Piecewise Linear Topology

P^ .

\e ,e .e P = P^ \ P^ 1 \ °' ' \ ^o '

then P^j

o

b y a n e l e m e n t a r y c o l l a p s e , a n d we w r i t e

c o l l a p s e s to t h e s u b p o l y h e d r o n

e x i s t s a finite s e q u e n c e

Remark.

B A P

t

to

then P

. o

o Definition .

P

i s s a i d to be c o l l a p s i b l e if

If t h i s i s t h e c a s e , we w r i t e

P

c o l l a p s e s to a s i n g u l a r point,

P^O.

By the p r e c e d i n g r e m a r k , e v e r y c o l l a p s i b l e polyhedron is c o n t r a c t i b l e . T h e c o n v e r s e i s f a l s e , h o w e v e r , a s t h e following e x a m p l e s h o w s . Consider a two-simplex:

Let

s p a c e obtained by m a k i n g the identifications shown.

D be the quotient

T h e s e c o n d d e r i v e d of t h i s

t w o - s i m p l e x is a t r i a n g u l a t i o n c o n s i s t e n t w i t h t h e i d e n t i f i c a t i o n s , consider

D to be a s i m p l i c i a l c o m p l e x .

a n d so w e m a i

M o r e o v e r , b y a t h e o r e m of W h i t e h e a d

-

D is contractible;

for

H^(D) = 0 , i > 0, a n d so

'Tr^(D) = 0, t h e o b v i o u s c e l l - d e c o m p o s i t i o n s h o w s ^^(D) = 0, a l l i.

Now D is not c o l l a p s i b l e . X 6 BB - B n D . o

Then

a r e not in K pose that K = K

o

Let

It t u r n s o u t t h a t

D X 1^0,

K, w h e r e

K = K ^ u {A} v j { a A } .

c o l l a p s e to

a

es

I = [O, l ] .

K .

We s a y t h a t

K = K^ \

2

.

l)

K , and s u p o

K collapses \

K^ ^

\

. . \

simplicially

K , and if \

o

o B e K i s c a l l e d a p r i n c i p a l s i m p l e x if

B i s not a p r o p e r face of a n y s i m p l e x of K.

Remarks;

aA

.

K is a complex,

face of no o t h e r s i m p l e x of

Suppose A and

i s a v e r t e x of

\

to K^ if t h e r e i s a finite s e q u e n c e xS this is the c a s e , we w r i t e K \ K

If

B u t no p o i n t of D

K c o l l a p s e s by an e l e m e n t a r y s i m p l i c i a l

K ^

\

Let

(We a l s o w r i t e t h i s c o n d i t i o n in t h e f o r m

T h e n we s a y t h a t

K , and we w r i t e

Definition .

D - D^ = B .

K C K be s i m p l i c i a l c o m p l e x e s . o

but a r e s i m p l i c e s of ^

+ A + aA.)

D^D^,

l i n k ( x ; D ) = l i n k ( x , B) = a p . l . b a l l .

has a p . l . ball as a link. Definition.

F o r suppose

K, t h e n

If t h e face

A of

A i s c a l l e d a f r e e fac e of

B is the p r o p e r B in

K.

An e l e m e n t a r y s i m p l i c i a l c o l l a p s e i s an e l e m e n t a r y c o l l a p s e .

\es

If K \

K a n d K = K + A + a A , t h e n aA i s a p r i n c i p a l s i m p l e x of K \ o o with f r e e face A . On t h e o t h e r h a n d , if B i s a p r i n c i p a l s i m p l e x of K w i t h free face

A, t h e n

B = aA; a n d if

K

= K - ({A} O {B}), o

\es

and K ^

K

is a subcomplex o

K^ .

, 3) It is f a l s e t h a t

|k|

L a s u b c o m p l e x of

K, i m p l i e s t h a t

K

L.

L e m m a 2. 1. if

a) A c o n e c o l l a p s e s s i m p l i c i a l l y to a s u b c o n e .

5 K a r e s i m p l i c i a l c o m p l e x e s , then v. K ^ v . K ^ , b) Say K ^ . K ^

K n K^ C K . 1 2 3 Proof. dimension. out

A^

Then

a r e s u b c o m p l e x e s of

K,

K^

a joinable pointj K^ ,

and

K U K^ \ K u 1 2 \ 3

L e t A ^ , . . . , A^ b e t h e s i m p l i c e s of K - K^ Then

v

Precisek

A^

in o r d e r of d e c r e a a

i s a f r e e face of the p r i n c i p a l s i m p l e x v . A ^ .

and v. A^.

Then

A^

i s a f r e e face of t h e p r i n c i p a l s i m p l e x

Collapst v . A^,

what r e m a i n s , etc. . . . b) It suffices to c o n s i d e r K^ = K^ + aA + A. Hence

Then

aA

K

\

K , with K A K C K . Suppose X ^ J ± ^ D a n d A a r e not in K^, s i n c e K^*^ K^ ^ K ^ .

K^ U K^ = K^ U K^ + aA + A defines a n e l e m e n t a r y s i m p l i c i a l collapse

K ^ U K^ ^ ^

K ^ U K^.

L e m m a 2. 2.

K K c o l l a p s e s to

s t e l l a r s u b d i v i s i o n of Unsolved P r o b l e m ;

K, t h e n

(r(K) i s a

""(K^)" It i s t r u e for

< 3.

In t h i s p r o o f w e do not d i s t i n g u i s h i n t h e n o t a t i o n b e t w e e n a s i m p l

and its a s s o c i a t e d s i m p l i c i a l c o m p l e x . complex

s i m p l i c i a l l y , a n d if

I s t h i s t r u e for n o n - s t e l l a r s u b d i v i s i o n ?

c o m p l e x e s of d i m e n s i o n Proof .

o-(K) \

K

If A i s a s i m p l e x , we w r i t e

A for th

A.

It suffices to c o n s i d e r e l e m e n t a r y s i m p l i c i a l c o l l a p s e s .

It a l s o suffices to

c o n s i d e r o n l y s u b d i v i s i o n s o b t a i n e d b y s t a r r i n g at o n e s i m p l e x . K = K^ + aA + A,

aA

a p r i n c i p a l s i m p l e x with f r e e face

So s u p p o s e

A, a n d s u p p o s e t h a t

o-(K) = K - B . l i n k ( B j k ) + b B l i n k ( B ; K ) , C a s e 1;

B n o t a face of

aA;

then

b e B,

B e K.

cr(aA) = a A , a n d

(r{K) = cr(K^) + aA + A . C a s e 2;

—

—

B
Let

[ P i c t u r e for C a s e 2]

A = BA^

1

(with A^ = ^ a p o s s i b i l i t y ) .

A.

Then

(r(aA) = b . B, a. A ^ . We h a v e : • xS . . . a.A^.biB a. A^. B + a A ^ . b . B , • • • since

A,.B+A..b. B 1 1

is a s u b c o m p l e x

of A ^ . b, B , and b y L e m m a 2. 1. •

But

•

.

aA = a . B A , + a B A ^ , 1 1

so

o-(aA) = a . B . A^ + a . b . B A^ . So,

o-(aA) ' ^ a ( a A ) .

K^n

= cr(aA) C K^, K u K

That i s , C a s e 3; that is,

Let

o-(K) B ^ aA

K^ ='(r(aA), K^ = crfaA),

Then

so

= (r(aA) ucr(K )

K_U K

= cr(K ) U(r( aA) = a-(K ).

cr(K^), b y L e m m a 2 . 1 . but

B C aA.

B^A

Put

(

= "not a face of");

B = aB^, A = A.B 1

aA = a A ^ B ^ = A ^ B .

So

1

Then

1

cr(aA) = A ^ . b . B = A ^ b ( a B ^ + B^) = a b A . B , + b. A f 1 1

l>y L e m m a 2. 1.

o-{K^).

^

\

a . b . A^B^ + bA 1 1

Now, abA^B +bA = abA^B + b(A.B^ + A ^ B ) 1 1 1 1 1 1 1 1 = a b A , B , + b A , B , X a ( b A , B^ + A , B^ ) + b A , B^ , 1 1 1 1 \ 1 1 1 1 1 1 b y L e m m a 2. 1 (both p a r t s ) . (r(aA) \

H e n c e we h a v e

a ( b A ^ B ^ + A^B^ ) + b A ^ B ^ .

Now, o-(aA) = (r(aA_,B, + a A ^ B , ) = (r(A,B + a A , B , ) 1 1 1 1 1 1 1 = a A ^ B , + b ( a B , + B j A , = a ( b A , B ^ + A^ B J 1 1 1 1 1 1 1 1 1 That is,

(r[aA) ^

(r(aA).

Case 4:

B = aA.

Then

b(aA + A) \

baA

and

+ bA.B, 1 1

.

Now c o n t i n u e a s i n C a s e 2). cr(B) = o-(aA) = b . B = b(aA + A).

abA \ aA = o-(aA).

Thus

(r(aA) \

But (r(aA).

Now proce<

a s i n C a s e 2). \e L e m m a 2. 3. plex

Let

K

\

L ,

L

a s u b c o m p l e x of t h e s i m p l i c i a l c o m -

K.

T h e n t h e r e e x i s t s a s u b d i v i s i o n K' of vS s u b d i v i s i o n of L , K ' \ L ' , a n d L ' i s s t e l l a r . Proof.

B.

Let

B = cl( K | -

L ) =

K-L

,

K

s u c h t h a t if

B H

A^ i s a f r e e face of the s i m p l e x

Write

B for t h e t r i a n g u l a l i o n K - L

of B and A r e s p e c t i v e l y , stellar;

such that

a p p l y L e m m a 1 . 1 0 to h

.

i s t h e indv

L ) = F , a f a c e of t h e ba

By C o r o l l a r y 1. 27, t h e r e i s a p. 1. h o m e o m o r p h i s m

where

L'

h : ( B , F ) —>

A (i. e . , d i m A^ = d i m A - 1 ) . of

B,

Let

B'

and

A'

be s u b d i v i s i

hr B ' — > A' i s s i m p l i c i a l a n d Note that as

h(F) = A^, B'

B'

is

contains a tri

-47a n g u l a t i o n of F , s a y F ' . d u c e d s u b d i v i s i o n on Let

B is

K'

b e a s t e l l a r s u b d i v i s i o n of

K whose in-

B'.

p: A —3> A^ b e t h e l i n e a r m a p w h i c h i s t h e i d e n t i t y on A^

the v e r t e x division

Let

v opposite

A " of A '

A^

to S'li i n t e r i o r p o i n t of

such that

and sends

Then t h e r e is a sub-

p : A " —5> A^' i s s i m p l i c i a l , a n d

A^'

is a stellar

s u b d i v i s i o n of A^ . Let

B"

hi B ' —> A '

b e t h e subdivi£ j n of

B'

making

was already simplicial,

e x t e n d s to a s t e l l a r s u b d i v i s i o n

L"

F" of

L " m e e t in t h e c o m m o n s u b c o m p l e x

h : B " —> A"

simplicial.

i s a s t e l l a r s u b d i v i s i o n of

L ,

F", K"

Put

K" = B" U L " .

F',

Since

Since and B"

and

i s a w e l l defined s u b d i v i s i o n of

K,

not n e c e s s a r i l y s t e l l a r . To p r o v e t h i s l e m m a , it suffices b y L e m m a Z. 1 t o p r o v e t h a t B " r i L« = F " . where

To p r o v e t h a t

B"

p : A " — > A" i s s i m p l i c i a l . 1

, in o r d e r of d e c r e a s i n g d i m e n s i o n ,

it suffices to p r o v e t h a t

B" ^ F " ,

as

xS A " \ A'^

Now l e t

{A.} be t h e s i m p l i c e s of A " 1 1 -1 \S -1 • p A. \ p A. o A. by c o l l a p s i n g t h e

-1

p r i n c i p a l s i m p l e x e s of p

A^ f r o m t h e i r t o p f a c e s in o r d e r .

gives t h e r e q u i r e d s i m p l i c i a l c o l l a p s e of

A

A"

onto

A'^ .

D o i n g t h i s in t u r n

-

Theorem 2.4. if

I K|

If L and

K are simplicial complexes,

| L | , then there exists subdivisions

L

K' a n d L' w i t h

C

K , and

L' C. K'

and

K Proof. most

By i n d u c t i o n , a s s u m e t h e t h e o r e m for a l l c o l l a p s e s c o n s i s t i n g of

(n-1)

elementary collapses.

T h e r e is a t r i a n g u l a t i o n of

P . , s a y K. , 1 1

K' n-1

\

\

K' . o

By i n d u c t i o n , t h e r e i s a s u b d i v i s i o n Now,

K' ^ extends to a s u b d i v i s i o n n-1

K" a n d K " , of K' n n-1 n s u c h t h a t K " \ K" ^ . n \ n-1

B y L e m m a 2. 2, K" n

\

X

K" . o

|K| =

> • • • ^ ^ ^Q ~

K ^ of K c o n t a i n i n g a s s u b c o m p l e x e s t r i a n g u l a t i o n s

there exist subdivisions K" ^ s t e l l a r , n-1

Suppose

K" , ^

K' , of n-1

K' of K . n n

^ n-1

with

By L e m m a '

a n d K ' . r e s p e c t i v e l y , with n-1 ^

K" = i n d u c e d s u b d i v i s i o n of o

\^

K

K' . o

Hence

-

2.

F u l l S u b c o m p l e x e s and D e r i v e d N e i g h b o r h o o d s Definition.

«

If K

o

i s a s u b c o m p l e x of t h e s i m p l i c i a l c o m p l e x

said to b e ful l if a n y s i m p l e x in simplex

of

K^ ;

i.e.,

L e m m a 2.5. deriveds, then

l)

K^

2) If then

K a l l of w h o s e v e r t i c e s l i e in K^

no s i m p l e x in K - K^

If K^

K' i s full in o

K and

K^ ^ K'

f

-1

Proof.

is a

i s a full s u b c o m p l e x of

K and K^ 9. K'

is any subdivision

K'.

i s full in

o

(0) = K ^ . 1) If '

1 < i ^ s, then ' s

is

a r e first

i s full in K and A e K - K , t h e n A n | K | O O O e m p t y o r a s i n g l e face of A. (And c o n v e r s e l y . )

such t h a t

o

K^ .

3) If K

4) K

K

h a s a l l i t s v e r t i c e s in K^ .

i s a s u b c o m p l e x of

i s a full s u b c o m p l e x of K^

K,

( L i n e a r m e a n s l i n e a r on s i m p l i c e s

o-e K ' , l e t '

A

there exists a linear map

K

0- = A , . . , A , 1 S

A

h a s a n i n t e r i o r point in K o

cr € K' . o 3) If A € K - K

1

< .. . < A and h e n c e

R

S

+

ft K

> R

= [0,oo).)

e K. A

is e i t h e r

If A. € K ' , 1 o

e K , s

So A. € K , o i o

i < s, and

meets O L e t A^ = s p a n { a , , . . . . a . } . 1 ^ 1

K

, l e t ( a , , . . . , a.) be t h e v e r t i c e s of A i n K . O 1 1 o Then A. e K and A . < A. S i n c e A H | K I i s 1 o 1 o

a l w a y s a u n i o n of f a c e s of A, e a c h of w h i c h i s s p a n n e d by i t s v e r t i c e s ,

A 1

= A n IK

. o 2) S u p p o s e

t a r y c e n t e r of

cr i s i n A .

A^ a face of A .

mi

K ^ ^ K i s full.

Therefore

Then cr r\

Let

ere K ' .

o" C A .

Choose

Ae K

such that the

Moreover, o - r \ [ K | C A n | K = A , o ~ o 1 = o" n A^ , w h i c h i s e i t h e r e m p t y o r a

face of

0".

one face.

T h u s , e v e r y s i m p l e x of

K'

which m e e t s

K ' m e e t s i t in exs

K' i s full in K ' . ( C o n v e r s e of 3 ) . ) o 4) If K^ ^ K i s a full s u b c o m p l e x , l e t f: K r"*" b e defined b y

setting

This m e a n s that

f(v) = 0 if v i s a v e r t e x of K

and

f(v) = 1

if v

i s a v e r t e x in

o K - K ^ , and e x t e n d i n g l i n e a r l y o v e r s i m p l i c e s .

Clearly,

| K^

+

C o n v e r s e l y , if

f: K—5> R

=f

(O).

-1

i s g i v e n and we s e t

K

= f

(O), t h e n

K

o a full s u b c o m p l e x .

It i s a s u b c o m p l e x b e c a u s e if x e a-, tr e K, t h e n

f(x) = 0 = > f((r) = 0. of

0" ,

then

o

It i s full b e c a u s e if

tr e K a n d f i s z e r o on t h e vertic

f((r) = 0 .

N ( L D; Le f)i n=i t i o n . rTL

Ssutpapr o( sve| Lt )h a t (union L^ i os vae rs uvbecr ot im c ep sl e) ,x of c a l lLe d. t hTeh ecnl ows e d define simplici

neighborhood

L^

of

Definition . Let

in

L.

Suppose that

K ^ C K b e a t r i a n g u l a t i o n of X C M; i . e . ,

K^

a fTill s u b c o m p l e x of

n e i g h b o r h o o d of

X in

M,

K.

Then

where

N = |N(K^;K')|

M aji m - m a n i f o l d , =X,

| K | = M;

X C with

is called a derived

K' C K' i s t h e f i r s t d e r i v e d s u b d i v i s i o n of O

a

C

K

X is a polyhedron,

K.

o

IS

(r) (] Definition . If K^C K i s a n y t r i a n g u l a t i o n of X C M a n d if K^ ' ^ K' t h e r^^ s u b d i v i s i o n , t h e n | N(K K^^^) | i s c a l l e d an r^^ d e r i v e d n e i g h b o r

f X in

M.

For

r > 2, a n

r^^ d e r i v e d n e i g h b o r h o o d i s a d e r i v e d n e i g h b o r h j

-

Remark.

T h e r e a s o n for t a k i n g full s u b c o m p l e x e s o r at l e a s t 2nd d e r i v e d s

a s d e r i v e d n e i g h b o r h o o d s i s t h a t we w a n t to be a b l e to p r o v e t h a t a d e r i v e d n e i g h b o r h o o d of

X c o l l a p s e s to X .

d e r i v e d n e i g h b o r h o o d of X i n

If

M is

M =

X = A^, then the first

M, w h i c h d o e s n o t c o l l a p s e to

X.

The

2nd d e r i v e d n e i g h b o r h o o d d o e s c o l l a p s e to X , h o w e v e r .

f

L e m m a 2.6. and K Then f

= f"^(0),

K

be a full s u b c o m p l e x of K.

f linear.

Suppose

0 < £ < f(v) ,

([0, E]) i s a d e r i v e d n e i g h b o r h o o d of

Proof . I'.,

Let

Let

K'

be obtained from

[K

= A

! N ( K ' ; K ' ) | = f'^([0, £]). a

A

< ... < A 1

r

f; K — >

v a n y v e r t e x in K - K , in K

K by s t a r r i n g e a c h s i m p l e x

€ A in o r d e r of i n c r e a s i n g d i m e n s i on, c h o o s i n g

^ . CUirH

Suppose

Let

A. 6 K. 1

A e f

) if

A

A r\ f

cr b e a p r i n c i p a l s i m p l e x of Then

at ^ N(K',K'),

A. e K' , so A. e K , s o m e i . 1 o 1 o

Take

i as l a r g e as possible with A

i+1' Hence

/ 1 I iC

= . . . = f(A^) = £

£(AJ = 0 o r £ .

, so

j < i*

f a r e g r e a t e r than

. . ^ ^ c f'^([O,

If f(A^) = 0, t h e n

A^

and l i e s in f"^([0, £ ] ) . N(K^,K')

£ ,

Therefore,

K ' ) C £"^[0. E ]. ]),

A^ < . . . < A^,

is a v e r t e x of

h a s a v e r t e x i n K^, s a y v , w i t h

£ ] C

=

1 ^ k < r - i , b y l i n e a r i t y of f.

Conversely, suppose

So

Then

have v e r t i c e s whose v a l u e s under

f~ ( £ ) n A

t h e n A^

A^ e K ^ .

K^ .

Then <

If f(A^) = £ ,

[ v | / A^, a n d so But v . A ^ . . .

e

K').

-

Ambient I s o t o p y Def inition. ij X X I land

An a m b i e n t i s o t o p y of a p o l y h e d r o n X i s a p. 1. h o m e o m o r p h i s m

X A I w h i c h c o m m u t e s w i t h p r o j e c t i o n o n I (i. e . , i s l e v e l p r e s e r v i n g )

has the p r o p e r t y that

h(x, O) = ( x , 0 ) , a l l

If h i s an a m b i e n t i s o t o p y , we w r i t e

x e X.

h^ for t h e p. 1. h o m e o m o r p h i s m of X

)nto i t s e l f defined by s e t t i n g h(x, t) = (h^(x), t ) . jontained

in X,

we say that h t h r o w s

X^

If X^

onto

X^

a n d X^ a r e p o l y h e d r a if b^(X^) = X^ .

Two

j l y h e d r a c o n t a i n e d in X a r e s a i d t o be a m b i e n t i s o t o p i c if t h e r e e x i s t s a n i m b i e n t i s o t o p y t h r o w i n g o n e onto t h e o t h e r .

The relation

"X^

is ambient

Usotopic to X " i s c l e a r l y an e q u i v a l e n c e r e l a t i o n . A homeomorphism

kr X

> X i s s a i d to b e a m b i e n t i s o t o p i c to t h e i d e n t i t y

t h e r e e x i s t s a n a m b i e n t i s o t o p y h of X w i t h h^ = k . If X C X, w e s a y t h a t t h e a m b i e n t i s o t o p y h of X k e e p s o

X

o

fixe d if

X I = i d e n t i t y m a p of X^ X I, Lemma 2.7. K| —> | K

Let

K C K b e s i m p l i c i a l c o m p l e x e s , and l e t o

be a p . l . h o m e o m o r p h i s m s u c h t h a t

1) h |

= identity.

2) h(tr) =0- , a l l

cr e K.

^Then h i s a m b i e n t i s o t o p i c to t h e i d e n t i t y v i a a n a m b i e n t i s o t o p y k e e p i n g

; fixed. Proof. ^dimension.

Let. c r 0 " I

Define

n

H on

b e t h e s i m p l i c e s of K - K , i n o r d e r of i n c r e a s i n g o

X I b y s e t t i n g it e q u a l to t h e i d e n t i t y .

Define

H

on K X 1 by setting defined on

H(x, 1) = (h(x), 1) a l l x 6 K.

cr. X I , a l l j < i . J

Then

H to

a point in

cr^ X I by defining

ar^ .

H h a s been^

H i s defined on t h e f a c e s of 1

Extend

A s s u m e that

cr. X I. J

1

^(o-^,— ) = (o"^, — ) and j o i n i n g l i n e a r l y ,

T h i s d e f i n e s a p. 1. h o m e o m o r p h i s m

H: K X I

a.

> K X I.

It

e a s y to c h e c k t h a t it i s l e v e l p r e s e r v i n g a n d i s t h e r e f o r e t h e d e s i r e d a m b i e n t isotopy. C o r o l l a r y 2. 8. and if

If h : B —> B,

h B = i d e n t i t y of B , t h e n

B a p. 1. b a l l , i s a p. 1. h o m e o m o r p h i s n i l

h i s a m b i e n t i s o t o p i c to t h e i d e n t i t y , keepii

B fixed. Proof .

Let

K = A, K^ = A

L e m m a 2. 9 .

Let

X in the polyhedron w h i c h i s fixed on Proof .

N

and apply L e m m a 2.7.

and N

M.

be two d e r i v e d n e i g h b o r h o o d s of t h e polyhec

T h e n t h e r e i s an a m b i e n t i s o t o p y t h r o w i n g

N^ onto |

X.

