscr fixed on X. As

^^

(Pa

and (Z) 2

f y^', Yj

. They ·

carries X onto X, IJLonto Λχ'

.' Q

be two simplicial presentation of P, containing

2 and N^ =

= X. Let

^ and Q

^ ^

and 1

β

Ν

"XL ^ neighbourhood of Ρ

η^^ο hjoh^ofy^' y|

Corollary. Let X be a subpolyhedron of a polyhedron P. Let

full subpresentations

Μ,:

ο h'ο h ο f γ^,

), h and h' are fixed on P -

also fixes X and Ρ -

6.2.6.

=

Then

onto N'. Secondly since P - 'XjI.'ρ - Cp~^([0, g}) and

Ρ - \Λ!.ζ Ρ - φ"^([θ,

g

p.

leaving X and Ρ -

respectively with 2

be centering-of

Ν N^ in P.

(p

and and

Then there is an isotopy of

fixed and taking N^

onto N^.

124

Propf; Take a cOTnmon subdivision

of dC ^ ^, Θ

and

and apply 6.2.5 twice. 6.3. Definition of "Regular Neighbourhood". Let X be a subpolyhedron of a polyhedron P. 6.3.1.

Definition.

A subpolyhedron Ν is said to be regular neighbour-

hood of X in Ρ if there is a polyhedral equivalence h of Ρ on itself, leaving X fixed, such that h(N) is a second derived neighbourhood of X. More precisely Ν is a regular neighbourhood of X if and only if i) there is a simplicial presentation (p of Ρ · with a full subpresentation

covering

X

and a centering T^ of (p ; and ii) a polyhedral equivalence 'h of Ρ X

fixed on

such that h(N) = J N'^ ( X>, Γ) )|,

Regular neighbourhoods -do exist and if Ν is a regular neighbourhood of X in P, then Ν is a neighbourhood of X in P. 6.3.2. Proposition.

If N^ and lil^

two regular neighbourhoods of _

X in Ρ and \X, a neighbourhood of N^UN^ an isotopy h of Ρ taking N^

in P, then there exists

onto N^ and leaving X and Ρ - Ίλ,

fixed. Proof J Let

(p^, %

Y)

i

1 > 2, be such that

= Ν.,

Let

i

be a subdivision of d( (p^,

simplicial vdth reference to

i and let

= 1,

2.

such that h^ is ^ be the subpresentation

125

of

covering X. Let "

Ί

Q^

be a centering of I

1

say /"Note that h

is fixed on X_7.

By 6.2,5, there is an isotopy f, fixed on X and Ρ - h^ \ ΛΛ) vdth f, taking ((N(p ,( f h^ . /V Ch^

η

onto I N^^C^j^, e

.

^hen

is an isotopy of Ρ fixed on X and Ρ - IjL and ~^ I , l~sl = h^ f^ h^

takes N^

onto N^

(where h^ is the isotopy

of Ρ in itself given by h^ (p, t) = (h^ (p), t)). Working similarly with φ

we obtain f' with =

f'

fixed on X and Ρ - "U. and

taking N^ onto N^. Now nJ

genuine second derived neighbourhoods, and

is a neighbourhood of

Nj U N^ . Hence by 6.2.6 there is an isotopy g of Ρ and Ρ

fixed on X

, with g^ (N^') = N^'. (.h2 f h^ )

6.3.3»

and N^ are

Proposition.

g (h^ f h^

If f : Ρ — ^

) is the required isotopy.

U

P' is a polyhedral equivalence

and Ν a regular neighbourhood of X in P, then f(M) is a regular neighbourhood of f(X) in p'. Proof; Let

^^ , (p ' be a simplicial presentations of Ρ and P'

with reference to which f is simplicial^ (P , f( ^ ) = that X

^

be a centering of

the induced centering on (p ' ·

contain full subpresentations

and x'; (by going to subdivisions

,

can assume covering

necessary),

f (|N(P ( % , η )1) = |Ν (p, ( X'^ n ' M · By definition^ there is a polyhedral equivalence jj of Ρ fixed on X such that

126

p(N) =

()ς , η ) . Let ρ' ^he the polyhedral equivalence of

P' given by f.p.R'.

Then (f. ρ · f" J(N') = (f . p) (N) =

) 1 ) = | Ν φ > ( χ ' , η ' ) | , and i f therefore

f . ρ.

χ'6 f(X),

) = f. ρ (f"Vx')) = f ·

) tX,

) = x'.

Q

6.3.4. Notation. If A is a subset of a polyhedron P, we will denote by int ρ Μ and bd ^ Ν the interior and the boundary of Ν in the (unique) topology of P. Ex. 6.3.6. If Ν is a regular neighbourhood of X in P, and Β = bd ρ N, then Χ ζ ^ Ν - Β .

Q

Ex. 6.3.7. Let X C n C ^ Q ^ P

be polyhedra, vdth Ν ζ"int ρ Q. Then

Ν . is a regular neighbourhood of X in Q if and only if Ν is a regular neighbourhood of X in P. Β Ex. 6.3.8. Let Χζ_Ρ

be polyhedra.

