The General Triangulation Theorem

9. The following proposition summarizes the results of t h e preceding chapters.

9.1. P R O P O S I T I O N . I n order to prove that every locally polyhedral space o/ dimension n has property T, it is su//icient to prove that property T is possessed by those spaces X of dimension n such that (I) X has no singular points, and (2) X = C I ( X - X ' - I ) .

Proo[. L e t X be a n y locally p o l y h e d r a l space of dimension n. L e t M = X - X "-1.

M is a locally p o l y h e d r a l space of dimension n; if 2~ r has p r o p e r t y T, so does X : L e t t h e points of X ~ be d e n o t e d b y xl, x 2 .. . . ; let (Ui, k,, J,) be a p o l y h e d r a l n e i g h b o r h o o d of xi;

a n d let (K, h) be a proper t r i a n g u l a t i o n of X "-1 which is locally compatible with (Ui, ki, Ji) at x~, for each i. N o w ( K , h ) induces a proper t r i a n g u l a t i o n of F r M =217 "-1 (by 4.6). I f x, is in 2]I, t h e n (U~ • M , ks, L~) is a p o l y h e d r a l n e i g h b o r h o o d of x i in 211, where L , is a sub- c o m p l e x of Jt. I f M has p r o p e r t y T, there is a t r i a n g u l a t i o n (L, g) of M which is compatible with (K, h) on F r M a n d locally compatible with (U i N 2~,/c,, L~) at x , B y 2.4, there is a t r i a n g u l a t i o n (K, h) of X such t h a t t h e induced triangulations of X "-1 a n d M are sub- divisions of (K, h) a n d (L, g), respectively. Hence (K, h) is proper, a n d it is easily shown t h a t (K, h) is locally compatible with (U~, k~, J~) at x~.

T H E T R I A N G U L A T I O N O F L O C A L L Y T R I A N G U L A B L E S P A C E S 8 9

Let ~]I be denoted b y Yn; Yn is a locally polyhedral space satisfying the hypotheses of Chapter I I I , for j = n - 1. Let Y~-I denote the composition space ( Y~)*; then Y~-I is a locally polyhedral space which also satisfies the hypotheses of Chapter I I I , for j = n - 2 (by 7.8 and 7.9). I n general, let Yk = (Yk*,)*; Yk has no singular points of index greater than k - 1. Consider the space Yo; it satisfies (1) and (2) of the present proposition. If Y0 has property T, it follows from a finite number of applications of 8.5 t h a t Y~ = M has property T also. Then X has property T as well.

9.2. THEOREI~. Let M be a 2-manifold with boundary. Given a triangulation of Bd M, there is a triangulation o / M which is compatible with the given triangulation o/ Bd M.

Proof. Let iL, g) be a triangulation of Bd M. There exists a triangulation (K, h) of M ([6], p. 167). Clearly there are subdivisions L', K ' of L, K respectively, such t h a t h-lg carries each simplex of L ' onto a simplex of K ' (not necessarily linearly). Let k be the linear map of L ' into K ' which agrees with h lg on the vertices of L', and let / = kg-lh on B d I K ] . Define / as the identity on the remainder of the 1-skeleton of K, and extend / to I KI b y means of radial lines from the barycenters of the 2-simplices of K ' . Let h = h/-1.

Then (K', h) is a triangulation of M which is compatible with (L, g) on Bd M.

9.3. DEFINITION. If X is a space, a subset A of X is said to be weakly locally tame in X if for each point x E A , there is a neighborhood U of x, a complex Kx and a homeo- morphism h of U onto a polyhedron in Kx such t h a t h ( U N A) is also a polyhedron in Kx.

If it is required t h a t X ~ I K1 for some complex K and K x ~ K, then A is said to be locally tame in K ([2], p. 146).

9.4. THEOREM.

1/

I f M is triangulated and 2t is a locally tame closed subset o / M , let there be given a triangulation of A U Bd M in which A is a polyhedron. Then there is a triangulation o / M which is compa- tible on A [J Bd M with the given one.

Proof. This is a theorem of Bing ([2], p. 156).

9.5. LEMMA. Let X be a locally polyhedral space of dimension n (n S 3), such that X has no singular points and X = C l ( X - X n - 1 ) . Let x E X ~ let (St x, h, L) be a polyhedral neighborhood on X , and let v =h(x). Then F r ( S t v) is a connected ( n - 1)-manifold with boundary.

Proof. F r ( S t v) is a complex J consisting entirely of (n - 1)-simplices and their faces.

E v e r y (n - 1)-simplex t of St v is a face of exactly one or two n-simplices of St v, since otherwise its points would have index n - 1 and St t - t would not be connected. As a result, every (n - 2)-simplex of J is a face of exactly one or two (n - 1)-simplices of J .