Let

K ^ Q J^

and

K^ Q J ^

b e t r i a n g u l a t i o n s of X C M, w i t h

K.l

full in J^ . L e t N = |N(K';J')|

p r i m e s denote first derived subdivisions,and suppose and N = |N(K' ; J ' ) | . Let K C J b e a c o m m o n subdivisio

of

K^C

JL

^

K^S

J^

simplicial. of J

o

±

and

Lt

Ct

,

C^

O

(Choose subdivisions making

They obviously a r e the s a m e . )

and, so

O

Then

K^

( p r i m e s denote first deriveds) N = ' o

1

• I-^q I —^

i s a full s u b c o m p l e x

N (J' ; K') o o o

is a derived

neighborhood. It c l e a r l y suffices to find an i s o t o p y t h r o w i n g throwing

N^ onto

N^.

N^ o n t o

N^

and an isotopy

We w i l l c o n s t r u c t an a m b i e n t i s o t o p y t h r o w i n g

N^onto

-

Let £: f'^(O) = fthat

—> R

K^ .

Then

0 < £ < f(v)

subdivisions " such that

K

be a m a p w h i c h i s l i n e a r on s i m p l i c e s , f i s a l s o l i n e a r on s i m p l i c e s of J ^ .

for a l l v e r t i c e s C J o ~ o

*

v in J ^ - K ^ .

and K, C J 1 1

of

K

o

Let £ be such

Then there exist first derived

C J o

f ' ^ ( [ 0 . £ ]) = I N(K *; J * ) | = | N(K

with

and K^ C J 1 1

respectively,

J * ) | = N*. by the p r o o f of

Lemma 2.6. L e t { A J = s i m p l i c e s of lA. € A. .

Say J^

J^.

Let

J^' b e o b t a i n e d by s t a r r i n g at p o i n t s

i s o b t a i n e d by s t a r r i n g

A^ € A^ .

I L e m m a 2.6, it i s c l e a r t h a t we m a y s u p p o s e simplicial homeomorphism linearly over simplices. ! identity, k e e p i n g

J^ —^^—> J^

A^ = A^ if A^ € K ^ .

>1< * N(K ; J ) .

throwing keeping

N

onto

o

| K^ |

By the L e m m a 2 . 7 , h is ambient isotopic to the

j K^ | fixed, for if

cr £ J ^ , h(o-) = cr, and h |

N

Let

£>0

fixed and t h r o w i n g

=

N(K

o

;J

o

) , a n d so N

1

onto

K^

fixed N , o

N.

is full in K. be such that

So l e t

K^

Then let £
M, a n d

in

M, t h e n N \ X . In v i e w of L e m m a 2 . 9 , it suffices to p r o v e t h a t

o

N^

i s a m b i e n t i s o t o p i c to

If X i s a p o l y h e d r o n c o n t a i n e d in t h e p o l y h e d r o n

|! d e r i v e d n e i g h b o r h o o d K

= identity.

S i m i l a r l y , t h e r e i s an a m b i e n t i s o t o p y k e e p i n g

if N i s a d e r i v e d n e i g h b o r h o o d of X

assume

and extending

fixed.

L e m m a 2. 10.

Proof.

Define a

by sending A. to

H e n c e t h e r e i s an a m b i e n t i s o t o p y k e e p i n g I ^ ^ | * N =

F r o m t h e p r o o f of

N \ X for o n e

K be a t r i a n g u l a t i o n of X 9 M, and

f: K

>R

all v e r t i c e s

+

be l i n e a r , with

v of K - K .

f

-1

(O) = K . o

We h a v e s e e n t h a t

N = f

is a d e r i v e d n e i g h b o r h o o d of K^.

So i t suffices to s h o w

£])\|KJ. Let

i = 1, , . . , r )

b e t h e s i m p l i c e s of K - K ^

d i m e n s i o n . T h e n C. = A^ n f Let

F . = A. n

-1

£ ), a f a c e .

U. = U VJ (U{C. j = 1 , . . . , i } ) . 1

O

1

in a f a c e .

'H ([0, £ ]) i s a c o n v e x l i n e a r c e l l a n d so a p. l . . | Now s e t Then

J

face of C . .

^ 1

Hence

U^ = (K^), a n d s e t

C. n U. 1

So c l { U . - U . J

1-1

U. \ 1

U. ^

1-1

i n o r d e r of i n c r e a s ^

= c l { C . - C. n U. J 1

But

1

= C . n A. = c l { C . - F . }

1-1

1-1

1

= C.

U^ = f" ([0,£ ]).

1

1

1

ig;

1

is a ball meeting

®

-

4.

E x i s t e n c e and U n i q u e n e s s of R e g u l a r N e i g h b o r h o o d s Definitio n.

L e t X b e a p o l y h e d r o n c o n t a i n e d i n the p. 1. m - m a n i f o l d

N C M i s c a l l e d a r e g u l a r n e i g h b o r h o o d of X in

M if

1) N i s a c l o s e d n e i g h b o r h o o d of X in 2) N i s a n m - m a n i f o l d , 3)

M.

M,

and

N^X.

T h i s s e c t i o n i s d e v o t e d to t h e p r o o f of t h e following t h e o r e m . T h e o r e m 2. 11.

Let

X C M,

M and m - m a n i f o l d ,

X a polyhedron.

Then

1) Any d e r i v e d n e i g h b o r h o o d of X i s a r e g u l a r n e i g h b o r h o o d ; 2) If N

and N

a r e r e g u l a r n e i g h b o r h o o d s of X in

BMSts a p . l . h o m e o m o r p h i s m

h : N^ —5> N ^

such that

M, t h e n t h e r e

h(x) = x if x e X ;

and

3) If X i s c o l l a p s i b l e t h e n a n y r e g u l a r n e i g h b o r h o o d of X a p . 1. m - b a l l . T h e o r e m 2. 11 i s p r o v e n b y i n d u c t i o n . i t e m e n t s , for e a c h i n t e g e r E(n)i

We c o n s i d e r t h e following t h r e e

n > 0:

If X i s a p o l y h e d r o n c o n t a i n e d in t h e m - m a n i f o l d

M, a n d if m ^ n,

e v e r y d e r i v e d n e i g h b o r h o o d of X i s a r e g u l a r n e i g h b o r h o o d . If N^ a n d N^

a r e d e r i v e d n e i g h b o r h o o d s of X i n m"^, a n m - m a n i -

a-iid if m ^ n , t h e n t h e r e e x i s t s a p . l . h o m e o m o r p h i s m identity on

h: N^ —> N^

which

X.

In a m a n i f o l d of d i m e n s i o n at m o s t •Biblepolyhedron is a p . l .

m-ball.

n, e v e r y r e g u l a r n e i g h b o r h o o d of a

L e m m a 2. 1 2.

Proof. of X in

Let

d i m M ^ n.

If X '^{X^}

^JID N i s a r e g u l a r n e i g h b o r h o o d j

M, t h e n X is a r e g u l a r n e i g h b o r h o o d of { x ^ } .

t r i a n g u l a t i o n of s t a r (x ; K) o of X .

U(n) i m p l i e s B(n).

=

M w i t h x^

a v e r t e x of K.

X . linkfx ; K) o o

Moreover,

Let

M = |K|

be

Then

i s a p. 1. m - b a l l , a n d a c l o s e d n e i g h b o r h o o d l '

| s t a r (x ; K)| ^ { x } .

So U(n)

i m p l i e s that

N is h o m e o |

to t h e p. 1. m - b a l l | s t a r ( x ^ , K ) | . L e m m a 2. 13. Proof .

E ( n - l ) and

B(n-l) implies

E(n).

L e t X ^ M b e a p o l y h e d r o n c o n t a i n e d in t h e m - m a n i f o l d

Let

K C K be a t r i a n g u l a t i o n of X C M, w i t h o ° ~

N=

N(K^;K') .

know t h a t N \ ^ X .

X

o

full in K.

M, m ;

Let

N i s c l e a r l y a c l o s e d ( t o p o l o g i c a l ) n e i g h b o r h o o d of X , and SO it r e m a i n s o n l y to show t h a t

do t h i s , it suffices to p r o v e t h a t

N i s a p . 1. m - m a n i f o l d .

N ( K ^ j K ' ) , for w h i c h we a l s o w r i t e

of n o t a t i o n , i s a c o m b i n a t i o r i a l m - m a n i f o l d .

N, by abiJ

Using induction and the formula ,

l i n k ( A B ; N ) = l i n k ( A ; l i n k ( B ; N)) w i t h a s i n g l e v e r t e x , it i s e a s y to s e e t h a t be a c o m b i n a t o r i a l m - m a n i f o l d if (and only if) for e v e r y v e r t e x

v of

N

N,

link(v; N) i s an ( m - l ) - s p h e r e o r b a l l . So l e t V be a v e r t e x of

N,

If v e K^, t h e n

s t a r (v; K') ^ N, a n d so

link(v; N) = l i n k ( v ; K ' ) = a s p h e r e o r b a l l of d i m e n s i o n ( m - l ) . S u p p o s e on the o t h e r h a n d t h a t

A € K.

Let

v e N-K' , o

Then v = ^

for s o m e s i m p l e

B = A r\ [ K^ [ ^ a s i n g l e (siiiiplicia^; face of A by f u l l n e s s of K^

(B i s c l e a r l y n o n - e m p t y ) .

-

Let

cr € K ' , a n d w r i t e

o" = A . , . . A , A^ < . . . < A e K. T h e n 1 s 1 s A^ < A < s o m e j , o r A < A^ o r A^ < A. So if

€ link(v;K')

= { B , . . . B . | A < B < . . . < B. € K}, t h e n 1 J ^ 1 J where

cr. e (A)'

and

cr € S,

|(We allow t h e p o s s i b i l i t y

(A)'

ere l i n k ( v ; K ' ) < ^ cr = cr o" , 1 ^

b e i n g t h e i n d u c e d s u b d i v i s i o n of K'

tr^ = 0 , i = 1, 2, a n d w r i t e

= ar^, .

on

A.

= cr^.)

Thus, l i n k ( v ; K ' ) = A . S . Let

Now A < B = > B / k ' . H e n c e s a k ' = Therefore o o r\ K' = A ' n K' = B ' . T h e r e f o r e , L o N c o n s i s t s of t h e s i m p l i c e s of L o o meeting

L=A.S.

B'

and t h e i r f a c e s .

T h e fact t h a t

B i s c o n v e x i n s u r e s t h a t it and i t s

faces f o r m a full s u b c o m p l e x of a n y s i m p l i c i a l c o m p l e x c o n t a i n i n g i t .

We h a v e

L r\ N = N ( B ' ; L ) = N ( B ' ; A ' S ) = N ( B ' ; A ' ) . S , the l a s t e q u a l i t y b e i n g a c o n s e q u e n c e of t h e fact t h a t N(B';A') manifold | A ' | N(B';A')|

B' £ A ' .

is a derived neighborhood.of the collapsible complex of d i m e n s i o n at m o s t ( n - l ) .

B'

in t h e

H e n c e b y B ( n - l ) and E ( n - l ) ,

i s a p . 1. b a l l w h o s e d i m e n s i o n i s

( d i m A - 1).

However,

S

is

p . l . h o m e o m o r p h i c to l i n k { A ; K ) .

In fact, if A < B and C i s t h e c o m p l e m e n t a r y A face, the m a p on v e r t i c e s w h i c h s e n d s B to C d e t e r m i n e s a s i m p l i c i a l h o m e o m o r p h i s m of ''

S onto

to m - d i m A - 1.

(liiik(A; K ) ) ' . «

Hence

A'.S

T h u s to c o m p l e t e t h e proof, link(v;N) = l i n k ( v ; K') A N. ^

Thus,

T. • T, 1 1

meets B'.

i s a p . l . b a l l of d i m e n s i o n e q u a l

i s a p . 1. b a l l fo d i m e n s i o n

m-1.

it r e m a i n s o n l y to s h o w t h a t

C e r t a i n l y , l i n k ( v ; N) C l i n k ( v , K ' ) A N,

0-€ l i n k ( v ; K ' ) O N = N ( B ' , A ' ) . S , t h e n ^

|s|

cr = (r^cr^ w h e r e

So v o - < v T . t r ^ w h i c h m 6 e t s 1 2

B',

« ^^ «

Conversely, ^ ^

So vo" e N, o-e l i n k ( v , N ) .

L e m m a 2. 14.

If M i s a n m - m a n i f o l d ,

B C M is an m - b a l l such that

F = B n M

t h e n t h e r e e x i s t s a p . 1. h o m c i o m o r p h i s m of

if X C M i s a p o l y h e d r o n , if i s a face of

B , a n d if B n X s

h: c l ( m - B ) —5> M w i t h h X = i^g

X.

Proof . and say

By i n d u c t i o n o n

m.

dim M = m,

1) c l ( M - B )

is an m-manifold.

Namely, triangulate c o m p l e x e s , and c o n s i d e r

y X^fyi^

,

M

so t h a t

B and F

l i n k ( x ; M - B ) , x a v e r t e x of

l i n k ( X ; M - B ) = l i n k ( X ; M), first

So a s s u m e 2. 14 for m a n i f o l d s of d i m (m-

an ( m - l ) ball o r s p h e r e .

x / F.

Then

a r e t r i a n g u l a t e d a s suli M-B .

If x e M - B , thJ

If x e M - B r» B,

suppose|

l i n k ( x ; M - B ) = l i n k ( x ; M) - link(x; B) i s a n (i

s p h e r e w i t h t h e i n t e r i o r of a n ( m - l ) - b a l l d e l e t e d , a n d so an ( m - l ) b a l l . then X € F .

[F O M - B = F ] .

(link(x;M))

= l i n k ( x ; M ) , a n d so

a f a c e of t h e ( m - l ) b a l l

So l i n k C x . F )

Moreover,

( l i n k ( x ; M)) n l i n k ( x ; B) = l i n k ( x ; M r i B )

link(x;B).

cl( I l i n k ( x ; M) I - | l i n k ( x ; B ) [ )

i s an {m-2) ball.

If x «

= linki

H e n c e by ina^'ction,

i s p . l . h o m e o m o r p h i c to | l i n k ( x ; M ) | , a n ( m - l ) t

-£)1Iberefore,

| link(x; M - B )

isp.l.

(m-1) ball.

This proves that

cl(M-B)

is

inanifold of d i m m . 2)

Let

boundary c o l l a r . jet

F^ = a ] ^ .

Choose

D = c ( F ^ X [0, £ ] ) .

c: a(cl(M-B) X I

such that

> cl(M-B)

c ( F ^ X [ 0 , £])

E x t e n d to a l l of

be a

does not m e e t

T h e r e i s a p. 1. h o m e o m o r p h i s m

is the i d e n t i t y on D - F . liiomeomorphism

£>0

Let

B uD—> D

X. which

M by t h e i d e n t i t y , g e t t i n g a p. 1.

M—> cl(M-B).

To s t a r t t h e i n d u c t i o n , we l e a v e it to t h e r e a d e r to v e r i f y t h a t in c a s e m = 1, icl(M-B) I

is a manifold,

I

Lemma 2.15.

It'-

,

,

—

I

and t h e n to p r o c e e d a s in 2).

E ( n - l ) and B ( n - l ) i m p l i e s

U(n).

!• • I . . I

Proof .

L e t N be a r e g u l a r n e i g h b o r h o o d of X i n M.

T h e n we w i l l s h o w

[that N i s p. 1. h o m e o m o r p h i c to a d e r i v e d n e i g h b o r h o o d of X in a-manifold, on X. Let r s o that

X

a polyhedron in

So l e t

K = K^ ^ ^ K^ ^ ^ ^ . . . ^

K" = b a r y c e n t r i c s e c o n d d e r i v e d of and U

an

U(n).

K C K C J be t r i a n g u l a t i o n s of X C N C M. o K^.

(M

M . ) , via a h o m e o m o r p h i s m which is the identity

T h i s t o g e t h e r w i t h L e m m a 2. 9 w i l l i m p l y

K ^

M.

K.

Let

i s a s e c o n d d e r i v e d n e i g h b o r h o o d of

We c a n c h o o s e

K^ b e t h e c o l l a p s e .

U^ = N ( K j ' ; K " ) . |K

|

K S K o

Then

Let

U^ = K " ,

in t h e n - m a n i f o l d

|Kj.

We

ft

I a r e going to c o n s t r u c t p. 1. h o m e o m o r p h i s m s •pointwise fixed.

We a s s u m e b y i n d u c t i o n t h a t

®o t h a t w e m a y a s s u m e in p a r t i c u l a r t h a t

> U. w h i c h l e a v e

K^

h a s b e e n c o n s t r u c t e d if i i s an m - m a n i f o l d .

r-1,

Now l e t u s o b s e r v e t h a t

U^ =

s t (o-jK").

Since

is a vertex

o-cK. 1 K'^ , t h e i n c l u s i o n

D is obvious.

S u p p o s e o n t h e o t h e r h a n d , t h a t T € U. A

then B

T I T . , where 1

<

I

< B

If B^ ^

€ K'.

s

T. m e e t s 1

KI*. 1

T h e n for s o m e

is a point, then

Suppose i,

B^ = tf-,
1

e K.". 1

( 1 o-eK. 1

B , . Then o - B , . . . B e K", 1 I s s t ( o - ; K " ) , a n d h e n c e so d o e s T,

Now let

U.^

1+1

B. e Kl , so B « 1 1 1 ir e K^ b e s u c h thai

So i n e i t h e r c a s e

K , = K. + A + B , A = a B , a e K . , A / K. . 1+1 1 1 1

c e n t e r s of s i m p l i c e s of A and

Hence

Otherwise, let

i s a v e r t e x of

T e

A

T. = B . , . . B , w h e r e 1 1 s

T h e n t h e o n l y bai

w h i c h a r e not b a r y c e n t e r s of s i m p l i c e s of

Therefore = U. + P + Q , 1

P = star(A;K"),

Q=st(B;K").

We n o w c l a i m t h a t t h e following two s t a t e m e n t s a r e t r u e : a)

U^ n P

i s a f a c e of

b) ( U ^ u P ) n Q To p r o v e a), l e t

P.

i s a f a c e of

Q.

L = l i n k ( A ; K") = A ' . S, w h e r e

(See L e m m a 2. 1 3 . )

Let

p: l k ( A ; K " ) — > L '

w h i c h i s defined o n v e r t i c e s b y s e n d i n g ^ C If

0- c P O U^, t h e n o" e l k ( A ; K " ) , a s

o-e l i n k { D ; K " )

for s o m e

D / it(A;K").

Therefore,

D e K.

as

S=

A < B^< . . . <

be simplicial h o m e o m o r p h i s m to

C for a n y s i m p l e x


and

A /
for s o m e

L.;

In a d d i t i o n ,

D e K.

(re P o u . = > o" e l':ak{A;K") O l k ( 6 ; K " ) ,

C o n v e r s e l y , it i s c l e a r t h a t a n y s i m p l e x of

C of

but some D

s u c h a n i n t e r s e c t i o n l i e s in

P

-63-

However, lk(Aj K") n

K " ) j/ ^ < = > AD e K ' .

F o r if

o- = B 1

< . . . < Bg « K ' , i s a s i m p l e x of t h i s i n t e r s e c t i o n , t h e n A a n d irertices of B , a n d c o n v e r s e l y . 1 D or

D < A.

But

So t h i s i n t e r s e c t i o n i s n o n - e m p t y if a n d o n l y if

A w a s a p r i n c i p a l s i m p l e x of

< A is the only p o s s i b i l i t y , in which c a s e

U De aB Since

P

D E aB.

^^ So we h a v e p r o v e n t h e

(link(A; K " ) n link(D; K " ) ) .

U. C l i n k ( A ; K " ) , w e m a y c o n s i d e r i t s i m a g e u n d e r t h e p . 1.

>omeomorphism p

B , 8 m u s t be

p: link(^; K « ) L ' .

lk{£);K")<S=^p(r€ •it"(D;L').

F o r if

If

cr € l i n k ( A : K " )

cr =

..

.

f r i t e B. = kr. . T h e n p( O" ) = . . . T £ L ' . But S' 1 1 I S « lk(D;K") b < B^ D^ < T^ po" e st(D; L ' ) .

and D € a B , t h e n

B^ < . . . < B^ 6 K ' ,

So w e h a v e :

-63(CI)-

p ( p n u )

=

M

.

=

DTaB T h e l a s t e q u a l i t y follows a n a r g u m e n t s i m i l a r t o t h a t u s e d in d e r i v i n g a s i m i l a r e x p r e s s i o n for However, i s full i n

L'.

U^ ( p a g e 62 ). (aB)' L'

is f\ill i n A '

L = A'.S.

Therefore, (aB)"

i s a p . 1. m a n i f o l d of d i m e n s i o n ( n - l ) .

E ( n - l ) ==^> N ( ( a B ) " , L ' ) aB|

a n d so a l s o i n

i s a r e g u l a r n e i g h b o r h o o d of

| aB

Hence But

in

s o b y B ( n - l ) , t h i s r e g u l a r n e i g h b o r h o o d i s a p . 1, ( m - l ) b a l l .

P n U^ i s a l s o a p . l . ( m - l ) b a U . b o u n d a r y of

Since

P n U^ C l i n k ( A ; K " ) , w h i c h l i e s

P = s t a r (A; K " ) , t h i s p r o v e s t h a t

To p r o v e b), l e t

Hence

P O U^ i s a face of

= link(:&;K') = B ' . S ^ , s a y .

P.

Define p^.lk(]S; K»') —5> L^

A

jy defining i t o n v e r t i c e s to s e n d € Q r\ (U. u P ) )r for

D = A.

D < B.

Since

>B8ibilities a r e

if a n d o n l y if

BC t o

C.

As before we have that

o- c link(]&; K " ) n l i n k ( ^ , K < ' )

for s o m e

D e K.

O n c e a g a i n , t h i s i n t e r s e c t i o n i s n o n - e m p t y if and o n l y if B < D B i s a f r e e face of t h e p r i n c i p a l s i m p l e x D « A or

P^(Qn(U u P ) )

D < B.

=

A, t h e o n l y

So t h i s t i m e w e find t h a t ) =

N((AB)";L').

Dt B or D = A ^Tjefore, w e s e e t h a t B(n-l)

i s full in L^ = B ' S ^

and i s c o l l a p s i b l e .

= > N ( ( A B ) " ; L j ^ ) i s a n ( n - l ) b a l l , a n d so

^

is a manifold.

Q = c l ( Q - F r Q ) , w h e r e t h e f r o n t i e r of

E(n-l)

Q O (U. ri P ) i s a face of Q.

| T o c o m p l e t e t h e proof, we a r e going to a p p l y L e m m a 2. 14. ive h y p o t h e s i s i m p l i e d t h a t

So

Recall that the

Moreover,

Q i s t a k e n w i t h r e s p e c t to

But F r Q = (U. U P ) A Q, a face of 1

Hence

Q.

is p. L h o m e o m o r p h i c to

m e n t g i v e s a p, L h o m e o n n o r p h i s m of

Hence cl {U

1+1

^ Q i s a l s o a face of

- Q} = U. vJ P .

U, O P

with

U. ^ u s i n g

A similar arj L e m m a 2.1^

again.

P r o o f of T l ^ e o r e m 2. 11. B(0), E ( 0 ) , ditid U(0), Then

Let

By t h e p r e c e d i n g l e m m a , it suffices to e s t a b i

M be a z e r o - m a n i f o l d ,

X a polyhedron,

M is a finite s e t of p o i n t s and X i s a s u b s e t .

hood of X i s a l s o X, a s if

P ^ X,

c o l l a p s i b l e , it i s a s i n g l e p o i n t , so Re'mark,

X u {P}

X C

H e n c e a n y d e r i v e d neig

d o e s not c o l l a p s e to X .