If A is any subpolyhedron of

P, let A' denote the polyhedron A - intp X. Then Ν is a regular neighbourhood of X in Ρ if and only if N' is a regular neighbourhood of X' in P'. α Ex. 6.3.9. Let A be any polyhedron, and I the standard 1-cell. Let 0 < eC < p) < y

<

1 be three numbers. Then, A A^O,

a regular neighbourhood of A in A y. I, and A neighbourhood of A

is ^ regular

^ in A X. I.

6.3.10. Notation and proposition. If (p is any simplicial presentation and ^ vertices of φ

, we denote by (p^

whose set of vertices is ^ δφ^"^ )

£

.

) (when

(p

any set of

the maximal-sWpresentation of(P is full in φ

. We write

is understood) for

U ^((flSj

.

127

This is of course vdth reference to some centering Σ ) is a regular neighbourhood

of ί (P

^ ' ^^

^^

a set consisting of single vertex x, we have the some what confusing situation

{χ

J where χ denotes the 0-simplex with

vertex x. In this case we wi.ll write

Let and

^ =

Given a centering of

). JYt' •= d((D ,

Si

of

as η

). If ^

is the

consisting of the centers of elements of

(p^ = ^

and'Jin:' =

for

(p' " ^^ (p' ^ ^

^ ) (still calling η

set of vertices of , then

^

be a subpresentation of a simplicial presentation

be a centering of φ ) . Let =

^

^

=

is full in

(p

), we define

C) j , for any G t (p u [ C-r. C iz'^lf^^is a regular neighbourhood

We use the same notation

(

even when

is not

subpresentation, but a subset of ^^ . These are Used in the last part of the chapter. As the particular centerings are not so important, we ignore them from the terminology whenever possible. 6.4. Collaring. To study regular neighbourhoods in more detail we need a few facts about collarings. This section is devoted to proving these. 6.4.1. Definition. Let A be a subpolyhedron of a polyhedron P. A is said to be collared in P, if there is a polyhedral embedding

128

h of A )<, jo, Q in P, such that i) h (a, 0)

c for all a e A

ii) the image o·*" h is a neighbourhood of A in P. And the image of h is said to be a collar of A. 6.4.2.

Definition.

Let Ν be a subpolyhedron of a polyhedron Ρ

and let Β = Bd ρ N. Ν is said to be bicollared in Ρ if and only if i) Β is collared in Ν ii) Β is collared in Ρ - Ν, 6.2^.

Clearly this is equivalent to saying that there is a poly-

hedral embedding h of Β

+

in P. such that

i) h (b, 0) = b , b f Β ii) h {B

(0, + Q ) C ^ - ^

iii) h

6})(y

iv) the image of h is a neighbourhood of Β in P. 6.4.3-

Proposition.

If Ν is a regular neighbourhood of X in P,

then Ν is bicollared in P. Proof; It is enough to prove this for some convenient regular neighboai>· I

hood of X. Let ^ full subpresentation

be a simplicial'presentation of Ρ containing a covering X and let ^ t Ρ — ^

the usual map. We take Ν to be Bd ρ Ν = φ

^

be

^((Oj έ^) clearly

(g). Let us denote this by B. We can now show that is polyhedrally equivalent to Β Κ '-1, -+ Q

following way:

1

in the

129

Β has a regular presentation Q^ consisting of· all nonempty sets

Cf\(p\s)

for Γ

Likewise


^ ) has a polyhedral presentation,

consisting of all non-empty sets of the following sorts: Γ α φ ik) Γ α φ " ^ α, έ) (ΓΛφ^ (έ) (έ, i) Γ (-S

ί· of

-1j

for

α φ (I)

, (Oj + Ο,

{+

is a regular presentation

· There is an obvious combinatorial isomorphism between

and (j3) X ^

/which determines, is a appropriate centerings, a

polyhedral equivalence between Β )<,

1, + i

and

This shows that Ν is bicollared in P. Ex. 6ΛΛ.

5 ) C P· D

If A is collared in P, then any regular neighbourhood

of A in Ρ is a collar of A. /"Hint! Use 6.3.7 and

6.3.9_7.

Q

Thus if Ν is a regular neighbourhood of X in Ρ and Β = Bd ρ N, a regular neighbourhood of Β in Ρ - Ν is a collar of B. Ex. 6.U. 5.

If N^ is a regular neighbourhood of X in Ρ and

Ν2 is a regular neighbourhood of N^ in P, then is collar of Β = Bd „ Ν . 1 Ρ 1 /"Hint! Use 6.3.8 and S.k.hJ.

Ο

-

N^- Intp N,

130

Ex. 6.4.6. If N^ . is a regular neighbourhood of X in P, and N^ is a regular neighbourhood of N^ neighbourhood of X in P. Ex. 6.4.7.

(i) If M^

of X in Ρ with N^ Q

in P, then N^ is a regular

Ο

and N^ ar'e two regular neighbourhoods ^"^ρ

N^ - N^

is collar over

B, = Bd ρ N.,. (ii) N^ is a regular neighbourhood of N^. /"Hint: Take two regular neighbourhoods N^'j N^' of X, such that Ν ' - Ν ' is a collar and try to push Ν onto N^' and Ν •c 1 2 2 1 N/J.

onto

D The following remark will be useful later·:

Ex.. 6.4.8. Let Ν be bicollared in Ρ

and n' be a regvilar

neighbourhood of Ν in P. Then there is an isotopy of Ρ taking Ν

onto N'. If X C II^T Ρ Ν, this isotopy can be chosen so as to

fix X.

•

6.4.9. Definition. A pair (B, C) of polyhedra with Β ^ C, is said to be a cone pair if there is a polyhedral equivalence of Β onto a cone on C, which maps

C onto C.

Clearly in such a case we can ass\ime that the map on C is the identity.

And if (B^ C) is a cone pair, C is collared .

in B. 6.4.10.

Definition. Let AQ Ρ be polyhedra, and

'a' a point of

A. Then a pair (L , L ) is said to be a link of a in (P^ A) if Jr Λ 1) L.CI..

131

ii) L^ is a link of - a in A iii) Lp is a link of a in P. If (Lp', L,') is another link of a in (P, A), then the standard mistake Lp

^ Lp' takes L^ onto L^', and there-

fore there is a polyhedral equivalence L^ taking L — ^ L ι Ρ Ρ Λ ' A We shall briefly term this an equivalence of pairs (Lp , L ^ ) — ( L p , , ^^i^i^·

that, upto this equivalence, the link

of a in (P, A) is unique. 6.4.IT. Definition. Let

Ρ be polyhedra.

A is said to be

locally collared in Ρ if the link of a in (P, A) is a cone pair for every point a ζ A. Clearly k

0 is locally collared in A

Q , and

therefore if A is collared in P, it is locally collared. We will show presently that the converse is also.true. 6.4.12. Definition. Let Β be a subpolyhedron of A ^ Β

is said to be cross section if the projection A )<, [o, l]] — ^ A,

when restricted to Β is

1-1

and onto and so is a polyhedral

equivalence Β ii A.

6.4.13. Proposition.

Let Β be a cross-section

of A

Γ] .

contained in Α χ ( θ , I). Then there is a polyhedral equivalence h :.A:?<{o,i3

^ A A ^ , Q , leaving A χ ο

fixed, and taking Β onto

and Α χ ί

point wise

and such that

h (a)C [p, i]) = a X ^ , ij for all a ^ A. Remark; There is an obvious homeomorphism with these properties, but it is" not polyhedral.

132

Proof; Let ρ : Α Χ

^ A be the first projection. Trian-

gulate the polyhedral equivalence ρ Β s Β

A. Let (j^ and (Jl

be the simplioial pr'esentations of Β and A. l^J ^

ti]

^ simplioial

presentation of [o, β . Consider the centering T| of (JL χ ^ given by "n ( δ" )<. Τ

) ^ (barycenter of ζ" , barycenter of

We will define another regular presentation ^

),

of A "A I

as follows: For each ζ" ^ (Jt > Ρ \ ^ ) is the union* of the following five cells: 0 , p"^ ( Γ ) Π B,

Fr· '

^ σ- '

where ^ ^ is the region between C ^ 0 and ρ \ ) Γ\ Β - 1 Ο is the region between ρ (β") Γ\ Β and 1 · (Note that Ρ

( Γ )Π Β ^ ( Β J.

We take ^

to be the set of all these

cells as ^

varies over

a centering θ of first

, Choose

, such that the

co-ordinate of each of the five

cells above is the barycenter of Now there is an obvious combinatorial isomorphism 5

^^

choose

the centerings described we obtain

.

133

h :Λ

I—^

A

I which is simplicial relative to d(

d( O ^ X ^ ί ^ ), and has the desired properties.U.13· In this situation, define (a, t)

, Q) and

Π

AB,

a ^Λ, t ζ: I, 3 b eB, b = (a, s) , t 4

i.e. this is all the stuff of the left of 3. Then h takes /^B onto A

1

, Β onto

A^^.

In particular Β is collared

in ?\b. 6.4.14. vertex

"Spindle Maps". Let L Q A,

with the cone on L and

'a' contained in A. . Call the- cone S, • Suppose

open in A.

(This is the case when

a is a vertex of a simplicial

presentation (Jl of A, and L = ^Lk (a, (JL) Let p

S - L is

and S = j s t (a:, C l )

= I — ^ I be an imbedding with

β (1) = 1. In

this situation wc define the "spindle map". m ( p , L , a) : A X I — ^ A X I thus: on L ίί- a Vv l|,

it is the join of the identity map .on

L

•a» with the map

(a, t)

^ (a, |^(t)) of a Κ I.

On the rest of

A /vl it is the identity map. A spindle map m

is an embedding, and commutes with the

projection on A. If Β is a cross section of A')<^I^which does not intersect A

then m (B) has these properties also.

6.4.15. Proposition. Let kQ_?

be polyhedra.

If A is locally

collared in P, then A is collared in P. Proof: In Ρ

Γ

, consider the subpolyhedron

Q = P X 0 U a x [ 0 ,

i3 . We identify

Ρ with

Ρ Κ 0 (^Q.