If Xj

B{0) i s a l s o trivial,.

In t h e c o u r s e of p r o v i n g L e m m a 2. 1, we a l s o s h o w e d t h a t given ar

regular neighborhood

N^

of X in

m"^, t h e r e e x i s t s a s e q u e n c e of m - m a n i f

N = V D ., , D V 1 r o with meets

V^ a d e r i v e d n e i g h b o r h o o d of V

1-1

X and

in a face a n d a l s o m e e t s

8V

cl(V^ - V^ m a face„

a n d m - b a l l , which:

-65-

15.

U n i q u e n e s s of R e g u l a r N e i g h b o r h o o d s w h i c h M e e t t h e B o u n d a r y R e g u l a r l y In S e c t i o n 3 w e p r o v e d t h a t d e r i v e d n e i g h b o r h o o d s of a p o l y h e d r o n in a

^manifold a r e a m b i e n t i s o t o p i c .

In t h i s s e c t i o n w e e x t e n d t h i s r e s u l t to a l a r g e r

l a s s of r e g u l a r n e i g h b o r h o o d s . Definition . ^manifold

A regular neighborhood

M i s s a i d to m e e t t h e b o u n d a r y r e g u l a r l y if e i t h e r

: r e g u l a r n e i g h b o r h o o d of X n 8M in |Note;

N of t h e p o l y h e d r o n

M m e e t s the boundary r e g u l a r l y .

+ C K - K .

o

f: K — R

-1 is linear,

o

^^^

f

(O) = K^, and f(v) > 6

f'^[0, £ ] =

H K N 8 K = ^ , 9 K

^'derived n e i g h b o r h o o d of

N O 9M i s a

9M o r b o t h of t h e s e i n t e r s e c t i o n s a r e e m p t y .

A d e r i v e d n e i g h b o r h o o d of X i n

F o r suppose

X in t h e p. 1.

9K in

9K.

Otherwise

for a l l v e r t i c e s

BK N f ' ^ [ 0 , £ ] i s a

T h e u n i q u e n e s s of d e r i v e d n e i g h b o r h o d s

shows t h a t t h e r e s u l t h o l d s for a l l d e r i v e d n e i g h b o r h o o d s . T h e o r e m 2.1

If N

ledron X in t h e m a n i f o l d imbient isotopy throwing

and N

a r e two r e g u l a r n e i g h b o r h o o d s of t h e p o l y -

M which m e e t s

9M r e g u l a r l y , t h e n t h e r e e x i s t s an

N^ o n t o N^, fixed on

X,

N a t u r a l l y to p r o v e t h i s t h e o r e m we w i l l n e e d s o m e l e m m a s . L e m m a 2. 17. fenifold.

Let

Let

N C M be m-manifolds.

X C N be a polyhedron,

ll^se B n Frj^(N) 1) B C Int M

i s a face of

Suppose

N n 9M i s a n ( m - l )

B C N and m - b a l l ,

B O X =

Sup-

B and e i t h e r

or

2) B n 9M = B^ i s a face of

B and

B^rs F r j ^ ( N )

i s a fac e of

B. .

'^en t h e r e e x i s t s a n a m b i e n t i s o t o p y of M, t h r o w i n g N onto c l { N - B ) , w h i c h i s i s t a n t o u t s i d e a n m - b a l l c o n t a i n e d in M not m e e t i n g

X.

-66-

Pictures:. l ) BCM.

2)

dN

-67-

Proof. triangulate

F i r s t of a l l , c l ( M - N ) M with

1) X e M - N .

cl(N-B)

a r e manifolds.

Nannely,

N a s a s u b c o m p l e x and let x be a v e r t e x of

Then

M-N .

lk(x; M - N ) = link(x; M) = s p h e r e o r b a l l of d i m m - 1 .

2) X € ( F r N ) n (Int M), But lk(x; M)

and

Then

link(x; N) a n d

link(x; M ^ F ) = l i n k ( x , M) - l i n k { x ; N ) .

l i n k ( x ; M) i s an ( m - 1 ) s p h e r e .

H e n c e l i n k ( x ; N)

is an ( m - l ) b a l l and t h e c l o s u r e of t h e difference i s a n ( m - l ) b a l l . 3) X € 8M n F r ( N ) . F r ^ ( N ) = 8N - N since we a s s u m e d t h a t

aM

, w h i c h is a p . l .

N n 9M w a s , a n d b y l ) and 2).

Now link(x; M - N ) = l i n k ( x ; M) - link(x5 N)

and

Link(x; M ^ n N) = link(x5 M) - link(x5 N) n link(xi N). B^ = l i n k ( x ; N ) , b o t h ( n - l ) b a l l s . SB.HB^ 1 2

i s a face of

This proves Let F

B^ and 2

cl(M-N)

= B n Fr{N).

Then

B ^ - B ^ n B^

i s a face of B ^ .

F

cl{N-B)

=BOcl(N-B).

F

is a face of

'2 = cl(B - F ^ U B^), a n d w e s a w t h a t

F^ ^

^et

n X = 0 .

Let

^

B , for i n

and in c a s e 2 , ® ^^

para-

M w i t h F ^ , B^ = B H aM, F ^ , B, N, and X a s s u b c o m p l e x e

C = s e c o n d d e r i v e d n e i g h b o r h o o d of F ^

^ngulation.

So

is a m a n i f o l d by L e m m a 2 . 1 4 .

F ^ = cl(B - F^);

Triangulate

B^ = l i n k ( x ; M),

B , - B^ i s an n - b a l l , 1 2

is a manifold. Let

Let

: a s e 1) of t h e s t a t e m e n t of t h i s l e m m a

5raph.

(m-l)-manifold,

in

M - N , w i t h r e s p e c t to t h i s t r i -

D = s e c o n d d e r i v e d n e i g h b o r h o o d of F ^

in N - B .

Note t h a t

Since

F ^ and F ^

are collapsible,

u n i q u e n e s s p a r t of T h e o r e m 2. 11. is a n m - b a l l .

E = C

B

D n B = F

C and

D a r e m - b a l l s , by t h e

C n B = F ^ , a c o m m o n face,

, a c o m m o n face, C*

so

Co

^

so D u B i s a n m - b a l l .

D is a s e c o n d d e r i v e d n e i g h b o r h o o d of

B in

M a n d so is an

m-ball. Now we c o n s i d e r t h e two c a s e s of t h e s t a t e m e n t of t h i s l e m m a . 1) B c int M. We define

f: E —» E

a s follows.

Put h | E = identity.

Now C n (B u D) = C n F r ( c l ( M - N ) ) = C n ( 8 c l ( M - N ) ) , a s But F ^ c F r N

and

boundary regularly. (C

C is a d e r i v e d n e i g h b o r h o o d in c l ( M - N ) Hence

C

B) (\ D = D n a ( c l ( N - B ) )

(B * > D)

h a v e i d e n t i c a l b o u n d a r i e s , b o t h c o n t a i n e d in E ,

h^:

C n (B u D)

morphism

> (C u B) n D.

h • C —> (C J B)'

morphism

Now, fixed.

> E.

h^

h, t h i s d e f i n e s a p . l . hoi

h

h : (B u D ) ' —> D|

e x t e n d s to a p . l . homeo-

h ^ : B U D —> D.

Let

( T h e r e a d e r i s a d v i s e d to c o n s u l t P i c t u r e 1 on p a g e

h(B u D) = B .

l e v e l for p o i n t s o u t s i d e E . X fixed.

Hence

e x t e n d s to

Moreover,

h is a m b i e n t i s o t o p i c to

Extend this ambient isotopy over

and l e a v e s

homeomorphism

a n d a p. 1. h o m e o m o r p h i s m

h ^ : C —> C u B a n d

h = h ^ u h^ : E

M o r e o v e r , t h e s e two ba

H e n c e t h e r e s t r i c t i o n of h

Together with

w h i c h a g r e e w h e r e t h e y a r e b o t h defined.

a n d so m e e t s

is an ( m - l ) b a l l .

i s a l s o an ( m - l ) b a l l .

t h i s c o m m o n b o u n d a r y e x t e n d s to a p . l .

C O SM =

keeping 9

M by l e t t i n g it be t h e i d e n t i t y at eve:

The resulting ambient isotopy t h r o w s

N onto

cl(I^

-69-

2) B n a M / 0 . a r g u i n g a s in l ) h : E^

> E^

Let

C^ = C n 8M,

D^ = D n 8M, E ^ = E

9M.

By

(one l o w e r d i m e n s i o n ) , w e m a y find a p. 1. h o m e o m o r p h i s m such that

( R e c a l l : BM =

)

h | aE^ = i d e n t i t y , h(c^) = C^ U B ^ , h(D^U B^) = D ^ .

Define

h on F r E

by setting h | F r E = l .

Then as before,

h i s defined on (C H (B U D))', w h i c h it m a p s h o m e o m o p r h i c a l l y onto ((C u B) n D ) ' .

( T h e s e a r e not e q u a l . )

to a - p . l . h o m e o m o r p h i s m of E

Once a g a i n , t h i s definition e x t e n d s

which is the identity on F r ( E ) .

Now h

is

a m b i e n t i t o t o p i c to t h e i d e n t i t y v i a a n i s o t o p y fixed on F r ( E ) , b y a c o r o l l a r y to 2.7 which we did not s t a t e . Notes;

l)

E x t e n d t h i s i s o t o p y a s in l ) .

The unstated c o r o l l a r y is;

If A^

i s a p r i n c i p a l face of A = v A ^ ,

«

any h o m e o m o r p h i s m the i d e n t i t y k e e p i n g

h : A — > A w i t h h vA^ = i d e n t i t y , i s a m b i e n t i s o t o p i c to vA^

fixed.

This applies because

E S vE^.

2) T h e m - b a l l o u t s i d e w h i c h t h e i s o t o p y i s c o n s t a n t i s L e m m a 2. 18.

If X C Int m"^

a n d N^

a n d N^

E.

a r e two r e g u l a r n e i g h b o r -

^h oods of X w h i c h l i e in Int M, t h e n t h e r e e x i s t s an a m b i e n t i s o t o p y t h r o w i n g N^ onto

N^.

Proof.

In t h e p r o o f of T h e o r e m

H

( s e e l e m m a 2 . 1 4 a n d the r e m a r k on

page 4 ^ ) , w e s h o w e d t h a t t h e r e e x i s t s a s e q u e n c e of m - m a n i f o l d s , Nj = V D V . TD . . . ' ^ V with ^ r r-1 o .W

ith B. = c l ( V . - V . J 1 1 1-1 Int M.

^

V

o

and m - b a l l w h i c h m e e t s

B o ( 9 V ) = B. n F r V . . i l l 1

a m b i e n t i s o t o p y of

a d e r i v e d n e i g h b o r h o o d of X in M and

Hence L e m m a 2.1

M, fixed o n X , t h r o w i n g

a-mbient i s o t o p i c to a d e r i v e d n e i g h b o r h o o d . P-re ambient isotopic.

V. . and 1-1

9V. in f a c e s . 1

Since

applies: there exists

V^ onto V^

Hence

N^

is

So i s N ^ , a n d d e r i v e d n e i g h b o r h o o d s

L e m m a 2. 19. of X r^ 8M in throwing

N^

Proof.

If X 9 M

a r e r e g u l a r neighborhood

9M, t h e n t h e r e e x i s t s a n a m b i e n t i s o t o p y of onto

Let

\ triangulation.

M, fixed on X-

N^ . M be t r i a n g u l a t e d w i t h N^

^^^

X n aM.

, and N^ and N^

Then

^o ~

a n d X a s s u b c o m p l e x e s with a

d e r i v e d n e i g h b o r h o o d of X w i t h r e s p e c t to thia

U^n BM i s a s e c o n d d e r i v e d n e i g h b o r h o o d in

9M of

We s a w in the proof of T h e o r e m 2. 11 ( s e e L e m m a 2. 15) t h a t in 91

t h e r e e x i s t s a c o l l e c t i o n of ( m - l ) m a n i f o l d s cl(V. -

is a ball meeting

9V. = Fr^^ V. , 1 oM 1

Therefore,

V. ^ and

Lemma 2.1

N = V 1 r ~ a(V.) in f a c e s .

As

a p p l i e s to e a c h p a i r

give an a m b i e n t i s o t o p y t h r o w i n g

V^ onto V^

b a l l in

X.

9M w h i c h d o e s not m e e t

=U o o 8(aM) = V.

,

V. , toJ 1 1 - 1

c o n s t a n t o u t s i d e of an ( m - l ) -

Call this ambient isotopy

b e t h e b a l l o u t s i d e of w h i c h it i s c o n s t a n t

such

(may take

H. , and let

Ej

E . = 2nd d e r i v e d neighboj

hood of

of

c l ( V . - V . J i n 9M). E . H X = Cf. 1 1-1 1 ^ Now t r i a n g u l a t e M w i t h X and E a s s u b c o m p l e x e s . L e t F . = 2nd d e r i l i 1 E . in M. F . n X = 0 . We e x t e n d H. to F . a s follows; P u t H. = ident 1 X ^ 1 1 1

on ^ r j ^ ( F ^ )

and e x t e n d

H^ and

H over

F.

S e c t i o n 3, L e m m a 2.7 and C o r o l l a r y 2. 8 . ) . M X 1.

T h i s d e f i n e s an a m b i e n t i s o t o p y of

a n d F . X I in t h e u s u a l w a y Now put

Similarly, X also.

N But

Ct

is a m b i e n t i s o t o p i c to

U^

U'ri o

is a m b i e n t i s o t o p i c to

H^ = i d e n t i t y on the r e s t

M throwing

p o s i n g t h e s e i s o t o p i e s d e f i n e s an i s o t o p y t h r o w i n g 9M,

N^

(see

V.

onto

onto U^

V.

Com-

9M, fixed on

U ' a d e r i v e d n e i g h b o r h o o d of o

U^, a n d a n y a m b i e n t i s o t o p y t h r o w i n g

-71-

onto

U' m u s t t h r o w o

U n 8M onto o

U' O 8Mp a s p . l . h o m e o m o r p h i s m s of o

manifolds p r e s e r v e boundary.

E. 1

SM

L e m m a 2. 20.

If N i s a r e g u l a r n e i g h b o r h o o d of X i n

M a n d if N

meets

)m r e g u l a r l y , t h e n N ^ X O (N n a m ) \ j X . Proof.

F i r s t s u p p o s e t h a t N i s a d e r i v e d n e i g h b o r h o o d of X , i. e . , ;K')| ,

iubcomplex.

Let

where

K^ ^ K i s a t r i a n g u l a t i o n of X C M w i t h be t h e s i m p l i c e s of

K-K^

which m e e t

K^ K^,

a full ordered

a s to s a t i s f y t h e following tow p r o p e r t i e s : o a) S i m p l i c e s of K

• p r e c e e d t h o s e of K .

b) A. p r e c e e d s i t s f a c e s , n | K I = B . , a s i n g l e face of A . . A. n N = | N ( B : ; Al) , a b a l l . O 1 ° 1 1 X I n N = N(BJ; a : ) , a face of t h i s b a l l . Hence X U. = K U { U ( N O A )} \ | K J U { U ( N n A )} = U.^^ , 1 o j=l '' j=i+l

-72r

,

as j=l

r

(N 0 A.) 1 (N p| A.) J ^ j=i+l J

a n d by L e m m a 2. 1,

d i v i s i o n in w h i c h t h e c o l l a p s e s a r e s i m p l i c i a l ) . K

C ( [J j=l

r\

p o i n t of K Now, U

r

^

L e m m a 2. 1 a p p l i e s b e c a u s e

X

( N n A )) = ^

|K

A.) C j K ^

j=l

is c o n t a i n e d in A . , s o m e J

U^ = N.

JL

U

In (

( a p p l i e d to a s u b -

In

(

U

A ), j=t+i ^

for if a

j, it i s c o n t a i n e d in a p r o p e r face of

C l e a r l y , t h e r e e x i s t s an i

such that

U^ = X u (N O 8M),

A., J

b y a)]

= X. Now s u p p o s e t h a t

regularly. Claim;

Then

N is a r e g u l a r n e i g h b o r h o o d of X w h i c h m e e t s the bour

N n DM i s a r e g u l a r n e i g h b o r h o o d of X n 9M in

N H 8M C 8N i s a r e g u l a r n e i g h b o r h o o d of X n 8N in

N n a M i s a n e i g h b o r h o o d of X H 8N i n

8N

w h i c h c o l l a p s e s to L e t N^ regularly.

9N.

X n F r N = ^ a n d N n SM is

N r\ 9M i s a n ( m - l ) m a n i f o l d

X n 9M = X O 9N,

b e a d e r i v e d n e i g h b o r h o o d of

X in M .

Then

Now, t h e r e e x i s t s a p . 1. h o m e o m o r p h i s m

h | X = i d e n t i t y and h(N) = N ^ .

in

i s o t o p y of N^, fixed on X, t h r o w i n g there exists a p.l, homeomorphism s u c h t h a t h'(Nry 9M) = N^ H 9 M . X,

since

N^

h: N

meets > N^

9M

such that

M o r e o v e r , h(N H 9M) and N^ O 9M a r e b o t h

r e g u l a r n e i g h b o r h o o d s of X r j 9N^

N ^ X U (N PI 9M)

SN.

because

X n 9N = (X O F r N ) u ( X n N o 3M) = X O 9M a s o b v i o u s l y a n e i g h b o r h o o d of X A 9M in

9M.

Hence t h e r e e x i s t s an ambient h(NA9M)

h'

But

o n N^ O 9Mo

of N onto

N^ w i t h

In p a r t i c u l a r , h ' | X = identity,

N^ ^ X U (N^ n 9M) ^ X .

(h') ^ p r e s e r v e s c o l l a p s e s .

Hence

-73-

P r o o f of T h e o r e m Z.16.

We a r e going to show t h a t any r e g u l a r n e i g h b o r -

hood w h i c h m e e t s t h e b o u n d a r y r e g u l a r l y i s a m b i e n t i s o t o p i c to a d e r i v e d n e i g h borhood.

Since d e r i v e d n e i g h b o r h o o d s a r e ambient isotopic, this will prove 2 . 1 6 .

So l e t N b e a r e g u l a r n e i g h b o r h o o d of X i n M m e e t i n g Then N ^ X u

(9M A N ) ^ X .

Let

K b e a t r i a n g u l a . t i o n of

N a r e t r i a n g u l a t e d a s s u b c o m p l e x e s , K^ and L , s a y . \s . \s vss es L ^ K^ U ( L n K) \ K^ . L e t L = K^ ^ . . . ^ with K

= K u ( L n K), s o m e o

s

d e r i v e d of that

K.

Let

s ^ r.

K^ = K^ ^ + A + B,

U. = U. ^ U P 0 Q, w h e r e 1

Let

M such that

and

t h e s e two c o l l a p s e s , K" =

2nd

T h e n we h a v e seen. ( L e m m a 2 . 1 5 )

U^ S U^ ^ U P = U^

• L e m m a 2 . 1 6 to s h o w t h a t i n fact

We a r e going to u s e

U. i s a m b i e n t i s o t o p i c to

I

U

1

^O P

X

We m a y s u p p o s e t h a t

U. = N(K'! ; K " ) , w h e r e 1 1 r

A = aB.

regularly.

P = •st~(A;K"). Q = s t a r ( ^ ; K " ) , and that t h e r e

1-1

e x i s t s a p . 1. h o m e o m o r p h i s m

8M

is a m b i e n t isotopic to

P

and

1-1

U^ , k e e p i n g

X fixed.

T h i s w i l l c o m p l e t e the

"proof. Either and

A and

B a r e both in

Q b o t h do not m e e t

9M.

ad we h a v e s e e n ( p a g e ^^ ) t h a t

9M o r n e i t h e r i s in In t h i s c a s e ,

Motopies t h r o w i n g

U^ onto

U^ ^ U P

S u p p o s e on t h e o t h e r h a n d t h a t ftar{A;K") n K" = s t a r ( A j K " ) a face.

We s t i l l h a v e t h a t

F r U^ i s a face of Q.

Q.

In t h e l a t t e r c a s e ,

P nFr(U._^U

P r\ 9(U^ ^ 'J P )

n F r ( U . ) = Q n 9(U.) i s a face of

9M.

P) = P n

i s a face of P .

P),

Similarly,

H e n c e b y L e m m a 2. 16, t h e r e a r e a m b i e n t and

A and

U^ i

^

^i*

B a r e both in

a n d s i m i l a r l y for P a Fr(U. ^ O P)

so P

9M, and

is a face of P ,

Then Q each meets and

H e n c e in o r d e r to c o n c l u d e t h e p r o o f by a p p l y i n g

9M

L e m m a 2 . 1 6 , w e m u s t show t h a t (Q n 9M) r\ F r U . Now,

a r e f a c e s of

N O 8M =

( P H 9M) n F r ( U ^ ^ ^

P r\ a M a n d

Q ndM,

and respectively.

K" I i s a r e g u l a r n e i g h b o r h o o d of | K^ .

Moreover,

>es \

.es

. K^oK.

K"

(We a r e a s s u m i n j •

h e r e that

i < s.)

Clearly, we have that

N((K^A

K)").

•

U. O 8 M = N(K:' n K" ; K " O K" ) =

A l s o , we j u s t n o t e d t h a t

P O 8M = s t a r ( A ; K ") an<

A •

/

Q ry 9M = s t a r ( B ; K " ) . a p p l y in

H e n c e , t h e a r g u m e n t s of L e m m a 2 . 1 5 ( s e e p a g e ( 3 )i

SM to show t h a t

( P n BM) n 8[(U^ ^ U P ) n BM]

(Q r\ 8M) n 9(U. PI 9M) a r e f a c e s of

P r\ 9M a n d

9[(U. ^ U P ) o 9M] = [ F r ( U . ^ u P ) ] n 9M, Thus

P ri 9M A F r ( U ^

face of

^

P)

i s a face of

and

and

Q n 9M, r e s p e c t i v e l y .

Bu

9(U. H 8M) = ( F r U ^ D 9M.

P O 9M a n d

Q

8M n F r ( U p

is

Q o 9M.

C o r o l l a r y 2.16. 1 (Annulus P r o p e r t y ) :

Say

B^ ^ Int B^,

B^ a n d B^ p . l j

e

m-balls.

Then

cl(B

- B ) i s p . l . h o m e o m o r p h i c to

B X I

C o r o l l a r y 2. 16. 2. ( G e n e r a l i z e d A n n u l u s P r o p e r t y ) : r e g u l a r n e i g h b o r h o o d s of X in

M with

regularly, then there exists a p . l . h: cUN^ - N^) Proof.

K | = M, w i t h are

and

^ ^ ^ ^^ ^ ^

N^ a r e |

meets

9MS

homeomorphism > (^^M^l^ ^ ^ •

C l e a r l y , 2. 16. 2 i m p l i e s 2. 1 6 . 1 , s i n c e a b a l l i s a r e g u l a r neighboj

h o o d of a n y i n t e r i o r p o i n t .

[0,1]

If N^

= X.

0 and 1) w i t h

To p r o v e 2. 16. 2, l e t Let

K^ be a full s u b c o m p l e x of

K — ^ [O, l ] b e a s i m p l i c i a l m a p = K^.

Choose

(vertices

T h e n b y 2. H . j

-75I

-1

[ t h e r e e x i s t s a p . l . h o m e o m o r p h i s m h : N^—S> ff [0,t^], h j K ^ = identity. -1 -1 N o w , h(N^) a n d ^ [0, f ^ ] a r e r e g u l a r n e i g h b o r h o o d s of X i n ^ [which

m e e t t h e b o u n d a r y r e g u l a r l y (in fact, is a derived neighborhood).