Let (p

be a simplicial presentation of P, in which a subpresentation (JL

134

covers A, Consider a vertex Lk (a, Ο )

'a' of Q\ ; let L^ and Lp denote

Lk (a,(p ) \ , Then (L , L ) is a link of a ' L Λ in (P, A) and there is a polyhedral equivalence"^ : L — ^ L. ·$(• 15 for some r, taking L onto L^. We can make identity on L A and

by composing vd.th (7 L^)

id^

. And so we suppose

^ L^

is identity. We can suppose that ν is so situated larger vector space) that

(for example in a

-it- ν and L * (a, l) intersect only

A in L^^. Thus we have via "y and the identity on L ^f. (a, l) , a polyhedral equivalence of L^ = L VJ L ^ _ Q Ρ A

n.

(a, 1) with L A

Ε

where Ε ·=νν, (a, ' Η j which is identity on L^ * (a/l). Now L^ -if a is a star of

'a' in Q, and via this p,e. is poly-

hedrally ec[uivalent to L^ if- Ε -}(· a.

/

We can find a polyhedral equivalence ^

of Ε ·)(• a (which is equivalent

to a closed

1-cell) leaving ν

and (a, 1) fixed and taking (a, 0) to (a, 5).

Such a

obviously takes a)<.' 0, h, 1 onto aX Take the join of

^

and the identity map L^, this

gives a polyhedral equivalence of

-if a which is the identity

on Lj^. Hence this can bp extended to a polyhedral equivalence of Q by identity outside L^ •«· a. Let us call this equivalence of

135

^^ Pa·

l^a

C A

A Χ I is a spindle map.

and

Now take the composition h in any order of all such with A=

'a' running over all the vertices of

0 into a cross section h(A)

not intersect A

)

or

. This maps

β of A >^[^0,

which does

A γ. 0. Finally h(P) Γ\ A χ I =

Β is collared in ^ B, and so in h(P). Then, taking h ^ we see that A is collared in P.

Π

6.4.16. Corollary; If Μ is a P.L. Manifold with boundary M, then

Μ is collared in M.

•

Now, an application of the corollary: 6.4.17r

Proposition. If h is an isotopy of ^ M, then h

extends to an isotopy Η of M. Proof; Let

^ : I X I — ^ I be the map given by

β (s, t) = Max (t - s, 0). This is polyhedral, e.g. the diagram shows the triangulations and the images of the vertices. ^ is, 0) ^ 0, Define Η = ( ^ Μ ) ^ Ι ) Χ Ι - ^ by Η ((χ, s), t) ^ ((h

t) = 0, 0 (0, t) = t. (0ΜΑΐ)Χΐ s), t).

This is polyhedral. (x), s), 0) = ((x, s), 0), (s,0) since h^ = Id. Hence H^ = Id of ^ M K I.

Η ((χ, s), 0) ^ = ((h

Η ((χ, 0), t) = ((h^ ^ (x), 0), t) = ((h (x), 0), t). Ρ (0,t) t Thus Η extends the isotopy ^ Μ Κ 0 given by h (identifying ^ Μ and

M)i. O).

136

And Η ((χ, 1), t) ^ ((h

^^ (χ), 1), t)

= ((χ, 1), t) since Hence Η

(Jb (ΐ, t) = 0.

Μ A 1 is identity. Hencfe the isotopy h of

Β Μ

extends to an isotopy Η of any collar so that at the upper end of the collar it is identity again, and therefore it can be-extended inside. Thus h extends to an isotopy of M.

Π

6.5. Absolute Regular Neighbourhoods and some Newmanish Theorems. 6.5. t. Definition.

A pair of a polyhedra (p, A) is said to be an

absolute regular neighbourhood of a polyhedron X, if i) X Q Ρ - A ii) Ρ

0 is a regular neighbourhood of

X Κ 0 in Ρ >s 0 U A

0 C Ρ X [?' Κ ·

Hence A is collared in P. I

·

Probably, it will be more natural,to consider X, Ρ and t A in an ambient polyhedron Μ in which Ρ is a neighbourhood of X

as in links and stars. But, after the definition of regular

,

neighbourhood, absolute regular neighbourhood is just a convenient name to use in some tricky situations. Ex. 6.5.2. If (p, A) Xs an absolute regiilar neighbourhood of .X and if h s Ρ •—^ p' is a polyhedral equivalence, then (P', h A) is an absolute regular neighbourhood of h X.

D

Ex. 6. 5· 3· If Ν is a regular neighbourhood of X ,in P, and Β = Bd ^ N, then (N, B) is an absolute regular neighbourhood of • 1

X.

Q

.

137

Ex. 6.5.4. Let F C Q} and suppose that ' (P, A) is an absolute reg^ilar neighbourhood of X, and Ρ - A is open in Q, and A is locally collared in Q - (P -A). hood of X in Q.

Then Γ is a regular neighbour-

•

Ex. 6.5.5. Let C (A) be the cone on A with vertex v. Then (C (A), A) is an absolute regular neighbourhood of v. In particular if D is an n-cell,. (D, ^ D) is an absolute regular neighbourhood of any point χ t D - ^ D. 6.5.6. Theorem. If D is an n-cell, Μ D

0

a PL-raanifold,

Int M, then D is a regular neighbourhood of any χ 6 D -

D

in M.. 6.5.7. Corollary.- If D is an n-cell in an n-sphere S, then S - D is an n-cell.. Proof of the theorem; The proof of the theorem is by induction on the dimension of M; we assume the,theorem as well as the corollary for η - 1. i) First we must show that

D-

D is open in M.

If we look at the links, this would follow if we know that a polyhedral imbedding of an (n-l)-sphere in an (n-l)-sphere is necessarily onto. And this can be easily seen by looking at the links again and induction, (see 4.4 in particular 4.4,. 14 and 4,4. 17(a)).. ii) If we know that "d D is collared in Μ - Int D (it is collared in D), we are through by 6.5.4. For this, it is enough to show that

'b D is locally collared in M- int D, Consider a

link of a in M, say

such that a link of

'a' in D is

138

an (n-O-cell D^"^ C S^ ^ , with d""^ Π a

*

a,

D - ^ d" \

a

It is

, 3 .

clearly possible to choose such links (see λΛ. 1? (b)). ~ Now, a link of

'a' in M ^ int D is a

- (d"^"^ a

). a

/. \ ^ -V, n-1 1 "S in U ; 2a - Ο is open in Sa and therefore the link ^ Da of a in Μ - int D is But by the corollary to the o.

cL

theorem in the (n-l)-case, this is an (n-l)-cell, say and it meets D in is equivalent to

=

(C ( ^ Λ

• And (

^Λ"^'^)·

^

Therefore

)

^ D is

locally collared in Μ - int D and we are through. Proof of the corollary assuming the theorem; Represent s"^, a standard n-sphere as a suspension of s'^ ^, a standard and observe that the lower hemisphere

(n-1 )-sphere,

(say D ) is a regular

neighbourhood of the south pole, say s. Let f be a polyhedral equivalence of S to

taking a point χ t D -

D to the south

pole s. By the theorem D is a regular neighbourhood of x, therefore f(D) is a regular neighbourhood of the s in s". By. 6.3.2 there is a polyhedral equivalence ρ of S'^ such that p(D ) = f(D). Therefore f (S - D) = f (S) - f (D) = S"^ - ρ (D^) = ρ (S") - ρ (D^) =p(S

- D s) = P ( D η) , where Dη denotes the upper hemisphere, -1

Therefore p~ . f (S - D) - D η

or S _ D is a n-cell.

Q

Ex. 6.5.S. Corollary. If Μ is a PL n-msnifold and D^, D^ are two n-cells contained in the interior of the same component of M, then there is an isotopy h of the identity map of M, such that h(D^) ^ D^.

Q

139

We usually express this by sayine that "any two n-cells in the interior of the same component of Μ are equivalent" or that they are "eq-uivalent by an isotopy of M". If Μ is a PL n-manifold, "^M its boundary, then by 6.5.8, any two

(n-l)-cells in the same component of

lent by an isotopy of d M.

^M

are equiva-

Since this is actually an isotopy of the

identity, by 6.Zt. 1? we can extend it to M. Thus 6.5.9. Proposition. Μ

Any two

(n-l)-cells in the same component of

are equivalent by an isotopy of M. • This immediately gives

Ex. 6.5.10.

If D is an n-cell and Δ

then (D, Δ

) is a cone pair (That is, there is a polyhedral

equivalence of

(D, Δ ) and

(c{

an (n-1 )-cell in ^ D,

), Δ ). And we have seen such

a polyhedral equivalence can be assumed to be identity on Δ

). Q

This can also be formulated as: Ex. 6. 5. 10^

If Λ ^ is an (n-l)-cell in the boundary of D^, an

n-cell, i - 1, 2, any polyhedral equivalence be extended to a polyhedral equivalence

D^ — ^

Λ ^

^ Δ 2

Dg. D

Also from 6.5.9, it is easy to deduce if Δ

is any

(n-l)-cell in ^ M, then there is at least one n-cell D in Μ such that D

= AC^i^'From this follows the useful proposition:

Ex. 6.5.11.

If Μ is a PL n-manifold and D an n-cell with

ΜΟ D=

D - an (n-l)-cell, then M O D

is polyhedrally

equivalent to M. Moreover, the polyhedral equivalence can be chosen to be identity outside any given neighbourhood of M U D

in M. -Q

140

The methods of the proof of the theorem 6.5.6, can be used to prove the following two propositions, which somewhat clarify the nature of regular neighbourhoods in manifolds: Ex. 6.5.12. Let Μ be a PL-manifold, ^ Μ its boundary (possibly 0), and Ν a regular neighbourhood of X in M. Then a) Ν is a PL-manifold with (non-empty) boundary unless X is a union of components of M. b) If X C Μ - δ Μ, then Ν ^ Μ - b Μ, the interior of Μ. c) If X A "^M / 0, Ν n ^ Μ is a regular neighbourhood of X Pv ^ Μ in

M.

d) In case c), Bd ^^ Ν is an (n-1)-manifold, meeting

^ Μ in an (n-2)-manifold

where N' = Ν A /"Note that int M

Ν and bd M

Ν denote the interior and

boundary of Ν in the topology of M. On the otherhand if Ν is a PL-manifold int Ν

and

"δ" Ν denotes the sets of points of Ν

whose links are spheres and cells respectively_7. Hint: Use 4.4.8.

D

Ex. 6.5.13. If Ν is a regular neighbourhood of X in M, a PL-manifold with X Ν

int M, and n' is polyhedrally equivalent to

and located in the interior of a PL-manifold M^ of the same

dimension as M^ then n' is a regular neighbourhood of x' in M^, where x' is the image of X mder the polyhedral equivalence Ν — ^ M'.