[ambient isotopic", CI

t^] 1

Addendum 2. 16. 3. leeting

atopy of f; P r o o f .

P C M -

k(h(N^)) = f \ o ,

a

S FrN h-^k-^Xl

(N^ U N^)

be a p o l y h e d r o n .

M , fixed o n P U N ^ , t h r o w i n g 2. 1 7 . 2 i m p l i e s

So XI.

N^

Then t h e r e e x i s t s an ambient

onto

N^.

c l ( N ^ - N ^ ) S ( F r ^ ^ N ^ X I).

:e N^ ^ N^ ( L e m m a 2 . 1 ) .

Similarly,

N^

N^ b e a s e c o n d d e r i v e d n e i g h b o r h o o d of r e g u l a r n e i g h b o r h o o d s of 1.2.)

homeomorphism

m N ,N ,N b e r e g u l a r n e i g h b o r h o o d s of X in M J> ^ O S u p p o s e N and N a r e (topological) neighborhoods I t

Let

9M r e g u l a r l y . Let

BN^ r e g u l a r l y and

H e n c e t h e s e two n e i g h b o r h o o d s a r e

such that

£ ] ^

f

N^.

meets

in p a r t i c i a l a r , t h e r e e x i s t s a p . l .

e^] —>

:1{N,-N ) 6 1

N^

N^ U P

meeting

Hence

i s a r e g u l a r n e i g h b o r h o o d of P.

Then 9M

N^ U N ^

regularly.

Hence t h e r e e x i s t s an ambient isotopy throwing

N^, k e e p i n g

N^ U P

fixed.

cl(N^-N^)FrN^. N^.

a n d N ^ "J N^

(N. O N^ = ^ ,

N^ U N^

onto

S i n c e a p . l . h o m e o m o r p h i s m i s c o n t i n u o u s and

i'PS c o n n e c t e d c o m p o n e n t s o n t o c o n n e c t e d c o m p o n e n t s , it follows t h a t t h i s t isotopy throws

N^

onto N^.

-76 C h a p t e r III - - P . L . S p a c e s and Infinite C o m p l e x e s 1.

Introduction. C h a p t e r s I a n d II h a v e b e e n c o n f i n e d to t h e s t u d y of c o m p a c t p o l y h e d r a a

p . l . manifolds contained in given E u c l i d e a n s p a c e s .

As in Differential

w h e r e one c a n i n t r o d u c e a b s t r a c t m a n i f o l d s , one c a n define P . L ,

Topold|

s p a c e s and

manifolds without r e f e r e n c e to an a m b i e n t E u c l i d e a n s p a c e and without the h y p o t h e s e s of c o m p a c t n e s s .

In t h i s c h a p t e r we p r o p o s e t o s t u d y a b s t r a c t P . L |

s p a c e s a n d m a n i f o l d s a n d to i n d i c a t e how to e x t e n d t h e p r e c e d i n g r e s u l t s to sv objects. One c a n a l s o define t h e n o t i o n of'a l o c a l l y finite infinite c o m p l e x containec in a given E u c l i d e a n s p a c e (possibly E°°).

We w i l l s h o w t h a t t h e n o t i o n s of P .

s p a c e a n d infinite c o m p l e x a r e e s s e n t i a l l y e q u i v a l e n t .

In p a r t i c u l a r ,

compact|

P . L . s p a c e s a n d m a n i f o l d s a r e no m o r e g e n e r a l t h a n t h e finite p o l y h e d r a and; p . l . manifolds which we have been considering.

2.

T r i a n g u l a t i o n of P . L . S p a c e s and M a n i f o l d s . Definitio n,

Let

X be a topological space,

a topological embedding maps

(f, P )

A co-ordinate map

f: P —> X of a E u c l i d e a n p o l y h e d r o n

P.

(f, P ) iSj Two such -

a n d (g, Q) a r e c o m p a t i b l e p r o v i d e d t h a t if f(P) r\ g(Q) i ^

there|

-1

exists a coordinate map are

p.l. maps.

(h, R)

such that

Equivalently, we say that

-1

f

h(R) = g(Q) O f(P) (f, P )

and

f

h and g

a n d (g, Q) a r e c o m p a t i b l e

-1

(gQ) i s a s u b p o l y h e d r o n of

(Put h = g|f"^gQ),

assuming

Q and

g

f: f

f(P) (\ g(Q) = j^.

(gQ)

> Q is a p . l .

map.

-77-

Definition .

A P. L. structur e

^

o n X i s a f a m i l y of c o o r d i n a t e m a p s

such that 1) Any tv/o e l e m e n t s of

^

are compatible.

2) F o r a l l x e X, t h e r e e x i s t s n e i g h b o r h o o d of x in 3)

^

(f, P ) e ^

such that

f(P)

is a topological

X.

i s m a x i m a l , i . e . , if

(f, P )

i s c o m p a t i b l e w i t h e v e r y m a p of

^

,

then If X i s a Z^^ c o \ i n t a b l e H a u s d o r f f s p a c e , t h e p a i r P. L.

(X, ^

) is called a

space.

Definition .

A f a m i l y of c o o r d i n a t e m a p s

:alled a b a s e for a P . L . s t r u c t u r e o n Lemma 3.1.

Every base

^

space X i s c o n t a i n e d i n a u n i q u e Proof. is

.

Let

^

g(Q) i

such that

s a t i s f y i n g 1) and 2)

is

X.

for a P . L . s t r u c t u r e on t h e t o p o l o g i c a l P . L, structure

^

are compatible.

S' •

F o r if

(f, P )

w e m a y find a finite c o l l e c t i o n f(P) O g(Q) C

IV

ipatible w i t h e a c h

on X

= t h e s e t of a l l c o o r d i n a t e m a p s in X c o m p a t i b l e w i t h t h o s e

T h e e l e m e n t s of

and f{P) n i n a p s in

^

^

U

h.(B.). 1

h . , so if w e l e t

RJ = h . ' ^ f P

and

(g, Q)

are

(h^, B^), . . . , (h^, B^)

By d e f i n i t i o n

f and

g

are

1

and

R." = h J ^ g Q ,

R^ and P-^'

| 8 u b p o l y h e d r a of A B . . L e t^ R^'' =• RJ H _r !1' . >!< T h e n U h . R [ = f(P) O g(Q).- 1 S'efore, P ^ = f ' (gQ) = f" (Uh^R^ ) = Uf h^R^ i s a p o l y h e d r o n , and g f -1 * ^ P ^ b e c a u s e in e a c h p i e c e f h^R^ it a g r e e s w i t h t h e p. L m a p

is

-78-1

g

-1

h^h^

f w h i c h a l s o i s defined o n t h i s p i e c e .

J^

It i s c l e a r t h a t

satisfies!

2) and 3) i n t h e definition of a P . L , s p a c e and i s t h e u n i q u e s t r u c t u r e containing ^

.

L e m m a 3. 2. maps,

If

f: P —5> X a n d

g: Q —» X a r e two c o m p a t i b l e c o o r t

X a topological space, then there exists -1

with

h(R) = f(P) u g(Q) a n d w i t h Proof.

Let

|k| = P

and

b e s u b d i v i s i o n s of K

and o

> X, a coordinate

-1

f and h

g p.l.

maps.

| l | = Q be triangulations with -

subcomplexes, triangulating

h

h: R

f L

1 1 gQ and g such that

o

fP

respectively.

g'^f; °

K' — L ' o o

K^

and

L^,

Let

I K^

and

is simplicial.

L«

N K'

and

L'

b e e x t e n s i o n s of t h e s e s u b d i v i s i o n s .

which h a s one v e r t e x

j(v) for e a c h v e r t e x v of

for e a c h v e r t e x v of K ' , a n d no o t h e r s . i:K'

L ' - L^

L'.

be a simplex

and one v e r t e x

i{v|

i a l r e a d y g i v e n o n v e r t i c e s and'

j : L ' —> A defined b y p u t t i n g j(v) = i(f

e x t e n d i n g l i n e a r l y to a l l of Let

A C F

Consider the simplicial homeomo]

> A d e t e r m i n e d b y t h e d e f i n i t i o n for

homeomorphism

Let

g(v))

if v e L ^

( j i s a l r e a d y defined on v e r t i c e s of

L ' - L^

R be t h e u n i o n of t h e i m a g e s of t h e s e s i m p l i c i a l h o m e o m o r p h i s m s ,

a simplicial complex. h(x) a

Define

fi"^(x)

h(x) = g o j

h: R

> X b y defining

if X e I m a g e

(x) if X e I m a g e

i. j..

T h e d e f i n i t i o n s a g r e e on t h e o v e r l a p , s i n c e if x 6 ( i m i) n I m (j) g o j " ^ ( x ) = g g " ^ f i ' ^ ( x ) = f i " ^ ( x ) . It i s not h a r d to s e e t h a t h : R —5> X i s a hoi -1

m o r p h i s m with image

f(P) o g(Q) , a n d t h a t

h

-1

f and h

g a r e p. 1. m a p s .

-79-

Corollary 3.3. there exists Proof.

If

(X, ^

(h, R) e ^ Let

) i s a P . L. space and C C X

with

then

C C Int h ( R ) .

(h^, R^), . . , , (h^, R^)

There exists a coordinate map with h c o m p a t i b l e w i t h e a c h

is c o m p a c t ,

be in

with

h ; R—5> X w i t h h.

(i.e., h

C C Int(h^(R^) U . . . U h^(R^)),

h(R) = h ^ ( R ^ ) u . . . ^ h ^ ( R ^ ) ,

h . : R.

>R

a r g u i n g a s in L e m m a 3 . 1 , it i s n o t h a r d to s h o w t h a t

i s p. 1. , a l l

i).

and

By-

h is compatible with e v e r y

element of ^ and so i n ^ . Definition . The P . L. space :for a l l x e X t h e r e e x i s t s Lemma 3.4. |then there exists

If

(X, ^

(h, R) e ^

(X, ^

) i s c a l l e d a P . L . m - m a n i f o l d if

h: a " ^ —> X w i t h

{h, A™) c ^

and x e I n t ^ h(A"^),

) i s a P . L . m - m a n i f o l d and C C X i s c o m p a c t , with

d) R i s a p . 1. m - m a n i f o l d , 2)

CClnt^h(R).

Proof, "

By L e m m a 3. 2, c h o o s e

p ) C Int g(Q).

Let

(f, P ) a n d

K^ b e a full s u b c o m p l e x of

et N b e t h e s e c o n d d e r i v e d n e i g h b o r h o o d of •manifold, for t h o u g h

|K|

of of L e m m a 2. 13).

in K.

with

= Q,

K^

Then

w h i c h i s a p. 1. m - b a l l .

i l s p h e r e o r b a l l for a l l v e K ^ , a n d A meeting

K^

K, | K

^

K n e e d n o t be a c o m b i n a t o r i a l m a n i f o l d ,

h a s a n e i g h b o r h o o d in

Splices

(g, Q) i n

K^.

C C Int f(P), = g

-1

fP.

N i s an e v e r y point

So l i n k ( v , K) = a n

link(A, K) is a s p h e r e o r b a l l for a l l

So t h e p r o o f t h a t

N i s a manifold g o e s t h r o u g h

(see

g | N —3> X i s t h e r e q u i r e d c o o r d i n a t e m a p of t h i s l e m m a .

-801

Note;

S t r i c t l y s p e a k i n g , t h e l a s t two l e m m a s h a v e b e e n u s i n g t h e fact t h a t

if in t h e P . L . s p a c e

(X, ^ ),

(h, P )

is a coordinate m a p such that

b e c o v e r e d by t h e i m a g e s of a finite n u m b e r of m a p s in ^ compatible, then

(h, P ) e ^

.

h(P)

with which h isj

T h e p r o o f i s left to t h e r e a d e r

(see Lemma

T h e n e x t l e m m a m a y be v i e w e d a s a f f i r m i n g t h e p o s s i b i l i t y of " t r i a n g u l a P . L , s p a c e s and m a n i f o l d s ,

a s we s h a l l s e e following t h e i n t r o d u c t i o n of loca

finite (infinite) c o m p l e x e s . Lemma 3.5.

Let

(X,

be a P . L . s p a c e .

s e t of s i m p l i c i a l c o m p l e x e s a n d s u b c o m p l e x e s , f. :

J,

—> X

T h e n t h e r e e x i s t s a count

K^

J^ , L^ C J^ a n d embedc

such that

1) x = U f.(|jJ). i=l 2) £ . ( | J j ) n y | J j . l ) = ^ if

4) f.

, f.:

1+1 1

If

L.

> K.. .

1

| i - k | a 2.

is a simplicial homeomorphism,

1+1

(X,

is a P . L . m - m a n i f o l d , w e c a n t a k e J , to b e a c o m b i n a t o r i £ m - m a n i f o l d and K. and L . to be c o m b i n a t o r i a l ( m - l ) m a n i f o l d s in 9 J . . 1 1 1 Proof. X is l o c a l l y c o m p a c t a n d oo Let X = C. , C. c o m p a c t . L e t i =l (h., R p 6 J

, i ^ 2,

such that

-1

( F r h.R.). 1 1

Let

e J" .

C.C

P . = c l ( R . - h." h. , R . ) , a p o l y h e d r o n . 1 1 1-1 1 ^ ' S. = h . 1 1

countable.

f. = h. P . . 1 1 1

Hence

Define i n d u c t i v e l y

C Int h.(R^). Let Let

X i s (r-compai

Let

-1. Q. = h. F r „ (h. , R . , ) , 1 1 X 1-1 1-1

K.,L. J , b e t r i a n g u l a t i o n s of 1 1 1

-81-

Q.S., 1

P, . 1

F o r e a c h i, let

LI a n d K1. . b e s u b d i v i s i o n s s u c h t h a t 1 1+1

i " ^ f.: LI > KI, ^ i s s i m p l i c i a L S i n c e K. O L . = t h i s defines a subi+1 1 1 1+1 1 1 d i v i s i o n of K^ U L^ w h i c h w e e x t e n d t o a s u b d i v i s i o n JI^ of J . T h e n J ' KI, LI a n d f. s a t i s f y t h e f i r s t p a r t of t h e l e m m a , i' 1 1 1 T h e p r o o f of t h e s e c o n d p a r t of t h e l e m m a i s s i m i l a r , u s i n g L e m m a 3 . 4 i n s t e a d of L e m m a 3. 2.

T h e d e t a i l s a r e left t o t h e r e a d e r .

To mcike t h e n o t i o n of a t r i a n g u l a t i o n of a P . L . s p a c e m o r e p r e c i s e , we i n t r o d u c e infinite c o m p l e x e s . (x . . . . . X ) w i t h * 1' ' n

(x., . . . , X ,0). 1 n

convex h u l l of a s u b s e t ; s e t of E ^ i

.

Let

fas aU ( o o ) - t u p l e s

F i r s t of a l l , w e v i e w E ^ C E ^ ^ ^

Note that u n d e r t h e s e identifications,

S of e " i s t h e s a m e a s i t s c o n v e x h u l l v i e w e d a s a s u b co = E^ , w i t h t h e w e a k t o p o l o g y . E ° ° m a y be v i e w e d i=l

E°°

m a y b e v i e w e d a s t h e t o p o l o g y of p o i n t w i s e c o n v e r g e n c e .

I The c o n v e x h u l l of a n y s u b s e t of

i s defined in t h e o b v i o u s w a y .

w e d e n o t e t h e c o n v e x h u l l of t h e p o i n t s

[(0,0,1,0,...), Definition .

the

(x^, . . . , x ^ , . . . ) w i t h a l l b u t a finite n u m b e r of x^ b e i n g z e r o ,

^and t h e t o p o l o g y of

ay

b y identifying

( 1 , 0 , . . . ),

In p a r t i c u l a r ,

(0,1,0,...),

etc. A l o c a l l y finite s i m p l i c i a l c o m p l e x

K in E°°

i s a c o l l e c t i o n of

|finite) s i m p l i c e s , K, s u c h t h a t 1) (r,T € K = > 2)

0-

€ K,


T < 0- = >

T

€ K.

3) F o r a l l X € | k | , t h e r e e x i s t s a n e i g h b o r h o o d l y f i n i t e l y m a n y s i m p l i c e s of K . K l i e s in s o m e

E^.)

u

of x in

meeting

( E x e r c i s e : P r o v e t h a t e v e r y finite subcompley:

Let

(X.

b e a P . Li. s p a c e .

U s i n g L e m m a 3. 5, a n d t h e t e c h n i q u e of

L e m m a 3 . 2 one c a n c o n s t r u c t an infinite l o c a l l y finite c o m p l e x

K

v e r t i c e s a r e v e r t i c e s of

> X of

and a h o m e o m o r p h i s m

h: | k |

whose |K| I

onto X s u c h t h a t t h e r e s t r i c t i o n s of h to finite s u b c o m p l e x e s a r e e l e m e n t s of ^ K

.

M o r e o v e r , if

is also;

c o n t a i n e d in

(X,

) i s a P . L . m - m a n i f o l d , t h e n we m a y i n s i s t tha

t h a t i s , e v e r y p o i n t of | K| .

K|

l i e s in t h e i n t e r i o r of a p . l . m-be

In t h e c a s e t h a t t h e r e i s a bound

on t h e d i m e n s i o n s of the

N s i m p l e x e s of L e m m a 3. 5, one c a n t a k e the complex

K CE

for s o m e finite

N.

In t h i s

K i s c o n s t r u c t e d w i t h i n a s u i t a b l e E u c l i d e a n s p a c e b y " b a r e han2

u s i n g t h e i n s t r u c t i o n s p r o v i d e d by L e m m a 3. 5. Definition .

The p a i r

(K,h)

l o c a l l y finite c o m p l e x ajid h : | K |

D e t a i l s a r e left to t h e r e a d e :

i s c a l l e d a t r i a n g u l a t i o n of

if

K is

> X i s a h o m e o m o r p h i s m s u c h t h a t the

r e s t r i c t i o n s of h to finite s u b c o m p l e x e s a r e e l e m e n t s of ^

.

-83-

3.

P . L). M a p s a n d S u b i d i v i s i o n T h e o r e m s Definition .

IS

Let

(X, ^

) and

c a l l e d a P . L . m a p if for a l l

(Y,

be P . L . s p a c e s .

(f, P ) e ^

e i t h e r e m p t y o r a s u b p o l y h e d r o n of

T h e n jZ(t X —» Y

and all

P , a n d if t h e l a t t e r ,

^ o f : f^f'^gQ

f'^^'^gQ

is

then

> Q

is, a p . 1. m a p . Notes;

l)

It i s e a s y to c h e c k t h a t a P . L . m a p i s c o n t i n u o u s .

2) B y a n a r g u m e n t s i m i l a r t o t h a t of L e m m a 3 . 1 , to s h o w t h a t a g i v e n m a p j i i s a P . L . m a p , it suffices t o c h e c k t h e c o n d i t i o n in the d e f i n i t i o n for e l e m e n t s (f, P ) of a b a s e of ^ Definition .

and e l e m e n t s

If ^ | K |

>

(g, Q) of a b a s e of h . ^

^

l o c a l l y finite s i m p l i c i a l c o m p l e x e s ,

/e s a y ^ i s P . L . if it m a p s e a c h finite s u b c o m p l e x p i e c e w i s e l i n e a r l y into a finite s u b c o m p l e x of Umark. id (X, ^

L,

T h e two d e f i n i t i o n s of P . L . m a p a r e c o n s i s t e n t . ) a r e P . L . s p a c e s a n d if

and Y r e s p e c t i v e l y , a n d if 0 and ttommutes

(X,:?)

(K, h) a n d ( L j j ) a r e t r i a n g u l a t i o n s of i^i a r e m a p s s u c h t h a t t h e following d i a g r a m

: X h IK

T h a t i s , if

A

L

0 i s a P . L . m a p if and o n l y if ijj i s a P . L . m a p .

Definition .

A map

f; X —> Y of t o p o l o g i c a l s p a c e s i s said, to b e a prop]

m a p if t h e i n v e r s e i m a g e s of c o m p a c t s e t s in Y a r e c o m p a c t . Definition .

A subdivision

K'

of a l o c a l l y finite c o m p l e x

K i s a loccdlyl

finite s i m p l i c i a l c o m p l e x s u c h t h a t K' 1) IK 2) E v e r y s i m p l e x of

K i s c o n t a i n e d in a s i m p l e x of

U s i n g L e m m a 1. 2 a n d l o c a l f i n i t e n e s s , of K i s a u n i o n of f i n i t e l y m a n y s i m p l i c e s of d i v i s i o n of K, t h e n K' s u b c o m p l e x of

K'.

M o r e o v e r , if

K'

i s a sub|

i n d u c e s a s u b d i v i s i o n ( i n t h e finite s e n s e ) of e v e r y

A.

If S i s a l o c a l l y finite f a m i l y of p o l y h e d r a

then there exists a subdivision

B.

i t i s e a s y t o s e e t h a t e v e r y simpt

K.

Theorem 3.6.

e a c h e l e m e n t of

K'.

K'

of

in

|K

K c o n t a i n i n g (finite) t r i a n g u l a t i o n s of

S.

If f: K —5> L i s a P . L . m a p of l o c a l l y finite c o m p l e x e s , t h e n t h e r e

exists a subdivision

K'

of K s u c h t h a t

h K' —> L

maps simplices linearl

into s i m p l i c e s . C. L'

If f: K — j > L

w i t h f; K' Proof .

A)

i s p r o p e r P . L . m a p , t h e n t h e r e e x i s t . s u b d i v i s i o n s K'j

> L'

simplicial.

Write

oo K = IJ i=l

'

s u b c o m p l e x e s , K. n K^ = ^ if

i-j| ^ 2 .

F o r e x a m p l e , if K i s c o n n e c t e d , l e t

and define

R^ = c l o s e d s i m p l i c i a l n e i g h b o r h o o d s of

R^ R^

b e a finite subcomplea for e a c h i .

Let

-85-

K = R . - R. . . i 1 1-1 to a v e r t e x of Each

The

R^

R. c o v e r 1

K b e c a u s e a n y v e r t e x of

K can be connected

b y a finite e d g e p a t h .

K j m e e t s f i n i t e l y o n l y f i n i t e l y m a n y p o l y h e d r a i n S.

induction subdividing

P r o c e e d by

K^ to c o n t a i n s u b d i v i s i o n s of i t s i n t e r s e c t i o n s w i t h m e m -

b e r s of S a n d w i t h t h e p r e c e d i n g s u b d i v i s i o n of K^

Then since

K^ i s not

changed after the ( i + l ) s t s t e p is o v e r , it is c l e a r that this defines the r e q u i r e d s u b d i v i s i o n of B), of K.

K,

S* = {(T n Let

K'

f"^(T)| or 6 K, T € L.}

b e a s u b d i v i s i o n of

i s a l o c a l l y finite s e t of p o l y h e d r a

K c o n t a i n i n g s u b d i v i s i o n s of t h e e l e m e n t s

of S. C), roper,

We m a y a s s u m e b y B t h a t { f(r | c 6 K}

f i s linea:r

n s i m p l i c e s of K.

i s a l o c a l l y finite f a m i l y of p o l y h e d r a in

Lve t h e s e p o l y h e d r a a s s u b c o m p l e x e s . ically finite c e l l s u b d i v i s i o n of

K.

Then

{(TAf

- 1

T|(r€K,

Let »

t € L'>

is L'

is a

A s i n t h e finite c a s e t h i s c e l l s u b d i v i s i o n

.8 a l o c a l l y finite s i m p l i c i a l s u b d i v i s i o n w i t h n o e x t r a v e r t i c e s .

[arning ;

|L| .

As f

C) i s f a l s e f o r n o n - p r o p e r m a p s .

le r e a l l i n e w i t h v e r t i c e s at t h e i n t e g e r s .