Q

141

Ex. 6.5.14. Λ is any polyhedron, and I the standard 1-cell (A yi.1, A y. 1) is an absolute regular neighbourhood of A 0 < cC

1J then

0. If

0, 1 [·) is an absolute regular neigh-

bourhood of A X ^

.

D

Ex. 6. 5. 1 T h e union of two n-manifolds intersecting in an (n-l) submanifold of their boundaries is an n-manifold. tX 6.6. Collapsing. 6.6.1.' Definition. Let (p be a regular presentation.

A free edge

of (p is some Ε ^ φ ) such that there-exists one and only one A ^ ^

with Ε < A. We may term A the attaching membrane of the

free edge E. It is clear that A is not in the boundary of any other element of (p ; for if A < B, then Ε <, B. It is easily proved that dim A = 1 + dim E. The set (p - • E, A | is again a regular presentation, and is said to be obtained from ^^ by an elementary collapse at the free edge E. 6.6.2.

Definition.

We say that a polyhedral presentation (p

collapses (combinatorially) to a polyhedral presentation write (p V

ί if there exists a finite sequence of presentations (P^ , ....

and cells Ε

Ε 1

(p

, and

with (P - (P^

and

(P^

d

, Ε. f fp s.t.fp is obtained from k-1 1 ^ vl i Vi

by an elementary collapse at E^ ^.

6.6.3. Proposition. If ^ collapse at E; and if

φ'

is obtained from (p -by an elementary is obtained from |p by bisecting a

]ΐα

cell C by a bisection of space (L; Η is the subpresentation with

H-) and if

ICH'

C (p 0^'·

/"Remark! Recall that, we have been always dealing with Euclidean polyhedra_7. 9·

Proof; If the bisection is trivial there is nothing to prove, so suppose that the bisection is non trivial. Then there are three cases. Case (i) edge of thus

C is neither Ε nor A, In this case, Ε is a free ^ ' with attaching membrane A, and

^

is obtained from

^

^p' - ^E,

by an elementary collapse.

Case (ii) C - E. Define E^ = Η + Π Ε, E2 = Η - fj Ε, F = L Π Ε. Then we have U

^

F

and no other cells of

Thus E^

is a free edge of

is a free edge of

^ A

are greater than F, E^, E^ or A.

^

with attaching membrane

A; F

o' - • E^, A . with attaching membrane E^.

The result of these two elementary collapses is

=

.

Case (iii)

C =A

·

Define A^ = H+ Π k, k^ ^ Hcontains

k, Β = L Π A. Mow ^ A^ 0

A, and therefore either ^ A^

Θ

or b A^ intersects E;

say, ^ A^ Π Ε / 0. Then, (p' being regular, we must have Ε <_ A^; then for dimensional reasons, dim Ε = dim B, we cannot have Ε ^ B hence Ε summary,

H+ ; and so it is impossible to have Ε < A^. In Ε < A^ > Β < A^.

Thus Ε is a free face of is a free face of

(p

(p' - ' E, A^

with attaching membrane

with attaching membrane A^.

The result of these two elementary collapses is I 6.6.4. Proposition. If (p

, and

|P

^.

Q

is obtained from

by a finite sequence of bisections of cells, and presentation of defined? by μ [0^1 ; then

(P

A^ ; Β

is the sub(p' V C ^ ' ·

Proof: The proof is by induction, first, on the number of collapses in

(p V

, and second, on the number of bisections involved.

The inductive step is 6.6.3.

D

6.6.5. Definition. We say that a polyhedron Ρ collapses (geometrically) to a subpolyhedron Q, if there is a regular presentation (p of Ρ with a subpresentation (3^ covering Q, such that (J) collapses combinatorially to (3> . We write Ρ ^ Q .

144

This notion is polyhedrally invariant: 6.6.6. Proposition. equivalence, then X

If Ρ V Qj and

·.

Ρ

f X is Β polyhedral

VOC(Q).

Proof; There are regular presentations φ , (p V Ρ

of Ρ

and Q, vath

combinatorially, and simplicial presentations Λ> J Xa

and X vdth J^ simplicial relative to /Λ and Xa . There is a

regular presentation

(p' of Ρ refining

from (p (also from A

^ and Λ

but we do not need it in this proposition) by

a finite sequence of bisections. Hence if of

(p' covering Q, then

(p' V

=[dC(c) (c

is the subpresentation by 6.6.4. Since ^ is

one-to-one and linear on each element of cL ( φ Ί

j and obtained

^

, the set

^(p'] is a regular presentation of X, which

m I·

I

is combinatorially isomorphic to

(p ' 5 and

cL (

) is a sub-

presentation covering «^(Q), which is combinatorially isomorphic to

Therefore

oC((p')

0^) or X VCC(Q).

6.6.7. Proposition. If Ρ V ^o ' ^nd

I Proof! Let ^ ^ , ^ ^ and (p

^

V P„ , then Ρ

di

Ρ.

I

be presentation of P^ , P^ with φ ^ ^(pg'

(p ^ be presentations of P^ , P^ with (P3

By 1.10.6 there is a regular refinement and subpresentations ^ ^^

Π

of ^ ^ ^ ^ ^ 0 (P 3 0(P 4' ^th (p . ^ =

obtained from (p ^ by a sequence of bisections. Clearly

= ό^3 therefore

and by 6.6.4, ^ ^2· ^

6.6.8. Proposition. then Ν ^ X.

^ V

3

If Ν is a regular neighbourhood of X in P,

145

Proof: By virtue of 6.6.6 and the definition of regular neighbourhood, it is enough to look at any particular M. Let (p be a simplicial presentation of Ρ vdth a full subpresentation let

φ : Ρ —> [_0, Q Let ^

both in ^

covering X; and

be the usual map. Take Ν -

denote all the simplexes of (p having vertices

and ( φ - % ) .

number of elements of ^

We prove Ν ^s/X by induction on the

. If

χ

= 0, then Ν = X, and there is

nothing to do. Hence we can start the induction. LetJft

=X

0

υ

ι ^

Then "Tt. is a regular presentation of N. If of maximal dimension, attaching membrane

^ Π φ ς;"·

principal simplex of φ

Vg)

is an element of

is a free edge of "Tt with

((O,

(Mote that CT is a

i.e. not the face of any other simplex).

After doing the elementary collapse we are left with

-OHC'· Now

(p' is a regular presentation containing the corresponding so, -Tt

^ .

= ^

-

, and

^ . Hence inductively

α

Ex. 6.6.9. Let N' be a neighbourhood of X in P, (all polyhedra). If Ν

V X, then there is a regular neighbourhood

M([_Int ρ n' such that N. 6.7. Homogeneous Collapsing. Let ^

Ν of X in P,

D

be a regular presentation, with E, A ^ffi , Ε < A

and dim A = dim Ε + 1. Recall the definition of ^

φ) E. This is defined.

145^

relative to some centering

of ^

, to be the fiill subpresentation

of d (p whose vertices are 6.7.1.

\® ^^

Definition. Let E, A, ^ ; be as above and

V^ be a

centering of (p . (E, A) is said to be homogenous in (p , if there is a polyhedron X and a polyhedral equivalence a suspension of X, such that

f(

f:

^X

A) = w.

It is easily seen that if this is true for one centering of

, then it is true for any other centering of

"(E, A) is homogeneous in ^

; hence

" is well defined.

6.7.2. Definition. Let % C

subpresentationsof ^

. We say

that "ATT collapses to 'Xa homogeneously (comblnatorially) in (j^ , if there is a finite sequence of subpresentations of (p ,

and pairs of cells (E^, A ^

, E., A. ^ ^ t ·

such that 1) Jit, =

, -^rt =

' 2)

. k is obtained from

i+1

by an elementary i

collapse at E^^ a free edge of "tc . with attaching membrane A^, for i = 1, and

3) (E^, A^) is homogeneous in φ) , for i = 1,.., k-lj

6.7.3. Proposition.

If

(p*is obtained from

cell C by a bisection of space with

obtained from

,

^

by bisecting a

(L; H+ , H-); and if X C ^ ^ CL(p >

by an elementary collapse at a free edge

Ε with attaching membrane A, where and if

k-1

(E^ A) is homogenous in ("b ;

are the subpresentations of

^

covering

146

and

; then

homogeneously in

^

.

Proof; If the bisection in trivial there is nothing to prove. If it is not trivial, there are three cases as in the proof of proposition 6.6^3· Case 1; C is neither Ε nor Α., In this case the only problem is to show that (E, A) is homogeneous in (p

. Let us suppose that

everything is occuring in a vector space V of dim n; and let dim Ε = k. Then there is an orthogonal linear manifold Μ of dimension (n-k), intersecting Ε in a single point

(E) = e, say..

It is fairly easy to verify that in such a situation if

If we now choose centerings of Q

D,. then

and

so that

whenever D fV Μ / 0,. we have the center of D belonging to M, then defining D A M D r^M ^ 0, D and

and

similarly with respect to

,

^

·

, we will have:

are regular presentations of

λ(ρ (Ε)

and

ί\|ρΐ(Ε)| are links of e in

hence polyhedrally equivalent

Γ\ M. Hence both (p Π

and

(by an approximation to the standard

mistake); if we choose a center of A the same in both case, we get a polyhedral equivalence taking hypothesis

/\Π(Ε)

A to

Y^ A.

Finally, by

is equivalent to a suspension with Y^ A as

a pole; and so

Λφ'(Ε)

homogenous in

'

has the same property, and (E, A) is

\hl

Case 2; C = Ε ; we define E^, E^, F as in the proof of 6.6.3· We have to show that

(E^, Λ) and

(F, E^) are homogeneous in

(p '·

I That fact

(Ε)

any D ^E^ ^

(E^, A) is homogeneous in

^

follows from the

appropriate centerings) because φ

is an element of Q) which is

and hence,

being regular ^ E. That

(F, E^) . is homogeneous in

, we see by the

formula:

(calling the appropriate centering of Case 3: C = A ; we define have to show that

^ ' also Y ).

A^, A^, Β as in the proof of 6.6.3· We

(E, A^) and

(B, A^) are homogenous.

There is a simplicial isomorphism taking Y" (A) onto we have

Γ|(Α^).

vAp' (E)

And as (E, A) is homogeneous in φ ,

(E, A^) is homogeneous in That

^^^

^

.