F o r example,

l o c a l l y finite s u b d i v i s i o n s to m a k e

f

triangulate

T h e r e is a P L m a p

ipping R h o m e o m o r p h i c a l l y o n t o t h e o p e n i n t e r v a l (O, 1). simplicial.

(See L e m m a 1 . 4 ) .

f: R — ^ [ 0 , 1 ]

It is i m p o s s i b l e t--

4.

P . L.

Subspaces

Definition .

Let

(X,

P . L . space with X C X. o — (X, ^ ) p r o v i d e d l ) X^

If

X,

(X^,

be a n o t h e r

(X , ^ ) i s c a l l e d a P . L . s u b s p a c e of o o

i(x) = X , i s a P . L . m a p .

(X , % ) i s a P . L . s u b s p a c e , t h e n

•

O

Examples;

Then

Let

h a s t h e r e l a t i v e t o p o l o g y i n d u c e d by X, a n d

2) i ; X Remark,

) be a P . L . space.

^

o

1) If

= { ( f , P ) e ^ | f(P)

^

<>0 C X i s o p e n a n d if

^ ^ = {(f,P) e S

vi I f(P) ^

the

(X , ^ ) is a P . L . s u b s p a c e of (X. 5 - ) . 2) E ^ h a s the n a t u r a l P . L . s t r u c t u r e g e n e r a t e d b y t h e i n c l u s i o n m a p s p o l y h e d r a in e'^.

A compact subspace

(with its n a t u r a l s t r u c t u r e ) .

F o r suppose

Then t h e r e is a coordinate m a p But X^

X^ of

(f, P )

E n" m u s t be a p o l y h e d r o n in C E^

i n t h e s t r u c t u r e of X ^ w i t h f(P) = Xj

i s a P . L . s u b s p a c e , so t h e c o m p o s i t i o n

Therefore

X^ = f(P)

3) In E'^,

4) If

d(x, X ) < 1} i s a P . L . o { x | d ( x , x^) < 1} i s n o t .

Lemma 3.7. —————

a r e p o l y h e d r a in E ^ , If

P — X ^ C E^

( X , % ) N O ^ ^O O

subspace,

^ -

i s a P . L . s u b s p a c e of

i s a P . L . s u b s p a c e of (X,

of X

.

K^ of K s u c h t h a t

E

and if X

i O

c l o s e d s u b s e t of X , t h e n t h e r e e x i s t s a l o c a l l y finite t r i a n g u l a t i o n and a s u b c o m p l e x

is a P . L.|

n is a p o l y h e d r o n in E .

{X

P^C P

i s a c o m p a c t P . L . subsr

h

K^

| K ^

X

o

h: K

is a triangul:

-87-

Proof . let k:

M

Let

h:

» X^

X

L

> X be a l o c a l l y finite t r i a n g u l a t i o n of X .

b e a l o c a l l y finite t r i a n g u l a t i o n of X ^ .

> X t h e i n c l u s i o n m a p . L e t M' and o r e s p e c t i v e l y , making the p r o p e r P . L. m a p

jet K

o

=

Image

Let

0 = h

K b e s u b d i v i s i o n s of (X

is closed)

fi

M

Then i

k.

and

simplicial.

-al

5.

Collapsing and Regular Neighborhood T h e o r y . Definitio n.

t h e n we s a y

If X^

X ^ X^

is a c l o s e d P . L . s u b s p a c e of t h e connpact P . L . s j

if t h e r e e x i s t s a finite s e q u e n c e of

X C X C . . . C X = X o 1 ~ — r isap.l.

(P. L. ) ball having

Definition .

If

l i n k ( h " ^ x ; K) i s a b a l l . Definition .

cl(X^-X.

Let

N ^ X

as a face. h; | K |

> M be a triang

We s a y x e 9M if

T h i s d o e s not d e p e n d u p o n the c h o i c e of

X^

(h, K).

b e a c o m p a c t P . L . s u b s p a c e of t h e P . L . m a n i f o l N of X i s a t o p o l o g i c a l n e i g h b o r h o o d

N, co

and N i s a n m - d i m e n s i o n a l P . L . s u b m a n i f o l d (i. e . ,

s p a c e w h i c h i s a m a n i f o l d ) of

X.

N m e e t s t h e b o u n d a r y r e g u l a r l y if

o r i s a r e g u l a r n e i g h b o r h o o d of X n 9M in T h e o r e m 3. 8 . M.

let

M (h i n t h e s t r u c t u r e ) .

Then a regular neighborhood such that

s u c h t h a t X. - X. ^ = c l ( X . - X. J 1 1-1 1 1-1

M is a P . L. manifold,

of a n e i g h b o r h o o d of x i n

P . L . s u b s p a c e s of

Let

as

N O6

9M.

X^ b e a c o m p a c t P . L . s u b s p a c e of t h e P . L . m - m ,

T h e n a r e g u l a r n e i g h b o r h o o d of X^ w h i c h m e e t s t h e b o u n d a r y r e g u l a r l y

exists.

If N

and N jL

a r e a n y two r e g u l a r n e i g h b o r h o o d s of X , t h e n t h e r e i ^

a P . L . h o m e o m o r p h i s m of N^

onto

N^ p o i n t w i s e fixed on X^ .

If

N^ ani

m e e t the bo\andary r e g u l a r l y , t h e n t h e r e e x i s t s a n a m b i e n t i s o t o p y H: M X I

>MXI

throwing

X fixed. X ^ O Proof . L e t (f, K) be an e l e m e n t of t h e P . L . s t r u c t u r e ^ of X s u c h ' X C int^^ f(K) and K i s a p . l . m - m a n i f o l d . Let N be the i m a g e u n d e r i O M a r e g u l a r n e i g h b o r h o o d of f

onto N

(X^) in

K.

and l e a v i n g

-89T h e i m i q u e n e s s t h e o r e m s follow s i m i l a r l y b y tciking N^ U N^ Q One c a n a l s o define r e g v i l a r n e i g h b o r h o o d s of n o n - c o m p a c t s u b s p a c e s of a P. L . manifold.

If X a n d X

are closed P. L. spaces, o

X

a s u b s p a c e of

we s a y X^ c o l l a p s e s t o X b y aji e l e m e n t a r y g e n e r a l i z e d c o l l a p s e if i s t h e \inion of a d i s j o i n t l o c a l l y finite f a m i l y of w h e r e , for e a c h i , B^ i s a p . 1. b a l l h a v i n g

P. L. subspaces

B^nX^

as a face.

if X c o l l a p s e s t o

cl(X-X^)

B^ of

X,

A generalized

c o l l a p s e i s a finite s e q u e n c e of e l e m e n t a r y g e n e r a l i z e d c o l l a p s e s . ^ X ^ X^

X,

o

We w r i t e

X^ b y a n e l e m e n t a r y g e n e r a l i z e d c o l l a p s e .

A g e n e r a l i z e d r e g u l a r n e i g h b o r h o o d of X ^

in t h e P . L . m - m a n i f o l d

M,

a closed P . L . subspace, is a closed topological neighborhood.which is an o i-submanifold and which c o l l a p s e s to

X by a n e l e m e n t a r y g e n e r a l i z e d c o l l a p s e .

T h i s d e f i n i t i o n g i v e s r i s e to t h e a n a l o g o u s e x i s t e n c e and u n i q u e n e s s 2orems a s for t h e c o m p a c e c a s e .

However, these generalized regular neigh-

Jorhoods h a v e h a d no i m p o r t a n c e so f a r .

C h a p t e r IV - G e n e r a l P o s i t i o n § 1.

Definitions Let

Then

K and

L b e P . L . s u b s p a c e s of t h e P . L . m a n i f o l d

K and L a r e in g e n e r a l p o s i t i o n ( o r

d i m (K n L) < d i m K + d i m L, - q .

Q, q = d i m Qj

K is in g e n . p o s . w . r . t . L) if .

(Note the s i m i l a r i t y b e t w e e n t h i s conditi

a n d t h e c o n d i t i o n in d i m e n s i o n s t h a t is n e c e s s a r y and sufficient for two subs p a c e s of a finite d i m e n s i o n a l v e c t o r s p a c e to s p a n t h a t s p a c e . ) -1 If f: P —?> Q is a m a p , a n d S (f) = S^(f) .

If

x e P| f

f(x)

h a s at l e a s t r - p o i n t a

P & Q a r e P . L . s p a c e s a n d f i s a P . L , m a p , then

follows f r o m t h e fact t h a t S'(f)

S'(f) =

P a n d Q m a y be t r i a n g u l a t e d to m a k e

i s a P . L . s u b s p a c e of

P.

If

f is p r o p e r , t h e n S^(f)

f linear t

i s a c l o s e d P„

s u b s p a c e , and d i m S^(f) = d i m S^(f). If f: P —> Q i s a m a p , P & Q P . L . s p a c e s of d i m e n s i o n p and q r e s p e c t i v e l y , we s a y t h a t

f i s in g e n e r a l p o s i t i o n p r o v i d e d

1) f is P . L . and p r o p e r . 2) for a l l

r,

(f) = 0

3) S

d i m S^(f) (i.e.

<

rp/(r-l)q

f is n o n - d e g e n e r a t e ) .

oo Let £ : P

>

f and ^^

t h e t o p o l o g y of

g be two m a p s

P — > Q,

P and Q P . L . s p a c e s .

be a p o s i t i v e , c o n t i n u o u s f u n c t i o n . Q,

t o ^ ) provided that

T h e n we s a y V x e P,

If f and g a r e m a p s ,

f

Let ^

b e a m e t r i c for

f i s an £ - a p p r o x i m a t i o n to ^

( f(x), g(x)) <

g (with r e s p

£(x).

f ( r e l K) m e a n s t h a t

a h o m o t o p y w h i c h i s t h e c o n s t a n t h o m o t o p y on

Let

K.

f i s h o m o t o p i c to

f

-91-

§2.

A p p r o x i m a t i o n of C o n t i n u o u s F u n c t i o n s by P . L .

L e m m a 4. 1. Let f: P — b e |p n P = ^ . 1 2

Let

s u b p o l y h e d r a of t h e p o l y h e d r o n P .

a continuous m a p , with

G i v e n £ > 0, t h e r e e x i s t s

1) f I P ^

is an

Proof. Implies > P 1

Let

p b e a m e t r i c for S/Z.

Let

C P, s u c h t h a t

K

B

Assume

w i t h t h e following p r o p e r t i e s :

f (w. r . t t h e u s u a l m e t r i c on I n . ) P

and choose

i s full in K,

5 > 0 such that

p(x, y) < 5

b e s i m p l i c i a l t r i a n g u l a t i o n s of r r e s h (K) < 5, a n d f|K

is linear.

^

f;

Chen f'|or„

-a any s i m p l e x of

| K | —>

o n v e r t i c e s of

b y f i r s t p u t t i n g f'(v) = f(v)

c.

If

'Ut f'l 0-= f I (T. F i n a l l y if e K , , cr^n K J = d, a s 1 Z 1' learly the m a p f

P—>

K^.K^.K^CK

iow define

f

f;

"^^P*

P3 .

6 - a p p r o x i m a t i o n to

d(fx, f y ) < P

^

i s a p. 1. m a p

2) fl P ^ ^ P 3 = f ' | P 3 ^ 3) f

Maps.

f

instruction, ggiark.

f

K^, b e defined by e x t e n d i n g l i n e a r l y t h e definition

o" r\ | K^ ( = 0 , h o w e v e r ,
a l r e a d y g i v e n on

K, is l i n e a r . 1 i s an

Since

for e v e r y v e r t e x v e K.

Vcr € K,

(i. e. , cr h a s no f a c e s in K^ ,

cr n | K^ | ^ jZ(, w e m a y put

T h e n we define o"^ and cr^.

f

-

cr by e x t e n d i n g

C l e a r l y , f ' j K ^ u K^ =

d i a m fa < 6 / 2

a n d d i a m f'cr < £ / Z

bv

< S - a p p r o x i m a t i o n to f.

f i s h o m o t o p i c to

f

( r e l P^,

P^)

by t h e h o m o t o p y

U) = tf(x) + ( l - t ) f ' ( x ) . T h e n d ( H J x ) , H (x)) < £ , a l l x e l " , a l l s , t e I. In 'j t s w o r d s , we c a n c h o o s e t h e h o m o t o p y H of f and f to a " a r b i t r a r i l y

L e m m a 4. 2.

Let

f: P —> Q be a c o n t i n u o u s m a p of t h e P . L, s p a c j

P

into t h e P . L. g - m a n i f o l d

of

P

on w h i c h

Q.

Then t h e r e exists

1) f ^ f ( r e l P ) , o 2) P (fx, f x ) < i5(x) a l l X / t o p o l o g y of Q),

Q C

stars.)

Q. P

f;

( p /

be a c l o s e d P , L .

Let

£ : P —> R

subspace

be a continuol

P—5> Q s u c h t h a t

s o m e g i v e n d i s t a n c e function for the

is a P . L. m a p .

Proof. with

P

f is a l r e a d y a P . L . m a p .

p o s i t i v e function.

3) f

Let

Let

{B^] be a l o c a l l y finite c o u n t a b l e f a m i l y of q - b a l l s in Q

I J Int B. . X U 1 Let

(For example, triangulate

Q and t a k e c l o s e d vert

Q. K be ( l o c a l l y finite) s i m p l i c i a l c o m p l e x e s t r i a n g u l a t i n g

P ^ ^ P , s u c h t h a t if

cr € K,

fcr C I n t ^ B^, s o m e j.

This is possible becau

t h e r e i s a U r i a n g u l a t i o n L of P , c o n t a i n i n g a t r i a n g u l a t i o n of P , s u c h tha oo L = 1 I L. , L. finite s u b c o m p l e x e s s u c h t h a t L . O L . = 0 if l i - i >2. iVi 1 1 1 J ' S u b d i v i d e e a c h L. to set 1 ®

LI s u c h t h a t 1

(T e L; i m p l i e s 1

T h e n f u r t h e r s u b d i v i d e ( p r o c e e d i n d u c t i v e l y ) to g e t L ' l and Let

a r e compatible; aJ

i = 1

s i m p l e x follows i t s f a c e s . then the 1 - s i m p l i c e s and

L'^

fer C Int B., s o m e j — J such that

for all i

t h i s g i v e s t h e r e q u i r e d s u b d i v i s i o n , K,

ooj- be t h e s i m p l i c e s of (For example:

of

K - K ^ , o r d e r e d so t h a t a

f i r s t t a k e t h e v e r t i c e s of L ^ -

L " - K and t h e v e r t i c e s of ( L " - L " ) - K , then o o ' o' o t h e 2 - s i m p l i c e s of L " -K , t h e 1 - s i m p l i c e s of ( L ' ' - L " ) - K and t h e z e r o o o l o o s i m p l i c e s of (L» - L " ) - K , e t c . . . . ) P u t K. = K u J A. . We a r e Z 1 o 1 o , j j= 1

-93-

to define i n d u c t i v e l y m a p s 1)

f.

2)

K.

f.: K —> Q

i s P . L.

f. f (rel P ) 1 o if cr € K, f^((r) c I n t ^ B^ , s o m e

3) 4) ^

(f.(x).f._^(x))

We s t a r t w i t h f iome j .

Let

K'

= f.

, all Suppose ^^

f. . i s defined. 1-1

b e a P . L. h o m e o m o r p h i s m .

N(AI;K'), R = R n K

U p . L . on R^ . R^ i s P . L . . x e R.

X .

T h e n f. ^(A.) C I n t ^ B., 1-1 1 Q J

b e a s u b d i v i s i o n of K s u c h t h a t N(A! j K') C f^ ^ ( I n t ^ B^).

.et h: B. —»

L= Fr

o

<

j.

H e n c e for e v e r y

. £>0

Then

Define

f.: K 1

Q

R = N(A! ;K'), R^ = A^ ,

R

R, = jZ! and

ther- exists

a l R ^ ^ R^ = h o ( f . _ j R 2 ' ^ R 3 ) .

f?

Put

and

a:R—>

h»(f.

|

R)

such that

p(a(x), h® f . )

<6

,

by f. R = h ' ^ a 1

T \ C1(|K|-R) = f . _ ^ I c l ( | K | - R ) . Now R i s c o m p a c t , so by c h o o s i n g

£ s m a l l enough we c a n e n s u r e that

L(x)-f^ ^(x)) < £ (x)/2^ for a l l XT | K | , and a l s o t h a t e v e r y .Bome I n t ^ B . . U J 4).

f^(o-) i s c o n t a i n e d

T h e n f. i s a w e l l - d e f i n e d m a p w h i c h c l e a r l y s a t i s f i e s 1), 3), 1

M o r e o v e r , f r o m the r e m a r k following l i e m m a 4. 1, w e s e e t h a t

— f.

. |R

(rel. R ^ R

^see t h a t 3) h o l d s .

) and s o , e x t e n d i n g the h o m o t o p y by the identity,

Call this homotopy

H^.

S y c o n s t r u c t i o n , f^ a g r e e s w i t h f^ ^ e x c e p t on a s i m p l i c i a l n e i g h b c r h r -c^ .

If cr € K,

0" m e e t s o n l y finitely m a n y of the R^,

T h e r e f o r e the f.

e v e n t u a l l y a g r e e on | K | —> Q. given

cr.

H e n c e putting

f -

Similarly, the homotopies

0" € K and so H = l i m

^

i s a w e l l defined c o n t i n u o u s ^

1 oo f ^ P (rel

a n d so

defines a P . L . m a p

H^ a r e e v e n t u a l l y the i d e n t i t y on

H. o . . . » H ,

•

K| X I — > Q ,

lim f. i —5> oo

K

). o

Remark.

U s i n g t h e r e m a r k following 4. 1, t h e r e a d e r c a n e a s i l y s h o w that

u n d e r t h e h y p o t h e s e s of 4. Z, we c a n find a h o m o t o p y such that x e P

§ 3.

f = H^

and

every

H of f, fixed on

s a t i s f i e s t h e c o n c l u s i o n s of 6. 2 a n d in a d d i t i o n , for e v e r i s,t

in[0,l],

d(H x, H x) < s t

£(x).

ApproxirngiLion of P . L . m a p s by N o n - D e g e n e r a t e P . L . Definition .

If X

E^

fi P — >

Let

f:

l)

f'jPj^

3)

f(P-P^)c!n

f2X be t h e unio

s p a n n e d by s u b s e t s of X . E ^ - i^X is d e n s e in

f2X i s a c l o s

E^.

P^^ and P ^ Q P be p o l y h e d r a , d i m P < n.

be a p. 1, m a p w i t h f j P ^ ^ P ^

e x i s t s a p. 1. m a p

P —> I^

non-degenerate.

Let

G i v e n £> 0 th

s u c h that

is n o n - d e g e n e r a t e

4) for a l l X 6 P , Note;

E^

of m e a s u r e z e r o , so

L e m m a 4. 3.

Maps.

is a finite s e t of p o i n t s in e " , l e t

a l l p r o p e r affine s u b s p a c e s of s u b s e t of

P

( f x , fx) < £

In g e n e r a l we c a n n o t shift

f to be n o n - d e g e n e r a t e on

c h a n g i n g it on P , if it is not a l r e a d y n o n - d e g e n e r a t e on P ^

example,

-95-

j l e t P 2 = 1 - f a c e of a 2 - s i m p l e x P , P^ = P , a n d s u p p o s e P r o o f of L e m m a 4. 3. (so t h a t

f:K—>

is linear.

land t h a t v , . . . ,v I VT i s = f(v.). |f(vp,

For

Let Let

be t h e v e r t i c e s of K^ n

r < i ^ s we m a y choose points jw^ , . . . , w. J

w^,

^ensure t h a t

f

For

i < r

If w e define i

4).

put

f ' : K —>

s, and f'(v) = f(v)

v , t h e n b y c h o o s i n g e a c h w^ c l o s e enough to

satisfies

K^

w^ a r b i t r a r i l y c l o s e to

, and w. e I^.

fto be t h e u n i q u e l i n e a r m a p s u c h t h a t f ' ( v p = w^, 1 lall o t h e r v e r t i c e s

is a p o i n t .

K^ , K^ C K b e t r i a n g u l a t i o n s of P^ . P ^ ^ P

be t h e v e r t i c e s of K - K K . 1 x 2 ,

such that w. /

flP^)

It c l e a r l y s a t i s f i e s 2) and 3).

for

f(vp, we m a y

To s h o w t h a t s u c h

,an f i s n o n - d e g e n e r a t e o n K^ , it suffices to s h o w t h a t i t s r e s t r i c t i o n t o e a c h cr e K [is.

T h i s w e p r o v e b y i n d u c t i o n on d i m cr.

Ls nothing t o p r o v e .

If

cr / K H K

y induction, f ' | v .

...v.

put

If

0 e K^O K^, f' | (r = f cr, so t h e r e

cr = v . . . . v .

is non-degenerate.

As

, j , < . . . < j. » j. > r . d i m P < n,

[spanifv. , . . . , f ' v . I / E ^ , so f'(v. ) i s n o t in t h i s affine s u b s p a c s . I ^t-1 •'t fe t h e p o i n t s

{f'(v. )

L e m m a 4. 4 .

Let

f'(v. )}

a r e independent;

o

f' j tr i s n o n - d e g i r e r a i c ,

f: P — > Q b e a P . L . m a p , Q a P . L . m a n i f o l d and

a P . L . s p a c e w i t h d i m P < d i m Q. Suppose f P

so.

There-

is non-degenerate.

Let

P^ Q P

be a closed P . L.

T h e n f:^ f ( r e l P ), w h e r e f o

Regenerate P . L . m a p a n d f ' ( P - P ^ ) C Int Q.

M o r e o v e r , given

subspace

is a non£ : P—> R ^

• p o s i t i v e c o n t i n u o u s f u n c t i o n , w e m a y i n s i s t t h a t ^ ( f ( x ) , f'(x)) < £ ( x ) , a l l x , a g i v e n m e t r i c for t h e t o p o l o g y of

Q.

Proof.

E x a c t l y a s L e m m a 4. 2, vising L e m m a 4. 3 i n s t e a d of 4. 1.

Remarks. 1) As in 4. 2, we c o u l d a c t u a l l y i n s i s t t h a t t h e r e be a h o m o t o p y H; f ^ f ( r e l P ) s u c h t h a t for a l l x e P o d(H

x , H x) < S

and a l l

s,t

in [O, l ]

(x).

T

2) In 4. 2 and 4. 4, one c a n i n s i s t t h a t if t h e g i v e n m a p t h e n so is t h e m a p §i4.

f is p r o p e r ,

f.

Shifting S u b s p a c e s to G e n e r a l P o s i t i o n . L e m m a 4. 5.

I^, w i t h P

Let

91^ r. P , o

P

o

x P

and

R, , . . . , R be p o l y h e d r a c o n t a i n e d i n j 1 r I

G i v e n c > 0 t h e r e e x i s t s an a m b i e n t i s o t o p y

h of

such that 1) h i s fixed on 2) h ^ ( P - P ^ )

i i P. o is in g e n e r a l p o s i t i o n w. r . t . e a c h

3) for a l l t,

d(h^x, x) < £ .

Proof. tions V I

Let

J

be a t r i a n g u l a t i o n of

R^

h a v i n g a s s u b c o m p l e x e s triangv

K C K, L , L of P S P , R , , w i t h K full in J . Let o 1 r o 1 r o V be the v e r t i c e s of K - K , a n d l e t X be t h e s e t of a l l the v e r t i c e s s o n ' / ^ Let w , . . . , w be p o i n t s in Int I , s u c h t h a t w, ^ X ' ' (w , . . . , w. I s 1/ I

a l l i; we m a y c h o o s e e a c h w^ to be l e s s t h a n any p r e a s s i g n e d d i s t a n c e fron n In p a r t i c u l a r , we m a y c h o o s e t h e vv^ so t h a t if I d e t e r m i n e d by p u t t i n g / ( v ^ = w^ a n d I

i (v) = v

is a m b i e n t i s o t o p i c to 1 v i a an a m b i e n t i s o t o p y

is the l i n e a r m a p

J —> I

if v e X, and v / v^ a l l h

s a t i s f y i n g 3) and 1) .

i, t

-97-

C e r t a i n l y we c a n m a k e I

i s o t o p i c to t h e i d e n t i t y by " s m a l l " m o v e s .

Then

see p r o o f t h a t i s o t o p y b y m o v e s i m p l i e s a m b i e n t i s o t o p y , C h a p t e r V, §1, emma 5.1.)

n

To c h e c k 2), l e t '

ere K - K , T £ R . . o 1

K^l = 0

possible).

=

0- = 0" . (w. . . . w. ). X 11 1 1 s

If

Let

Write

cr = cr CR 1 2

CR e K 1 o

and

o-^ = Vi^. . . v i ^ , i^ < . . . < i ^ .

I cr and T s p a n

Then

E , then

dim(jeo- Pv T) < d i m cr + d i m r - n ^ d i m P + d i m R^ - n . itr and T do not s p a n

E'^, t h e n s i n c e

3T the affine s u b s p a c e s p a n n e d by n T = j^.

Since

P-P

= |K

W^ /

W , . . . , W^

cr . w . . . . w^^ ^ h £-1

and

T .

^i

This implies

(T , t h i s s h o w s t h a t

K o-eK-K

d i m r ( P - P )'^R.] < d i m ( P - P ) + dim R.- n , o 1 o 1 all i,

1 < i < r.

L e m m a 4,

.

Let

g - m a n i f o l d Q, w i t h itive function. I'

P

o

P n aQ c P

Let

Q—$>lR

be a c o n t i n u o u s

dQ u P^ ,

i s in g e n e r a l p o s i t i o n w. r. to e a c h R^ ,

I 3) d(h^x,x) < £ ( x ) Proof.

.

T h e n t h e r e e x i s t s a n a m b i e n t i s o t o p y h of Q s u c h thats

1) h fixes t h e p o i n t s of 2} h ^ { P - P ^ )

C p, R . .. , , R b e c l o s e d P . L . sub s p a c e s of the 1 r

Let

for all x (d a m e t r i c for the t o p o l o g y of Q. ). be a l o c a l l y finite countable family of q - b a l l s such the'-.

.CO

Int^ B.. U 1 "" ® K,

Let

K Q K b e t t f i a n g u l a t i o n s of P C P o o

s u c h t h a t , forr

0- C I n t „ B. for s o m e i. L e t [A. be t h e s i m p l i c e s of K - K . Q 1 ^ J ^ . o so t h a t a n y s i m p l e x follows i t s f a c e s . L e t K.. == K u I I A. . We X o .O J

a r e going to define P . L . h o m e o m o r p h i s m s isotopies

H^^^ of Q

1) h / ' ' .

(i>l)

fixing

h.

(i > 0 )

of

Q and a m b i e n t

9Q vj P ^ , s u c h t h a t

= h. ,

2)

Vo- e K , V t,

3)

Vx,

d(H

H^^V) C Int^ B . , some J Q j < e(x)/2\

c

all

t.

4) h.( K^ - I K^ ) i s in g e n e r a l p o s i t i o n w. r . t e a c h of t h e We s t a r t by p u t t i n g s o m e i ^ 1. Let

V

Let

A. C Int „ B. , 1 Q J ,K. J ^ ^ B . ) , let 1-1 i - r j"

o

W = a(R

k

h^ = i d e n t i t y .

B.). j'

Note t h a t

Let

Now s u p p o s e

a : B —>

R^ .

h^ ^ is c o n s t r u c t e d ,

be a P . L,

homeomorphisr

V = a(h. , K . n B . ) = V W ah. A . , and le| ' 1-1 1 j' o 1-1 1 I

V O

c V , — o

By L e m m a 4. 5, for e v e r y £ > 0 t h e r e e x i s t s an a m b i e n t i s o t o p y k of fixed on each

W

U ic

such that

k^(V-V^)

a n d s u c h t h a t , for e v e r y

i s in g e n e r a l p o s i t i o n w i t h r e s p e c t

t,

/>(x,k x) < t

.

Now define

H^^^ by

H^^^fB. X I = ( a " ^ X 1) 0 k o ( a X 1) J H^^^l cl(Q - B.) X I = i d e n t i t y . J Put

h^ = H^^^o h^

.

By c h o o s i n g

£ s m a l l e n o u g h we c a n e n s u r e t h a t

x) < £(X)/2^ for all x e | K | , t e I, a n d a l s o t h a t , given cr € K, t £ Iil x(i) (or) C I n t ^ B^, for s o m e H^

j.

To c o m p l e t e the proof, we o b s e r v e t h a t , by t h e c o n s t r u c t i o n of t h e w e m a y h a v e t h a t e a c h i s t h e i d e n t i t y o u t s i d e t h e i n t e r i o r of s o m e if C is a n y c o m p a c t s u b s e t of H (i)

are the identity.

B^.

H

(i|

Hence

Q, t h e n on C X I a l l but a finite n u m b e r of t b |

H e n c e it m a k e s s e n s e to define

-99-

h=

lim

H^'Ih^'-^K

. . . CH^ .

i —> CO T h e n h i s an a m b i e n t iBotopy and by c o n s t r u c t i o n s a t i s f i e s 1), 2), and 3)

in

the s t a t e m e n t o£ t h e l e m m a . §5.

S h j i t i n g m a p s to G e n e r a l P o s i t i o n . L e m m a 4. 7.

Let

K b e a ( l o c a l l y finite) s i m p l i c i a l c o m p l e x a n d l e t

f: K — > Q be a P . L . m a p w h i c h e m b e d s e a c h s i m p l e x .

Let

K, a n d

l e t R, , . . . , R b e f c l o s e d P . L . s u b s p a c e s of t h e P . L . m a n i f o l d 1 n f( K) - | K

I) C Int Q.

ffhen t h e r e is a m a p

Let

f:

K

>

Q.

Assume

be a positive continuous function.

K —> Q and a h o m o t . ,

H: K X I —» Q of f and f

, such t h a t 1) H i s t h e c o n s t a n t h o m o t o p y 2) H i s a P . L . m a p 3) f 4)

e m b e d s e a c h s i m p l e x of K and f ' ( | K | - j K^ | ) C Int Q

Vo-,, 1

(T

in

K-K

r

o

r dim(n 1 dim[( n

f'S-.) ^

6)

d(H x , f x ) < £ ( x ) f'( K

-

r 2 )

d i m (r. - ( r - l ) q ^

f 5- ) n R.] < ^

5)

< 1

i

I

d i m (T. "+ d i m R. - r q ^ J

for a l l x and

K I) C Int Q . o ~

s,

(d a m e t r i c on

,

all

Q)

j.

Proof.

Let

[aJ

i = 1, 2, . . . ]

s i m p l e x following i t s f a c e s .

be the s i m p l i c e s of K - K ^ , w i t h eac i K. = K^ U U A^ , a s u b c o m p l e x . We

Let

going to define, i n d u c t i v e l y , P . L . m a p s

f^ ,

i ^ 0, a n d P . L ,

homotopies

i > 1, s u c h t h a t 1)

V 0" € K,

f. I 0" i s an e m b e d d i n g ;

2)

H^^^ i s a h o m o t o p y of f^ ^ to f^ w h i c h l e a v e s

3)

Vo-, , . . . , 0- € K . - K , 1 r 1 o dim{ n f.®-.) 1 S J =1 ' J r d i m ( f~] j=l

f. ^ ^

4) 5) '

Put

^

< g(x)/2^ f.(|K| 1

f

o

= f.

- |K

o

R.

- (r-l)q J

r S ^ ^ ^ ""j j=l

^^^ \

all

•

k.

,

I) C Int Q . -

Now a s s u m e

f. , i s defined, 1-1

a l l t h e following P . L , s u b s p a c e s of a)

diin

fixed;

i>l.

Let

L ,..,,L be 1 IN

Qs

, 1< j < n , r

j=l , all

cr, , . . . , 0- i n K. , a n d 1 r 1-1

1^ k ^ n .

J=1 (Note:

r

not

fixed, )

Now w e a r e going to a p p l y L e m m a 4. U . P

o

= f. J A , . L), a n d l e t l-i 1

P = P U f. , (A.). o 1-1 1

Let

L = l i n k (A^; K).

Note that

P O 8Q C P

o

Let .

-101-

By L e m m a 4 . 6 , t h e r e e x i s t s an a m b i e n t i s o t o p y h of Q, fixed on P

9Q, s u c h t h a t

h. ( P - P ) i s in g e n e r a l p o s i t i o n w . r . t . e a c h

o

L . , and

o

y s and

1

Vx, d(hgX, x) <

Define

m i n [ e ( y ) | y e A.. L

H^^^ on (A.. L) X I b y p u t t i n g

|by t h e c o n s t a n t h o m o t o p y o u t s i d e Clearly

H^^^^(x) = h^ f(x).

(A^. L) X I, and put

H^^^ and f^ s a t i s f y 2) a n d 4 ) . f. ^

fbecause

o n l y o n s i m p l i c e s of

differs f r o m

Iposite of f.

f^

Extend

f^ =

H^^^ to

K XI

= h^^o f^ ^^ .

C o n d i t i o n 5) h o l d s b e c a u s e

| h j ( l n t Q) C Int Q a n d b e c a u s e f.

.

s a t i s f i e s 5).

C o n d i t i o n l ) h o l d s for f^ A^. L, w h e r e it is the c o m -

and a h o m e o m o r p h i s m . o

To c h e c k 3), we f i r s t o b s e r v e t h a t i'or s u p p o s e

x e P ^ O f^

Then

: = f. j ( z ) , z e A . . L e t y e p.T , Jf K and so i s e m b e d d e d by f^ fore

P

|v

K

o

n f. , ( A . ) C f. , ( A . ) . i - r i' - i - P 1

f^

- ^"^o

^^^ ^^^^ h a v e =).

x = f^ ^y, s a y , w h e r e

p e A^ a n d Therefore

Therefore,

as

y e A^L, and

T e L . T h e n A^T i s a s i m p l e x y = z, so x e f^ f. ,

1-1

embeds

A., 1

o

I. J^(A^) r\ P ^ =

C o n d i t i o n 3) now follows for

L lition for f. ^ and t h e fact t h a t the L . . 1 To c o m p l e t e t h e proof, p u t fhese a r e w e l l defined

hi .

f^ f r o m t h e c o r r e s p o n d i n g c o n -

® i s i"- g e n e r a l p o s i t i o n w i t h r e s p e c t to

H = lim H^^^ and f = H = l i m H^^^ = linri i 1 . 1 1 -5> CD i-$> CO l - > OO p . 1. m a p s b e c a u s e H^^^ | A. X I = j A. X I for

F i n a l l y we put s o m e of t h e a b o v e r e s u l t s t o g e t h e r to g e t ; L e m m a 4. 8.

Let

Q be a P . L , m a n i f o l d such that

fjP^

s u b s p a c e s of there exist

P

, d i m P < d i m Q.

isP.L.

Q.

Let

be a P . L. space, Let

and n o n - d e g e n e r a t e . £ ; P —> IR

P^

a closed subspace.

f; P —> Q be a c o n t i n u o u s Let

be c l o s e d P |

be a p o s i t i v e c o n t i n u o u s f u n c t i o n .

gs P —> Q a n d a h o m o t o p y

Let

H; f

g (rel P^)

Thei|

such that

1) g i s a P . L . , n o n - d e g e n e r a t e m a p , 2)

i s in g e n e r a l p o s i t i o n ,

3) g ( P - P ^ )

i s in g e n e r a l p o s i t i o n w. r.t* e a c h

4)

g ( P - P ^ ) C Int Q ,

5)

Vx,

d(H X, fx) < £ (x) s

Vs€[0,l]

(d

R^ ,

s o m e m e t r i c for the

t o p o l o g y of Q-). Proof .

By 4. 2 a n d 4 . 4 w e c a n find

f ^ f (rel P^)

and a homotopy

|

b e t w e e n f and f r e l a t i v e P 5 v/ith f P . L . a n d n o n - d e g e n e r a t e , f ' ( P - P ) C Int Q , a n d d(H' ox , fx) < £ ( x ) ' 7 . L e t K C K b e t r i a n g u l a t i o n s j O

S

of Q, so t h a t p l i c e s of

K.

f:

K —> L i s l i n e a r on s i m p l i c e s .

Let

H"

a) g i s P . L . b) c)

o

be a h o m o t o p y of f

to a m a p

embeds the siml

g, r e l a t i v e

P ^ , satisfj

non-degenerate;

g l P - P ^ ) C Int Q r ^ r d i m Q g ^^ < ^ ^

d) d i m ( g o - n R . ) J e) d(H"x, f x ) s

Then f

^

d i m o"^ - { r - l ) q , cr^, . . . , o"^ in K - K ^ '

d i m crf d i m R. - q, J e(x),

all

x.

c e K-K

;

-103-

Then

c) and 4) i m p l y

2) and 3) in the s t a t e m e n t of the l e m m a . H'(x, It)

0 < t <

H"(x;2t-1)

Y
Put

-

H(x,t) = <

By the t r i a n g l e i n e q u a l i t y ,

Definition . codimension > r

Let in P

P

H s a t i s f i e s 5).

C e r t a i n l y , g s a t i s f i e s l ) and 4).

C P b e poiybpidara.

We s a y t h a t

P^ i s of l o c a l

if, for any t r i a n g u l a t i o n K^C K of

any s i m p l e x A of K^, t h e r e i s a s i m p l e x

B of

K with A < B

P , and for and

dim B - d i m A > r . Lemma 4.9.

Let

Q be a P . L . manifold,

j e c t i o n on the f i r s t c o o r d i n a t e . |XC(9QXI) = X . ' ' o

the p r o -

S u p p o s e X i s a p o l y h e d r o n in Q X I

If dim X < m - r ,

r>l,

jis a l e v e l - p r e s e r v i n g P . L , h o m e o m o r p h i s m the i d e n t i t y , s u c h t h a t

and p: Q X I — > Q

S2(p|hX)

and

dim X

o

with

< m - r - 1 , then there

h: Q X I—> Q X I , a r b i t r a r i l y close

i s of l o c a l c o d i m e n s i o n > r

in

hX.

Furthermore,

if S ( p | X ) i s a l r e a d y of c o d i m e n s i o n > r in X , we ^ o o 'cBxi i n s i s t t h a t h 9Q X I i s the i d e n t i t y . jgte;

' L e v e l - p r e s e r v i n g ' m e a n s t h a t h c o m m u t e s with p r o j e c t i o n onto the

»econd f a c t o r . B e f o r e p r o c e e d i n g with the proof of l e m m a 4. 9 we n e e d a n o t h e r t e c h n i c a l Sttima.

Lemma 4.10.

Let

K

o

b e a full s u b c o m p l e x of

d i v i s i o n of K o b t a i n e d by s t a r r i n g a l l s i m p l e x e s of decreasing dimension. then

K.

Let

K-K^

be the sul

in o r d e r of

i s a s u b c o m p l e x of K' and if A e K ' - K o o l i n k ( A ; K ' ) O K^ i s e i t h e r e m p t y o r a s i n g l e s i m p l e x . Proof.

Then K

One m a y r e a d i l y c h e c k , by i n d u c t i o n o n d i m e n s i o n , t h a t a genej f\

A

A

s i m p l e x of K' m a y be w r i t t e n in t h e f o r m C e K and i

B < C

1

< . .. < C . r

D € l i n k ( A ; K ' ) r> K^ if and o n l y if AD = BD. C . . . C and 1 r so

K'

Cj^ n K

BD <
Now if A e K ' - K AD e K'

o

B e K^,

i s w r i t t e n in the abcve foi

and D e K^.

B D < C < C ^ < . . , < C . 1 2 r

is a single simplex

if and o n l y if

where

In w h i c h c a s e

Now K

o

is full in K,

or s a y , and t h e a b o v e c o n d i t i o n s a r e satisfied

So link{A; K') H K

= l i n k ( B ; o") = a s i n g l e s i m p l e x (if it

i s not e m p t y ) . P r o o f of L e m m a 4. 9. C a s e 1:

F i r s t c o n s i d e r the c a s e when

a l r e a d y of l o c a l c o d i m e n s i o n > r

Q = A*^ a n d w h e n S^CpjX^)

in X ^ ; a n d we w i s h to k e e p

Let

K

Let

K' C (3'(A X I) be o b t a i n e d f r o m

is

9Q X I fixed.

C K C/p(a"rX I) be t r i a n g u l a t i o n s of X C X C a X I, w i t h

K^ full in

e

t h e s i m p l i c e s of Let F o r every

K-K^

X , ...,x i s

K C ^(A X I) b y s t a r r i n g at i n t e r i o r poi

in o r d e r of d e c r e a s i n g d i m e n s i o n . be the v e r t i c e s of

K , o

v , . . . , tr t h e v e r t i c e s of i t

K'-K

£ > 0 l e t v ' , . . . , v ' be p o i n t s in A X I s u c h t h a t t h e following hold:

-105-

1) t h e r e i s a l i n e a r h o m e o m o r p h i e m to V.' a n d

z to i t s e l f if

2) V. and 1 1

v.'

P(A X I) —> A X I, s e n d i n g

z i s a n y o t h e r v e r t e x of

a r e on t h e s a n i e l e v e l ;

v^

P(AXI);

d(v.,vl)
for all

i,

and 3)

Note;

for e v e r y

i, pv.' / ^^{px^ , . , . , px^, pv^' . . . . , pv.'_^ ^ .

C o n d i t i o n 3) doe.s not in fact d e p e n d upon the o r d e r of t h e v e r t i c e s

(Vj

v^). Let

h: (3(A X I)

We c l a i m t h a t

h

> A X I be t h e l i n e a r h o m e o m o r p h i s m of l ) a b o v e .

is t h e d e s i r e d h o m e o m o r p h i s r r .

To p r o v e t h i s c l a i m , l e t .We c o n s i d e r . C a s e 1;

{pjtr)

po" and

-1,

pr

cr, r

be s i m p l i c e s of t h e s i m p l i c i a l c o m p l e x

(pT). together span

Em .

Then

dim(pcr n P ") < d i m per + d i m p ;Therefore y a s e 2;

d i m ( p o") ^pT per and px

A) 0-

A X I.

LE v e r t i c e s of fore

hK',


do not span Write

- m < d i m per - r .

e'^.

cr = pcr^ ,

T = p T^ , cr^ N T^ = 0 .

By 3) on p a g e 102,

pCcr^^) a r e l i n e a r l y i n d e p e n d e n t of t h e v e r t i c e s of

P(T).

There-

(pj 0-)' ^(p T ) = p = a- n T . B) '

cr a n d Put

T both m e e t

K

o

and

0- = cr, ff,, (T f K . (T O K 1 2 2 o l o

cr O T € K . o = 0.

(K

o

is full. )

Put t = t t

,

O K^ = 0 , T^ £ K^. T h e n the v e r t i c e s of p(o-^) a r e l i n e a r l y i n d e p e n d e n t of ^se of per a n d p r t o g e t h e r . T h e r e f o r e (pjcr) ^(pt) = (p^cr^) ^ ( p r ) C K^ . 2

And so i s of c o d i m e n s i o n ik r

in K

by the g i v e n c o n d i t i o n s . o C) 0-, T b o t h m e e t K , cr r\ T / K . o o L e t p = cr O T. o" = per cr w h e r e cr O K = 0-, € K . 1

T =

I

Z

T, O K = l o

T, e K . o

Z

a s i n g l e s i m p l e x , p^^ s a y .

^

X

Now

p € K'-K . o

By 3), t h e v e r t i c e s of

of t h e ( s p a c e s p a n n e d b y t h e ) v e r t i c e s of po'2> Therefore p|ppj^

per o p r = pCpcr^) o p(pT2).

is one-one because

Therefore

S2(p|K^)

O

^

Let

O

Hence link(p;K') O K

pcr^^ a n d ^^^

But p o - ^ . p r ^

pr^^

^^^

o

s

a r e independl each other.

a r e f a c e s of pp^, and

has loc. codim > r

a n d so

S^(pjK^) =

(p cr) ^(px) = p.

Now,

. S^ip | h K ' ) n

=

U

CL[(P[

a)" pT -

0-

n

T],

T

where

T ranges over

hK'.

So S (p h K ' )

is of l o c a l c o d i m e n s i o n > r

in hi

P r o o f of L e m m a 4. 9 c o n t i n u e d - - T h e G e n e r a l C a s e . 9

Let

K triangulate

simplicial.

Let

p | X : K' —5> J '

K'

X,

J

and

triangulate J'

Q, be s u c h t h a t

p X? K —5> J

be first derived subdivisions such that

is still simplicial. *

Let

A,,..., A be t h e s i m p l i c e s of J . L e t A. = d u a l c e l l of I n 1 L e t K. = ( p | X ) " ^ A ' ' ' r r | k | o (A* X I). K. i s a s u b c o m p l e x . Claim: d i m K. < d i m A. - r . 1 F o r let

is

0- € K. . 1

Put

or = B , . . . I r

, B l

< ... < B , r

A. in J 1

T h e n po" = pfe, . . . p

*

(with p o s s i b l e r e p e t i t i o n s ) . d i m A. < d i m p B , ^ d i m B , . 1 1

Now, po* e A.

if and o n l y if A. ^ p B ^ .

Therefor

-107-

H o w e v e r , d i m B , il d i m B - ( r - l ) = d i m B - d i m tr < d i m X - d i m a. 1 r r Therefore But

d i m o" < d i m X - d i m B^ < d i m X - d i m A^.

So d i m tr < ( m - r ) - d i m A ^ .

m - d i m A. = d i m A. . H e n c e d i m cr < d i m A. - r . 1 1 1 S u p p o s e t h a t A , . . . , A , s < n, a r e t h e s i m p l i c e s of t h e b o u n d a r y 1 s if

5.

•

Let A. = d u a l c e l l of A. in J ' . T h e n , s i n c e d i m X < m - r - l , if 1 1 o -1 # # L. = (p X^) A , t h e n d i m L^ < d i m A^ - r , i < s , b y t h e s a m e a r g u m e n t as in the l a s t p a r a g r a p h .

# Now l e t

Bj^, . . . , B^ be t h e d u a l c e l l s

A^ a n d A ^ in o r d e r of i n c r e a s i n g

K. = ( p | X ).-1.B . , c h a n g i n g n o t a t i o n . We r e c a l l f r o m t h e J J t h e o r y of d u a l c e l l s t h a t t h e B, c o v e r J | , thyl t h e i r i n t e r i o r s a r e d i s j o i n t , and d i m e n s i o n , and l e t

that

9B.

1

i s t h e u n i o n of s o m e of t h e

B. w i t h

i < i.

3

Now we c o n s t r u c t i n d u c t i v e l y p. 1. h o m e o m o r p h i s m s

h^: B^X I —>

^

such t h a t 1) if

B. C 8B. , h. B. X I = h . . J 1 1 J J

2) S^(p h . K . ) 11

i s of l o c a l c o d i m e n s i o n > r

S u p p o s e t h a t h . i s defined for J

j < i-1.

in K. . 1

Then the m a p s

h., j < i-1 J

iefine a p. 1. h o m e o m o r p h i s m h ' s BB. X I 1 Since B^ i s a b a l l , into i t s e l f ,

h'

> 9B. X I. 1

e x t e n d s to a p . l . h o m e o m o r p h i s m of (9B^ X I)

a n d t h i s h o m e o m o r p h i s m e x t e n d s in t u r n to a p . l .