(B, k^) is homogeneous in

^

we see by a formula

like that in case 2:

(Β) - /\(p(A) 6.7· 4. Proposition.

If --ίγ^ ^ ^

k^, η A^

α

homogeneously in (p , and

Ο

is obtained from ^^ by a finite sequence of bisections of space, and are the subpresentations of then

homogeneously in

^

(p

covering (vJK | and \ Xj ,

.

This follows from 6.7.3, as 6,6.4 from 6.6.3· 6.7.5.

Definition.

O-

Let Ρ be a polyhedron, and X, Ν subpolyhedra

148

of P. Ν is said to collapse homogeneously (geometrically) to X in P, if there are regular presentation

C. "Tt.

covering

X, Ν and Ρ respectively such that ^ ^ collapses homogeneously to combinatorially in ^^ We write

X

homogeneously in P. This definition is

again polyhedrally invariant: 6.-7.6, Proposition.. If Ν

X homogeneously in P, and dC '· Ρ — ^ Q

is a polyhedral equivalence, then

(N) — ^

homogeneously in

Q. This follows from 6.7Λ as 6.6.6 from 6.6.4. 6.7.7. Proposition. If Ν is a regular neighbourhood of X in P, then Ν

X homogeneously in P.

Proof; As in 6.6.8j we start with a simplicial presentation φ Ρ

of

in which a full subpresentation 'Xj , covers X, and take

Ν

where C^ : Ρ — ^ [o, 1

is the. usual map. By .

virtue of 6.7.6, and the definition of regular neighbourhood, it is enough to prove that this Ν V X homogeneously. Let "-^IX be the regular presentation of Ν consisting of cells of the form: simplexes of k)), for
Q

Αφ^(έ)

, for

^-th φ (

- (0,1) - (0,1)

to consist of all simplexes of

,

all simplexes of Q!) which have no vertices in

.

149

ίΓ

((Ο, i)) for Γ ^ Ρ

<Γ Λφ((2))

with Ci) ( Ο = (0,1)

^"AmVgj Ο J Αφ ^

is a re^g-olar presentation of F which refines (p , and has as

subpresentations

and

.

^Yl^ and

are the same as in

proposition 6.6, δ, and therefore we know that "^YX. claim is Ε

iy '^C. homogeneously in -'(έ), A =

. Now, the

(p . In otherwords, if

i)), where <Γ

with(p (tT) =

we have to show that (E, A) is homogeneous in denoting by (j^ the subpresentation of

. In fact

covering

"''(g),

we have Γ (p,(E) -

)

'Ad^^^) Γ

τ

where

0).

α

6.8. The Regular Neighbourhood Theorem. We have seen that if Ν is a regular neighbourhood of X in P, then 1) X C

int ρ Ν

2) Μ is bicollared in Ρ 3) Ν

X homogeneously in P.

Conversely. 6.β.1. The Regular Neighbourhood Theorem. If X

Ν 0

Ρ are polyhedra such that xQnt. pN

2) Ν is bicollared in Ρ

150

3) Ν ν Χ homogeneously in ' Ρ then Ν is a regular neighbourhood of X in

P.

The proof will start with some technicalities which exploit the homogeneity of the collapsing

(The X'S, P's etc. occuring mean-

while should not be confused with the X, Ρ of the theorem). 6.8.2. Proposition. Let Y

X be polyhedra, and let Ρ = X ^^

w

a suspension of X. Then a regular neighbourhood of Y * ν in Ρ is a regular neighbourhood of ν in P. /"in other words, a regular neighbourhood of a subcone of a suspension is a regular neighbourhood of one of the polesJ7. Proof; Let C^ (X) denote X * ν be the join of the maps X

and let ffi ; C^ (X)

^

^ 1 and ν — ^ 0. For any Ζ

l"] X,

C ^ (Ζ) for 0 < cC < 1 will denote the set of points < (1-t) v + t z

ζ;ζ;Ζ,

. If Ζ is a subpolyhedron

c ^ (Z) = (Z ^^ v) r\ cf\\0, dC] ). By 6.3.7? it is enough to prove the proposition for some regular neighbourhood of X -ίί- v. Hence, by a couple of maps, it is enough to show that ^

i® ^ regular neighbourhood of Ci (Υ) in

C^ (X). (It is clearly a regular neighbourhood of ν in C^ (X)). Let ^ subpresentation

be a simplicial presentation of X, containing a covering Y. We define a regular presentation

of G^ (X) to consist of: vl

_ for (T € Ai for

C- "Xi""^

151

Then

(p

^ W

Π

φ " ' ((Ο, i ) ) ]

γ Η

π

φ - ^ (έ)

α

φ

covering C ^(Υ), and for each

with A η Ci (Y) / 0

(g, l). Choose a centering η

^Ίι

ik, 1 )

has a subpresentation

(p -Q^

for ^

φ (A) includes the interval

of (p

, so that for all A ^ (p

with A Π Ci (Y) ^ 0, φ ( ^ A) Then d((p , η ) has the property that if f vd-th vertices both in d

and in d φ

contains

and d

is full in d (p . Choose a centering 0

of d ^

so that for ^ ζ d (p

ύΡ

=

and thus