B. X I —5> B. X I, w h i c h i s l e v e l p r e s e r v i n g . ' a s e 1 of t h i s p r o o f w i t h X = h " K .

and

Q = B. .

To define

(B. X 3')

homeomornhisi.-

h . , w e now a p p l y t j e

Clearly

morphism,

S (pjh K|) = U s i.

(p|h. K.'), where

h: | j | X I — > | j | X I , defined b y t h e

h i s t h e p. 1. h o m e o -

h^.

Therefore

h satisfij

t h e r e q u i r e m e n t s of t h e f i r s t p a r a g r a p h in L e m m a 4. 9. T h e proof i n c a s e is n e a r l y the s a m e .

S^(p|X^)

i s a l r e a d y of l o c a l c o d i m e n s i on at l e a s t j

We s t a r t out b y defining

h to be t h e i d e n t i t y on

(9J) X I and t h e n e x t e n d t h e d e f i n i t i o n i n d u c t i v e l y in o r d e r of i n c r e a s i n g d i n j sions over the dual c e l l s

A^

of

J

(not

J) u s i n g C a s e 1.

C h a p t e r V: 51.

Sunny C o l l a p s i n g a n d Unknotting of S p h e r e s and B a l l s

S t a t e m e n t of the P r o b l e m Suppose t h a t

Then the p a i r

S^ C s"^ a r e P . L . s p h e r e s of d i m e n s i o n

(S*^; S^)

i s c a l l e d a s p h e r e p a i r of t y p e ( q , n ) .

c a l l e d t h e s t a n d a r d p a i r of t y p e (S^, s " )

n and q

(q, n).

respectively

The pair The sphere pair

i s c a l l e d u n k n o t t e d if it i s P . L . h o m e o m o r p h i c to t h e s t a n d a r d p a i r ;

i . e . , if t h e r e e x i s t s a P . L , h o m e o m o r p h i s m

h: S*^—>

. A*^"" such that

»n+l h(S ) = A Question;

Is a s p h e r e p a i r always unknotted?

Answer;

Y e s if No if

q-n > 3 q - n = 2 ( e . g . , T r e f o i l knot in 3 - s p h e r e . )

Unknown if

q-n = 1

(Schoenflies

Conjecture. )

We a r e going to s h o w in t h i s c h a p t e r t h a t the a n s w e r to t h i s q u e s t i o n is indeed a f f i r m a t i v e if

q - n > 3.

A r e l a t e d q u e s t i o n i s t h a t of t h e u n k n o t t i n g of b a l l p a i r s .

A p r o p e r ball

>air (B^, B^) of t y p e (q, n) i s a P . L . m - b a l l B™ c o n t a i n e d in P . L . g - b a l l B° such a w a y t h a t .the p a i r

(a"^. A^

= B"^ O 9B®.

T h e s t a n d a r d ( p r o p e r ) p a i r of type (q, r.'

a n d a p r o p e r b a l l p a i r i s s a i d to be unknotte d if

1^18 P . L . h o m e o m o r p h i c to t h e s t a n d a r d p a i r . J£®tion: ^acerj

Is a p r o p e r b a l l p a i r a l w a y s \inknotted ? Y e s if q - n > 3 - - w e w i l l p r o v e t h i s . No if q - n = 2 ? if q - n = 1.

-1 In o r d e r to p r o v e t h a t p a i r s of c o d i m e n s i o n > 3 ( i . e . q - n > 3)

are

u n k n o t t e d , we s h a l l a l s o h a v e to c o n s i d e r t h e Factorization Question; an m-manifold,

If K^ C_ K C M

a n d if K ^ K ^

and

a r e c o m p a c t P . L . s p a c e s , with

M^K^,

soes

M ^ K

In s o m e c a s e s t h e a n s w e r is a l w a y s a f f i r m a t i v e t L e m m a 5. 1.

If, in a d d i t i o n to t h e h y p o t h e s e s of t h e f a c t o r i z a t i o n questi

K C Int M = M - 3M, t h e n Proof.

L e t N b e a d e r i v e d n e i g h b o u r h o o d of K in M.

and

T h e n N C Int

So N i s a r e g u l a r n e i g h b o u r h o o d of K^, m e e t i n g the bound

regularly. M

M^K.

\K,

By t h e g e n e r a l i z e d a n n u l u s t h e o r e m , so

M

M - N ^^ ( F r N ) X I,

Therefo

\K.

H o w e v e r , t h e r e s u l t we w i l l n e e d for t h e u n k n o t t i n g q u e s t i o n i s : Theorem 5.2. m - m a n i f o l d , t h e n if

If K C K C M a r e c o m p a c t P . L . s p a c e s , o ~ M\K

and K \ K

t o

^

M

and if d i m ( K - K ) < m - 3 , o

an

thenM\l

o

H e r e d i m ( K - K ^ ) = l a r g e s t d i m e n s i o n of s i m p l i c e s of K not in K^. T h e proof of t h i s t h e o r e m o c c u p i e s t h e n e x t few s e c t i o n s . §2.

Sunny C o l l a p s i n g Definition .

and If

Say X ^ C X C M X I

are compact P . L. spaces.

( x ' , t ' ) 6 M X I, we s a y (x, t) i s d i r e c t l y b e l o w ( x ' , t ' ) U = M X I, t h e s h a d o w of

b e l o w a point of

U

.

if x = x '

If (x, t) and

t<

U is defined to be t h e s e t ^ y e M X l | y i s d i r e c

We w r i t e

sh(U)

for t h i s s e t .

-111-

Picturej Sun

/ / r

3h(u)

M

Definition . tions li I; and

X

K C K of X o o

s u n n y c o l l a p s e s to X and

X^

in M X I if t h e r e e x i s t t r i a n g u l a -

J of M s u c h t h a t

1) T h e i n c l u s i o n K —> J X I is l i n e a r (on s i n n p l i c e s ) , 2) t h e r e e x i s t s a s e q u e n c e of e l e m e n t a r y s i n n p l i c i a l c o l l a p s e s :

IK = K

r

,es ves \ K , \ ., . V r-1

\es \ K \ o

such that

( Kj X

K.

1-1

I ) . , sh(K.) = 1

I Picture: If K then

o-'

e n t i r e figure i n s i d e the box, (KI

sunny c o l l a p s e s to !K

j.

L e m m a 5. 3.

Suppose

XC MXI

X^ = X r\ [(M X 0) u (9M X I)]. MXL

Then Proof.

-

L e t M = | J , X = ' o Let

J ) n sh(K^) = j2f

S t e p 1):

Suppose that

X

s u n n y c o l l a p s e s to X^

in

M X I ^^ (M X O) u ( 9 M X I) U X .

c o n t a i n e d l i n e a r l y in J X I. (|kJ

Let I

a r e compact P . L. spaces.

K | , X = | K L o ,es K =

^r-1

where .es

K C K, and o es

\

\

^o

K

^^^^

be the sunny c o l l a p s e .

I J | X I ^ (I J | X 0) u (I 9 J | X I) u | K | U s h ( | K | ) .

Let

(3{J X I)

and

\j{J) b e s i m p l i c i a l s u b d i v i s i o n s u c h t h a t

c o n t a i n s a s u b d i v i s i o n of K and coordinate, is simplicial.

Let

o r d e r of d e c r e a s i n g d i m e n s i o n . of A. X I.

Consider

P^:

(3(J X I)

> >/(j), p r o j e c t i o n on the fij

{ A ^ be t h e s i m p l i c e s of F o r each

i,

(3(J X I) |

Y(J) -

(3(J X I) c o n t a i n s a t r i a n g u l a t

c l ^A^ X I - (A. X I) o (K u sh(K))^ .

Now, if t h i s set

n o n - e m p t y it i s a c o n v e x l i n e a r c e l l w i t h A^ X 1 a s a p r i n c i p a l f a c e . c o l l a p s e s to t h e c l o s u r e of t h e d i f f e r e n c e of i t s b o u n d a r y a n d A. X 1. A. X I y A . X I) u (A. X 0) U [(A. X I)

(K \ j s h (K)}

So doing t h e s e c o l l a p s e s in o r d e r of i n c r e a s i n g JXI| S t e p 2):

y

of|

Hence; So

. i we find t h a t

| J | X 0 ) u (|K| u sh(|K|)) u ( | 3 J | XI)

.

(J X 0) U ( a j X I) O K vj s h K ^ ( j X O) U (9J X I) u K.

In t h i s s t e p we u s e t h e e x i s t e n c e of t h e s u n n y c o l l a p s e .

We a r e going

show t h a t (J X 0) u (9J X I)

K u s h ( K p ^ (J X 0) u ( 9 J X I) u K o sh

-113-

L e t K. = K. ^ ^^ A

with

A = aB,

A

K. ^ = a B .

Therefore

A n sh(K.) C a B . Let B.

B be an i n t e r i o r p o i n t of

T h e n for

to A because Isince

b n e a r enough to

B,

B.

b be a point d i r e c t l y b e l o w

b. A n K^ ^ = a B ; n o t e t h a t

A can c o n t a i n no v e r t i c a l l i n e s e g m e n t s .

bA < sh(K^) V K^, t h i s innplies t h a t

ffrom the face

Let

bA

K = A.

b is joinable

So b. A n K. = A . So, c o l l a p s i n g bA

bB

K vj sh K \ K ' • sh K. - Int bA - Int bB = K u sh K. , j sh b a B U baB 1

Jsing t h e fact t h a t Step 1.

1

1-1

(baB) n (K u s h K . _ ^ ) C a B C

collapse vertically as

K u sh K. , u sh b a B \ K u s h K. . 1-1 ^ 1-1

icturer

U the definition:

if P and Q a r e ( c o m p a c t ) P . L . s p a c e s , P ^ Q, we s

I of l o c a l c ^ d i m e n y o n g r e a t e r t h a n o r e q u a l to ciangulation K C K of o , and dim cr

d i m T -C.

P C Q, and s a y

c in Q p r o v i d e d t h a t , fr -

cr 6 K , t h e r e e x i s t s o

t p K v "r!

L e m m a 5. 4.

L e t F : X X I —> M X I b e a P . L . e m b e d d i n g , X and

compact P . L. spaces, such that F " ^ ( ( M X 0) u ( 8 M X I) = X X 0 . Let

-TT : X X I —> X

, p: M X I —> M

b e p r o j e c t i o n s o n t h e 1st f a c t o r s .

Suppose that 1) S^ ( p ® F ) 2)

i s of l o c a l c o d i m e n s i o n ^ 2 i n

-rr S ( p o F )

XXI

is n o n - d e g e n e r a t e .

Ct

Then

F ( X X I) Proof.

s u n n y c o l l a p s e s to

By i n d u c t i o n o n d i m K.

triangulating

X and

of K X I

K, r e s p e c t i v e l y ,

and

1)

F ( X X O) in M X I . Let

M, r e s p e c t i v e l y .

K and J

Let

be s i m p l i c i a l c o m p l e x e j

a(K X I) and

p(K) b e subdi\

such that

Q:(K X I) c o n t a i n s a t r i a n g u l a t i o n

L of

F).

2) -IT : a(K X I) —5> (3(K) i s s i m p l i c i a l . Let \ L

b e a s u b d i v i s i o n of

for a s u i t a b l e s u b d i v i s i o n >/(LnKXO) Let B L

1

.,B

of

hn

J'

such that of

J.

p

F j ^/L: w/L—> J '

Note that

and l e t

A^, . . .

-yL c o n t a i n s a s u b d i v i s i o n

b e t h e r - s i m p l i c e s of

b e t h e ( r - 1 ) s i m p l i c e s of \ L - ^/(L

i s a face of, a n ( r + l ) s i m p l e x of

s o m e s i m p l e x of a{Ai X I), s o m e

or(K X I). j. S i n c e

J

(K X 0)).

Hence each trs Qr(A. X I) J

e a c h B^ i s c o n t a i n e d in A^ X I, s o m e

j.

pK.

Let

Any ( r - 1 ) s i m p l e x B^ l i e s in a face of A. i s s i m p l i c i a l , i J

t

m e a n s that

is simplicial

(KXO).

dim K = r s

L

-115-

N o w we a r e going to c o n s t r u c t " b l i s t e r s " o n t h e B^ a s follows. A

each

i, let

A

B. = b a r y c e n t e r of

B^.

Choose

it (how n e a r w i l l be s p e c i f i e d in a m o m e n t ) . and d i r e c t l y a b o v e

A B.. i

Choose

be a point o n t h e s a m e l e v e l a s shortly).

Let

We c h o o s e .

and, if

For

E. = 1

A. s u c h t h a t J A

X. If

directly below

B^i^ X X 1, c h o o s e B. C A. X I, a n d l e t ^ J

Y^

near

o Z. e A. X I ^ J

B^ and n e a r it (how n e a r to be s p e c i f i e d X Y Z B i 1 i 1

if

B. C X X 1 1

X.Z.B. I l l

if

B. C X X 1 1 -

X . , Y . , and Z. n e a r enough to 1 1 1 B ^ X X 1, E . O (X X 1) = B. H (X X ), and it' 1 1

We o b s e r v e t h a t

B^ and n e a r

so t h a t

(i. e. B. / X X 1) ; 1

E.n

E. = B.f^B. i j i j E . O (X X O) = B . a ( X X'O), 1 1

X. and Y. a r e not in 3 . b e c a u s e S ( Tr( S (p" f)) = 1 1 1 CO Z land so no s i m p l i c e s of \ ( L ) m a y c o n t a i n a v e r t i c a l l i n e s e g m e n t . Picture (of 4 b l i s t e r s ) :

Let

E!

o

E.

'1

be the b l i s t e r s w h i c h m e e t

R(j)

a b a l l of d i m ( r X 1) and m e e t s an ( r + l ) - b a l l . E. n h

Since

E. n Jz

8{A. X I) in a f a c e . J

E. = B. n Ji Jz

8(cl(A. X I - E . )) = E . n ^ h h

Hence

E. ) is an ( r + l ) - b a l l . Jz

cl(A. X I - E

... ^ E

Each bl

cl(A. X I _ J

B. , it i s not hard to s e e that Ji

a(A. X I) = a face of ^

cl(A. X I - E. o J Ji

A^ X I.

E.

.

Hence

h

C o n t i n u i n g t h u s l y , we at l a s t findl I

) i s a n (r+l)-ball.

A s i m i l a r a r g u m e n t si

that

c l ( A . X 1 - (A. X 1) o ( E .

. . . o E.

clfA

^ X I - E

^R(j) H e n c e t h e c l o s u r e of t h e c o m p l e m e n t of

J

^ w . . . v.-* E . ]. Jl jR(j)

face i s a l s o a face of c l [ A . X I - E . J hedron collapses.

So,

o ••••

], to w h i c h t h i s l a s t pel ^R(j)

A. X I V[(A. X 0) w (3A. X I)] J

Let A = (r-1)

)) i s a f a c e of

'

s k e l e t o n of

J

( E . o . . . vJ E

J

pK = (3K-{A. J .

J]

T h e n , by what we have

proved pK X I \ (pK X 0 ) u ( A X I)

(E^U

...

E^) ,

a n d so R = F ( p K X I) \ j F { { p K X 0) U (A X I) ^ ( E ^ v j . . . u E^))= 5. Moreover

sh(R) r^ R C S.

X 6 A X I,

Since t h e r e exist subdivisions m a k i n g the c o l l a p s e

it follows t h a t

F o r if F ( x ) € sh(R) n R, t h e n x e S ^ { p o F )

R s u n n y c o l l a p s e s to

S.

r\|S

and

simplici^

-117-

Now let

B. C X X 1 .

Z.X.B.

U. = 1

1

1

1 X

Z.X.B. u Z.Y.B. ^ X l l 1 1 1

te. V. = rx.B.

B. X X 1 i"*^

B. C X X 1.

X . B . o Y.B. ^ 1 1 11 rhen

~

B. i 1

X X 1 .

b a l l a l w a y s c o l l a p s e s to a f a c e . Recall that

loFrYL—> J'

B , . . ,B a r e t h e ( r - l ) s i m p l i c e s of yl and t h a t 1 s

is simplicial.

j r d e r e d so t h a t if F { B p

We m a y s u p p o s e in a d d i t i o n t h a t t h e

overshadows

^(Bj)

r(B.) in i t s s h a d o w and t h e r e f o r e a l l of J )te t h a t s i n c e S^CpoF)

B^

are

(i- s . h a s i n t e r i o r p o i n t s of

F(B.) J

in i t s s h a d o w ) t h e n

i < j.

i s of l o c a l c o d i m . at l e a s t two, n o n e of t h e p o l y h e d r a

m a y contain a v e r t i c a l line segment. ) Since

:x 0) o

E . \ U. a l l j , w e h a v e : J ^ J i-1

(A X I) -

(J

I

r

f.

i

i-1 V

+ J

U + 1 J

i

CXO)vj(AXl) - U V + U

I

1

s

U

J

1

ME

c o l l a p s e s to

i s

U + y E J

Hence

i+1 J

i-1 i-I s F [ ( K X 0 ) U ( A X I ) - I J V. + M U + U 1 J I J i i, F [ ( K X 0) u ( A X I ) -

U 1

feover,

r

\ E ] J

\

i V. J

+U J

s U. + J

(J

F.]. J

F ( I n t E.) = Int F ( E . ) m i s s e s the s h a d o w of ^ i-1 ^i-1 s (A X I) V. + U. + ( ^ E . ] . F o r o t h e r w i s e , we would n-. I J 1 J i ^

Int F ( E . ) E

meeting

sh(F(E^)), s o m e

j > i.

F r o m the c o n s t r u c t i o n of t h e bliat

this i m p l i e s that

F ( B . ) o v e r s h a d o w s F ( B . ) , an i m p o s s i b i l i t y for i < j . J ^ It now follows t h a t a n y s i m p l i c i a l s u b d i v i s i o n s w h i c h m a k e (l) a s i m p l i c i a l col l a p s e m a k e it a s u n n y c o l l a p s e .

Hence we m a y conclude that s s c o l l a p s e s to F((K X 0) v^ (A X I) - U V + i, J U ). s Now l e t k: A X I —5> A X I sends M X I.

B. to

Z.

Then F '

0); t h e r e f o r e

a n d so S 2 ( p ° F ' )

Tr|S ( p o F ' )

m a p and so i s n o n - d e g e n e r a t e . F'OK(AX

U. be t h e p . l . h o m e o m d r p h i s m wh

s a t i s f i e s t h e h y p o t h e s e s of t h i s l e m m a .

at l e a s t two in A X I, a n d

to

s V.

a n d i s t h e i d e n t i t y on cl(A X I - '-'V^).

S ^ C p - F ' ) C YL --[B^ I J = 1, . . . , sj-

F ( K X I) sunny S

Let

F' = Fok: A X

For

has local co-dimension

i s t h e r e s t r i c t i o n of a n o n - d e g e n e r a t e l

H e n c e by i n d u c t i o n F ' o k(A X I)

s F(A X I - U 1

s V. + M U.) ^ Y ^

s u n n y collad I

s u n n y c o l l a p s e s to

F(A|

This m e a n s that F((A X I) u (K X 0) - UV. + S i n c e F ( K X O) C (J x O)

UU.)

F{(A X 0) w (K X 0)) .

X I), t h i s c o l l a p s e is a l s o a s u n n y c o l l a p s e .

T

c o m p l e t e s t h e proof.

§3.

F a c t o r i z a t i o n of C o l l a p s e s - - P r o o f of T h e o r e m 1. 2. L e m m a 5. 5.

Suppose that Then

(Q X I)

Let

B C Q X I b e an n - b a l l , Q a c o m p a c t g - m a n i f o l d .

B H [(Q X 0) vj (9Q X I)] is a face of ( Q X 0) \J (3Q X I) U B .

B.

Suppose that

n < q-2.

-119-

Proof.

Let

F = B f\ [(Q X 0) u (SQ X I)].

h o m c o m o r p h i s m with h(x,0) = x. morphism

L e t h; F X I —> B b e a P . L .

By L e m m a 4. 9, t h e r e i s a P . L . h o m e o -

k: Q X I — > Q X I, l e v e l p r e s e r v i n g ,

I local c o - d i m e n s i o n ^ 2 i n k B . Ifirst c o o r d i n a t e ) . Iwith (F X I)

Consider

It i s of l o c a l c o d i m

K«h

Imorphism, such that

i s of

(p » pro}, on t h e

2 in F X I, a n d so i t s i n t e r s e c t i o n in ( F X I) ^ ( F X O).

Hence

k ' : F X I —> F X I, a l e v e l p r e s e r v i n g h o m e o -

Sjirlk'CK))

jection of F X I o n t o

S^CpjkB)

^(S^(pk|B)).

X 0) i s of l o c a l c o d i m e n s i o n

|we m a y apply L e m m a 4 . 9 to find

Let

such that

h a s l o c a l c o d i m ^ 1 in k ' ( K ) , -n- t h e p r o -

F.

= koho(k')"^: F X I—» Q X L

Is of l o c a l c o d i m e n s i o n ^ 2} i n

F X I.

T h e n S^(p

= k't h " k " ^ ( S ^ ( p | k B ) )

M o r e o v e r , S ( """jS (pc
I

^

codimension > 1 in S {po
hence

-irlsjpoip)

This is b e c a u s e

k'

is n o n - d e g e n e r a t e .

and k a r e l e v e l p r e s e r v i n g

b o u n d a r y p r e s e r v i n g , a n d b e c a u s e of t h e definition of

smma 5 . 4 ,

Finally,

h.

H e n c e by

k h ( F X I) s u n n y c o l l a p s e s to k h ( F X I) n ((Q X 0) u (aQ(X I)).

* ence by L e m m a 5. 2,

(Q X I)

(Q X O) u (9Q X I)

k h ( F X I).

A p p l y i n g k"

1

•both s i d e s of t h i s c o l l a p s e , w e s e e t h a t (Q X I) ^^(Q X 0) u ( a o X I) U B. T h e o r e m 5. 2. m-manifold IK.

M.

Let

K CT K C M , o ^

Suppose

K , K P . L . e u b s p a c e s of t h e c o m p a c t o K

a n d d i m ( K - K ^ ) <, m - 3 .

Then

Proof. r - b a l l and

It suffices to s u p p o s e t h a t

B o K

= F , a face of B .

o

triangulated as subcomplexes. in M.

Let

K

i.e.

Subdivide

M with

K, K , and o

B

N be a 2nd d e r i v e d n e i g h b o r h o o d of

M is a l s o a r e g u l a r n e i g h b o r h o o d of

regularly.

c l ( K - K ^ ) = B, a

K^

and N a l s o m e e t s t h e

H e n c e , by t h e g e n e r a l i z e d a n n u l u s t h e o r e m , t h e r e e x i s t s a p . l j

homeomorphism h: c l ( M - N ) — > F r N X I with h(x) = (x, 0)

if X € F r N.

Now, N n B i s a r e g u l a r n e i g h b o r h o o d of F So N O B

i s a n r - b a l l and N

of c o l l a p s i b l e s e t s .

Therefore

in B m e e t i n g

9B

rega

B i s an ( r - 1 ) b a l l , b e i n g r e g u l a r neighbo: ( F r N) n B i s a l s o a n ( r - l ) b a l l . '•I

Let

B^ = c l ( B - N ) .

b e t h e r e s t r i c t i o n of [i: Q X I ^

h above.

Let

Q = F r N.

—>

We m u s t now c o n s t r u c t a p. 1. h o m e o m o r p l

t, and

b e a p. 1. h o m e o m o r p h i s m s u c h t h a t

\ (I X 0) = ((I X O) u (O X I)).

Set X

\(l,t) = (l,t)

(Exercise: Construct

be a bo\andary c o l l a r .

li: Q X I — 5 > Q X I

X

T h e n define

by jji(c(x, s), t) = (c(x, \ ^ ( s , t)), ^ ^ ( s , t))

l i ( y . t ) = (y, t) T h e two d e f i n i t i o n s a g r e e on t h e o v e r l a p ( w h e r e The map

L e t h: F^ —> Q

Q X I t h r o w i n g Q X 0 into (Q X 0) ^ ( 9 0 X I),

Let fox e v e r y

F^ = B n F r N .

p. i s p. 1.

if X 6 9Q if

y € c l ( Q - I m c).

s = 1 in t h e f i r s t definition

F o r on I m ( c ) X I, it i s t h e c o m p o s i t e s

-121-

Im(c) X I

""

^ ^ >

ao X I X I ^ ^ ^

> aQ X I X I

^ ^ ^

> Im(c) X I .

This a l s o s h o w s t h a t i s i s a h o m e o m o r p h i s m . Now, iJL(hB^) i s a b a l l in Q X I JjihF^. I Hence t

meeting

d i m B^ < d i m M - 3 < d i m Q - 2. c l ( M - N ) \ ( F r N) v B , a p p l y i n g \ 1

/Therefore [Note: ^

m \ n v

If

B^.

Let

QXI

X O) . (9QXI)w jihB^.

- 1 - 1 : |J. to the p r e c e d i n g c o l l a p s e .

h

so

are simplicial, L

liirst deriveds, then Proof.

Therefore

NL-B^ = N U B ,

L C L C J o

(Q X O) i.; (8Q X I) in t h e face

o

M ^N

B.

But

N

full in J a n d L ' C L ' o

are

; J') u L' i A^^ =>J3implices of

J-L

which meet

L^, in o r d e r of

i e c r e a s i n e d i m e n s i o n . Then A.P. N(L' ; J ' ) A. , N(L' ; J ' ) , F o r fr 1 o 1 o f 1 N(L^5 J ' ) i s a r e g t i l a r n e i g h b o r h o o d of A^ L,^ w h i c h m e e t s A^ IND so A. 1

J'

N(L';J') o

i s a face of t h e b a l l A. M N ( L ' ; J ' ) . 1 o

regularly,

)

Unknotting of B a l l P a i r s a n d S p h e r e P a i r s

jptation;

If P = (B^, b'^) i s a p r o p e r b a l l p a i r , t h e n o P and v P

Jfe t h a t v P

(VB^. VB'^),

V a j o i n a b l e point.

is p r o p e r .

L e m m a 5.6. |t

denotes ball pair

denotes the sphere

Let

P a n d Q b e two u n k n o t t e d b a l l p a i r s of t y p e ( q , t t n ) .

Q be a p . 1. h o m e o m o r p h i s m .

^I'pHsm k : P

Q w i t h k|=P = h .

Then there exists a P . L. homeo-

Proof .

^

So t h e r e a r e P . L. h o m e o m o r p h i s m s h: P —5> Q

P —> v P , Q —> v 6

exter

a n d we c a n

conically.

L e m m a 5. 7.

T h e c o n e a n d s u s p e n s i o n ( j o i n w i t h a s p h e r e ) of an

ball o r sphere pair is an unknotted ball o r s p h e r e p a i r . Proof.

Exercise.

By B

we d e n o t e t h e s t a t e m e n t :

a r e unknotted.

Let

S

a l l p r o p e r b a l l p a i r s of type (q, •

= " a l l s p h e r e p a i r s of t y p e (q, m )

a r e unknottec

q, m L e m m a 5.8.

Proof.

Let

B

q, m

implies

P = (S^, s " ^ ) .

L e t V be a v e r t e x of

S

q, m

Let

K be a t r i a n g u l a t i o n of

S*^!

K

. L e t P . = ( s F ( v ; K ) , i F ( v , K )). Let o 1 o P = c l ( P - P , ) = ( K-st(v;K) , K -st(v;K ) ) . T h e n P and P a r e both! ^ 1 o o J. ^ r

»

«

p r o p e r b a l l p a i r s , and

P^ = P ^ .

homeomorphism

^ vP^

The identity

•

P ^ —> P ^

e x t e n d s t o a p. 1.

»

P^

•

a n d a p . 1. h o m e o m o r p h i s m »

P

P^—> v'P^-

,

So

,

i s p. 1. h o m e o m o r p h i c ( a s a p a i r ) to v P ^ u v ' P ^ , a s u s p e n s i o n of

P^ an*

so u n k n o t t e d . Definition.

A fac e of the p r o p e r b a l l p a i r

pair

F =

fine

c l ( P - F ) = (aB^ - A^"^, Lemma 5.9.

with A ^ ' ^ C

9B^ a n d

P = (B*^, b"^)

is a p r o p e r ba

=

- a " ^ " ^ ) , w h i c h i s a l s o a face of

Wed P.

L e t P and Q be u n k n o t t e d b a l l p a i r s of t y p e (q, m )

in a c o m m o n f a c e . T h e n if

B

which

^ j i s t r u e , P VJ Q i s a n u n k n o t t e d b a l l P' q-1, m - 1

-123-

Proof .

Let

F be the c o m m o n face.

Let

P^ = cl{P-F), Q = cl(Q-F). *

B

. J implies q-l,m-l

F,P.

1 2

hornoemorphisms preserve to p - L

a r e unknotted.

boundaries.

Then

F

i s u n k n o t t e d a s p. L

B y 5.6, t h e i d e n t i t y

F—3> F

extends

homeomorphisms: aF , H^.

-> b F

F -

-> c F u h,:P "1

>aF<^bF

e x t e n d s to

'extends t o

abF

k.: P

2

and

h

1

k : Q — > b c F , both h o m e o m o r p h i s m s ,

k^l P U Q

5> a b F i. > bcF '-X a F v i c F

L e m m a 5. 10, aeighborhood of

b"^

Let in

(B*^, B"^) B^.

h : Q —> b F u c F J

^

is

So •iknotted.

be a p r o p e r b a l l p a i r .

Then

B^ ^

Let

and

N be a r e g u l a r i m p l y (N, b " ^ ) ,

proper ball p a i r , is unknotted. Proof.

Let

K C K triangulatt

b " ^ ^ B*^, a n d s u p p o s e t h a t

iy u n i q u e n e s s of r e g u l a r n e i g h b o r h o o d s , w e m a y a l s o s u p p o s e t h a t „ \es \es = N(K ; K " ) w i t h o u t l o s s of g e n e r a l i t y , L e t K = L V . . . \ L K E . = (N(L'! ; K") , N(L." 5 K " ) , w h e r e 1 1 1 o = (N, K^

),

M o r e o v e r , E.

o = V £ K .

K" = 2nd d e r i v e d s u b d i v i s i o n .

is a b a l l p a i r , by r e g u l a r n e i g h b o r h o o d t h e o r y ,

fis e a s i l y s e e n to b e p r o p e r , E ^ = ( s t a r ( v ; K ), s t a r ( v , K ' ^ ) = vAlink(v;K",), l i n k ( v ; K"^)), a c o n e on a i e r e p a i r of t y p e (q l . m - l ) . R

E. IS u n k n o t t e d . 1-1

P\U

H e n c e E^^ i s •.nknotted. L

- L 1 1 - 1

'

A

U,

S u p p o s e by i n d u c t i o n

A = aB,

Then

E.

E. ^ P u Q, 1 1 - 1

E. = E. 1

1-1

U P u Q, w h e r e P = (st(A,K"),st(A.K'^)) Q = (st(B;K"),st(^;K^))

(See r e g i i l a r n e i g h b o r h o o d t h e o r y , C h a p t e r III^ Now P = A ( l i n k ( A ; K " ) , l i n k ( A ; K J J ) ) . A

or a ball pair, according as 9(link(A;K")) o

The link p a i r is e i t h e r a sphere

"

A

A € Int K^

•

of A € K^

.

Since

= l i n k ( A ; k " ) c l i n k { A ; K " ) = a(link(A; K")), in t h e e v e n t o

t h i s p a i r i s a p r o p e r b a l l p a i r o r a s p h e r e p a i r of t y p e ( q - 1 , m - l ) .

A e k! Hence

i s Tinknotted. N o w we a r e g o i n g to p r o v e t h a t Let

L = ( l i n k ( A , K ' ) , link(A; K^)) .

P = AP^.

Let

p t P ^ — L

P ( ^ ) = 0-

if

P n E^ ^ i s a face of

Let

P

and

E^

P^ = (link(A; K " ) , link(Ar K ^ ) ) .

That

b e t h e p s e u d o - r a d i a l p r o j e c t i o n g i v e n by 0- € l i n k ( A ; K ' ) .

(See r e g u l a r n e i g h b o r h o o d t h e o r y . ) . We n o w i n t r o d u c e s o m e n e w n o t a t i o n , b y w r i t i n g

P = (P, , P ) D

(P"big" and P " s m a l l " ) .

etc.

onto t h e d e r i v e d ideighborhood of

(aB)

Then

in L

P

t h e d e r i v e d n e i g h b o r h o o d of

(aB)

in

L

.

sends

and s e n d s D

S

P

(E. S

)

onl

1— X S

Using the s u b l e m m a appearing

s t h e end of t h i s p r o o f , w e s e e t h a t t h e i m a g e o£ P r> of t y p e

(q-1, m - l )

unknotted p a i r . P u E.

a n d so a face of P

S i m i l a r l y , ( s e e r e g . nbhd. t h e o r y ) ( P

. a n d of Q;

1-1

and of

hence E.

1

is unknotted,

is a p r o p e r ball p Theref-re

P IS a fac<

-125-

S u b l e m m a 5. 10. 1.

Let X C M C Q ,

MCQ

a manifold p a i r ,

jvl r> 8Q = 8M.

A s s u m e e v e r y t h i n g is t r i a n g u l a t e d so t h a t

M and Q.

N = d e r i v e d n e i g h b o r h o o d of X i n Q.

I

Let

Proof.

iitriangulates I* °

First.

F r , o M

XCMCQ,

M) = F r ^ ( N ) n M. U

with

X i s full in b o t h

Then

a(N n M) = ( a N ) n M . LQ

F o r say

L full in K , K full in K. o o

|>e f i r s t d e r i v e d s u b d i v i s i o n s , and s u p p o s e

N = N(L';K').

K

o

C K

L e t L ' C K' ^ o Then

k'

N n M = N(L';K^).

-Say A € K' . T h e n A e F r ^ J N n M) if and o n l y if A A L = ^ but t h e r e e x i s t I, o M B € L w i t h BA € K' . A e F r ^ ( N ) O M if and o n l y if A € K , A n L = ^ and o Q o 1 t h e r e e x i s t s B € L ' w i t h AB e K ' . It i s c l e a r t h a t t h e s e c o n d i t i o n s a r e e q u i •valent.

Therefore

Now,

Fr.iN M) = F r _ ( N ) a M. M U (8N) rv M = ( ( F r ^ N ) o M) u (N ^ M M) =

[But M r \ dQ =

r^ M) u (N r^.^M).

dU.

C o r o l l a r y 5. 11. i Proof .

If

If g - m > 3, t h e n ^

B

and

m-l,q-l

q - m k 3 , t h e n by T h e o r e m 5. 2,

[(since b o t h c o l l a p s e to a p o i n t . ) (B^, b " ^ )

30).

S

m-l,q-l

imply B

m, q

B^

H e n c e B*^ i s a r e g u l a r n e i g h b o r h o o d of

b"^.

i s u n k n o t t e d by 5. 10.

T h e o r e m 5. 12.

If q - m > 3, t h e n e v e r y p r o p e r b a l l p a i r o r s p h e r e p a i r

pf type ( m , q) i s u n k n o t t e d . Proof. B

We a l r e a d y h a v e t h e following S

m, q

m, q

and

start the induction, a s s u m e

S

m, q

==> B

implications:

., ,, m+l,q+l

m = 0, q > 3.

,

if

q-m > 3 ,

So we h a v e a p o i n t , P

s a y , in t h e

-i; i n t e r i o r of B ^ .

Triangulate

regular neighborhoods

§5.

B*^ w i t h

P

[ P C B*^] ^ [ P (2, s t a r ( P , K ) ]

By t h e u n i q u e n e s s of

w h i c h is c l e a r l y unknottec

Unknotting of E m b e d d i n g s of B a l l s in B a l l s . Now we a s k t h e following q u e s t i o n :

f, g : B " ^ — w i t h

L e m m a 5. 13. , m u 9B^ h : B"^

If B

f(x)

Proof.

^

k : b'^ — > B ^

Therefore

g(x), a l l x e B"^ ?

extending

k" =

k"(B"^ = k'h"^.

Let

: A"^

be t h e s u s p e n s i o n of Then S p

such

(B^^; b " " ) —> > A"^.

Let

P ( i . e . j o i n up p w i t h the

i s the i d e n t i t y on

-

> B ^ i s t h e i d e n t i t y on

SB^.

A^""^). Moreover,

T h e n k | dB^^ = k ' | a B ^ = h | 9 B ^

= h.

=

Let

f, g: b " ^ Assume

a r e ambient isotopic keeping h auch that

k ' : ( B ^ b " ^ ) ( B ^ ; B"^)

P = Qfk'h

Let k = (k")"^k'.

L e m m a 5. 14. =

Let

(Sp ) or: B*^

k|9B"^ =

if

h.

So k ' h " ^ I dB^^ = i d e n t i t y .

>

i d e n t i t y on A*^'"^).

= g'^aS^l), is there

is a P . L . h o m e o m o r p h i s m , then t h e r e exists a

be a P . L . h o m e o m o r p h i s m . 2 |3

onto

By L e m m a 5.6, t h e r e e x i s t s

t h a t k' I dB"^ = h | 9 B ^ .

embeddings

C B ^ i s an u n k n o t t e d p r o p e r b a l l p a i r and

> B""u

P . L. h o m e o m o r p h i s m

given P . L. dB^^ =

f

an ambient isotopy throwing

topy

as a vertex.

h^o f = g a n d

?> B ^ be P . L .

embeddings,

q - m > 3 and f | BB"^ fixed. h leaves

g]

T h e n f and g

( T h a t i s , t h e r e e x i s t s an a m b i e n t i s o 9B^

fixed.)

-127-

Proof.

There exists a P . L. h o m e o m o r p h i s m

h(fB"^) = The m a p

as

(B^.fB^")

fg ^h: fB™ —=> fB^^

f g ' ^ l f l a B " " ) = hlfCaB""). homeomorphism. kjB^

and

[|6.

and

So hWfg'^h: aB'i . f B " " - ^ dB"!,.

By 5 . 1 3 , t h e r e e x i s t s a P . L .

> b'^ w i t h k | a B ^ = h

s u c h that

(B^, gB"^) a r e u n k n o t t e d p r o p e r ball p a i r s .

i s a P . L. h o m e o m o r p h i s m ,

a | f B ' " = gf

.

fB""

isaP.L.

homeomorphism

and k | fB^" = fg"^h.

is a P . L . h o m e o m o r p h i s m , and a I

h j B^ —> B ^

The m a p a = h k " ^ : b'^ ^ So

Off = g.

B^

Moreover,

= i d e n t i t y , so a i s a m b i e n t i s o t o p i c to t h e i d e n t i t y k e e p i n g

BB'^ fixed.

Unknotting C o n e s We s t a t e t h e following w i t h o u t proof: ( L i c f c o r i s h ' s T h e o r e m ) If f and

g a r e P . L . e m b e d d i n g s of v . K into

|and V a j o i n a b l e p o i n t , w i t h f'^CSB^^) = >dimv;K ^ q - 3 , t h e n

B^,

K a polyhedron

= K, and if f|K = g | K , and if

f and g a r e ambient isotopic keeping

aB*^ fixed.

-128] C h a p t e r VI: § 1.

Isotopy

C o n c o r d a n c e , I s o t o p y , A m b i e n t I s o t o p y , a n d I s o t o p y by M o v e s . Definition .

The e m b e d d i n g s

f and g of

a r e c a l l e d i s o t o p i c if t h e r e e x i s t s a P L m a p 1)

=

E q u i v a l e n t l y , we s a y t h a t p r e s e r v i n g embedding ( F(x,t) = ('F(x),t)). We s a y t h a t

F: M X I

> Q such that

= ^ ( x . t ) .) f and

F : MX I

g a r e i s o t o p i c if t h e r e e x i s t s a level]

> Q X I such that

The relation between

f and

F

and F

^^ = f and F^ = g, i s "Flx.t) = ( F ( x , t ) , t ) . i

g a r e a m b i e n t i s o t o p i c if t h e r e e x i s t s an a m b i e n t

h: Q X I —> Q X I w i t h h o f = g.

We s a y t h a t F:MXI X e

Q (PL spaces)

F^ = g

2) F^ is an e m b e d d i n g ,

isotopy

M into

f and g a r e c o n c o r d a n t if t h e r e e x i s t s

a P L embedding

- > Q X I w i t h F ( x , 0) = (f(x), 0) a n d F ( x , 1) = (g(x), 1) for a l l

M. »

Definition . morphism,

h|Q-X

sup{h) C X C Q.

h; Q —> Q is a P L h o m e o -

= s u p p o r t of h.

Then

We s a y h

^

h i s s u p p o r t e d b y X if and o n l y j

is the i d e n t i t y .

If Q i s a P L in

Q is a P L s p a c e and

sup(h) = j^xe Q | h x / x |

s u p p o r t e d by X if if

If

q - m a n i f o l d and

Q as a P L subspace, then h

a p r o p e r m o v e if e i t h e r with sup(h) C B*^,

h is s u p p o r t e d by a P L q - b a l l containec

is called a m o v e .

We c a l l t h e m o v e

h | 8 Q = identity or there exists

such that

B ^ o 9Q i s a face of

B^.

B ^ C Q,

h

B ^ a q-bal

-129-

Definition . we s a y t h a t h^, . . .

If f a n d g a r e e m b e d d i n g s of

Q,

f and g a r e i s o t o p i c b y m o v e s if t h e r e e x i s t s a finite s e q u e n c e of p r o p e r m o v e s of

Q with

h, o . . . o h Of 1 r L e m m a 6. 1. below it

M into t h e q - m a n i f o l d

=

g. ®

E a c h of t h e following s t a t m e n t s i m p l i e s t h e o n e s

(f and g e m b e d d i n g s

M

Q*^).

a) f and g a r e i s o t o p i c by m o v e s . b) f and

g a r e ambient isotopic

c) f and

g are isotopic.

d) f and g a r e c o n c o r d a n t . Proof, with

b) =?> c ) .

h^f = g . c)

Define

d) .

F: MX I

-> Q X I b y F = h o ( f X l ) .

It suffices to s h o w t h a t a n y m o v e i s a m b i e n t i s o t o p i c to t h e

So l e t

C a s e 1;

h: Q X I —5> Q X I be an a m b i e n t i s o t o p y

Clear.

a) ==> b). identity.

Let

h; Q —> Q be a m o v e . Sup(h) C B*^ C Q a n d h | 9 0 = i d e n t i t y .

Then h l a s ' ^

identity, so h b'^ i s a m b i e n t i s o t o p i c to t h e i d e n t i t y k e e p i n g Hence h i s a m b i e n t i s o t o p i c to t h e i d e n t i t y ( k e e p i n g C a s e 2;

Supp h c B ^ c Q,

I F .J == c l ( 9 B ^ - F ) .

a PL define

Then by continuity,

h o m e o m o r p h i s m sending k: A^ X I

B ^ n BQ = a face

F

F

of

B^.

Let

Let a

B^

into a p r i n c i p a l face

X 1 = a h a " ^ , k | c l ( A - A^) X I = i d e n t i t y ,

aB^^ fixed.

Q-B*^ fixed).

h F^ = i d e n t i t y .

> A^ X I b y f i r s t p u t t i n g

is the

A^

— A

of A*^.

k | a'^ X 0 = identity, k(A^; l / 2 ) = (A^, i / 2 )

and

-130A^ = b a r y c e n t e r of A

q

XI.

Then

A^; t h e n e x t e n d i n g

k , b y j o i n i n g up l i n e a r l y , to

k is an a m b i e n t i s o t o p y e n d i n g in aha

C1(A - A^) fixed. ing 01(6*^ - F )

Therefore

fixed,

T h e o r e m 6. 2.

and so

h|B^

-1

and

i s a m b i e n t i s o t o p i c to t h e i d e n t i t y k e e p -

If Q i s a c o m p a c t q - m a n i f o l d and H: Q X I — Q X I

p r o p e r m o v e s of Q s u c h t h a t

K

Let

,n+l XI C E .

embedding. , let

H

K triangulate

= h.o 1 1

Q.

Given a l i n e a r m a p

... oh

Assume

h^ , . . . , h

. r

(KJ ^ E " , and v i e w

^ : K —> I,

K

K XI

i s an

p^ : K X I -—-> K be p r o j e c t i o n on the f i r s t f a c t o r .

Q
and p ( K X I) b e s u b d i v i s i o n s m a k i n g H J A / K X I )

0- 6 p(K X I).

L e t i C cr

whose projection under simplex

H ^(cr ).

of l e s s t h a n

Since

-n/Z

of l e s s t h a n

be a v e r t i c a l l i n e s e g m e n t in

P : K X I —3> I i s a point). H is level preserving,

with the v e r t i c a l .

an upward pointing v e c t o r , then

Given

H

such that

simpl

cr (i. e. a l i n e

i ) i s a l i n e in the H \ i ) m a k e s an angle

( M o r s p r e c i s e l y , if i

is v i e w e d a s

i ) is a v e c t o r which m a k e s an angle

H ^(f ) i s p o s i t i v e . ) M o r e o v e r , b y l i n e a r i t y of

H on s i m p l i c e s , t h i s a n g l e i s i n d e p e n d e n t of t h e c h o i c e of f Since

(3(KXI):

ir/Z w i t h , s a y , t h e v e r t i c a l unit v e c t o r ; e q u i v a l e n t l y , t h e l a s t

c j - o r d i n a t e of t h e v e c t o r

tical.

of

= Let

Let

Let

i

h i s i t s e l f a m b i e n t i s o t o p i c to t h e i d e n t i t y .

i s an a m b i e n t i s o t o p y , t h e n t h e r e e x i s t s a finite s e q u e n c e

Proof.

keeping

^(KXI)

H ^(i)

i s a finite s i m p l i c i a l c o m p l e x , t h e ^ c - x i e t i

m a k e s an a n g l e

l i n e in a s i m p l e x of

ir- a, (

p(K X I).

-

ii/< 'r, '

< (p w i t h t h e v e r t i c a l I'f H i s tn\ ^ - -rical

-131-

On t h e o t h e r h a n d , t h e r e e x i s t s < 5, then if (i,^)(cr)

has diameter

cr e K, a n y l i n e s e g m e n t c o n t a i n e d in the ( c o n v e x l i n e a r c e l l )

m a k e s an a n g l e of at l e a s t

Now meets

5 > 0 s u c h t h a t i£

separates

(l X

(p w i t h t h e v e r t i c a l .

K X I.

in at l e a s t o n e point.

s u c h a path and \

1

=

[s | K s ) <

KXi

T h i s is b e c a u s e if

t h e n if \ ( I ) n

^

(s (X^(s) > ^ ( s ) / and

That is, a path from

=

I —> K X I i s

the sets

f o r m a s p l i t t i n g of

I by disjoint,

n o n - e m p t y o p e n s e t s , c o n t r a d i c t i n g t h e c o n n e c t e d n e s s of I. "broken line"

H ' ^ ( X X I), X £ K, m e e t s

However,: ^

and

(i,^)K

X I)

and IQ h a s t

m e e t in at m o s t one point.

away f r o m | ) m a k i n g an a n g l e of at m o s t T|

lies inside

and (when w h e n d i r e c t e d