90 J A M E S M U N K R E S

l J ] is obviously connected, since otherwise S t v - v would n o t be connected a n d x would be a singular point of X (by 5.7). These facts are sufficient, for n _< 2, to ensure t h a t I J ] is a connected (n - 1)-manifold with b o u n d a r y .

If n = 3, we m u s t also show t h a t if w is a v e r t e x of J a n d S t w denotes its star in J , t h e n S t w - w is connected. This is e q u i v a l e n t to t h e s t a t e m e n t t h a t for e v e r y 1-simplex s of S t v, S t s - s is connected. If the points of s h a v e index 1, this follows f r o m 5.7. If t h e y h a v e index 2, S t s - s consists of t h e connected set S t s - h ( X 2) plus some of its limit points, so t h a t S t s - s is connected. I f t h e points of s h a v e index 3, so does e v e r y point of S t s, so t h a t e v e r y 2-simplex of S t s is a face of e x a c t l y two 3-simplices of S t s. Ac- cordingly, those closed edges of t h e 3-simplices of S t s which are opposite to s f o r m a finite n u m b e r of h o m e o m o r p h s of a circle. Since each point of s has a'3-cell neighborhood, there is only one such circle. T h e n S t s - s is connected.

9.6. THEOREM. L e t X be a locally p o l y h e d r a l s p a c e o / d i m e n s i o n n (n < 3). T h e n X h a s p r o p e r t y T .

P r o o / . B y 9.1, we m a y assume t h a t X has no singular points a n d t h a t X = Cl ( X - X ~-1).

L e t x 1, x e . . . . d e n o t e t h e points of X ~ a n d let (Us, k~, Js) be a p o l y h e d r a l n e i g h b o r h o o d of x i. L e t (K, h) be a proper t r i a n g u l a t i o n of X ~-1 which is locally compatible with (Ui, ks, Js) at xi, for each i. W e assume t h a t t h e sets U s are disjoint. W e also assume t h a t for each i, there is a n e i g h b o r h o o d V~ of h -1 (x~) in I K ] such t h a t V~ is a subcomplex of K a n d t h e m a p k i h is a linear isomorphism of this subcomplex into Js. This involves no loss of generality, since we m a y o b t a i n this situation b y passing to a p p r o p r i a t e subdivisions of t h e complexes K, J i , J2 . . .

L e t Ys d e n o t e the v e r t e x k i ( x s ) of Js, a n d let L~ denote t h e subcomplex C l ( S t Ys) of J~. L e t L d e n o t e t h e complex which is t h e disjoint union of t h e complexes Ls, a n d let g be the m a p of ILl into X defined b y setting g equal to kV 1 on [L~]. T h e n (L, g) is a t r i a n g u l a t i o n of a subset of X. I t induces a t r i a n g u l a t i o n on g ( L ) ( ~ X n-1 which is equivalent to the one induced b y (K, h). B y 2.4, there is a t r i a n g u l a t i o n (K, h) of g (L) U X ~-1 which induces subdivisions of (L, g) a n d (K, h) on g (L) a n d X n 1 respectively.

L e t z i d e n o t e the v e r t e x h -1 (x~) of K', let K ' d e n o t e the first b a r y c e n t r i c subdivision of K, a n d let S t z s a n d S t ' z~ d e n o t e t h e stars of z~ in K a n d K ' respectively. L e t L d e n o t e t h e s u b c o m p l e x

h -1 ( X n - l ) U ( U~ e l ( s t ' zi) )

of K'. L e t Y = h(L). L e t N = X - U ~ h ( S t ' zs). T h e n N a n d Y are closed subsets of X, a n d N fl Y = Z = Y - U ~ h ( S t ' z , ) = ( N fl X ~-1) U ( ( J s F r ( h ( S t ' z s ) ) ) .

N o w (L, h) induces a t r i a n g u l a t i o n of Z; we seek a t r i a n g u l a t i o n of N which is compatible

T H E T t t I A N G U L A T I O ~ OF L O C A L L Y T R I A ~ G U L A B L E S P A C E S 91 with (L, h) on Z. These triangulations of N a n d Y will fit together (by 2.4) to give a triangula- tion of X, which is t h e n a u t o m a t i c a l l y compatible with (K, h) on ) ~ - 1 a n d locally compatible with (Ui, k~, J~) at x~, for each i. OUT t h e o r e m will t h e n be proved.

The crucial fact here is t h a t for n < 3, N is an n-manifold with b o u n d a r y , a n d B d N c Z . I t is clear t h a t each point of N which is n o t in a n y set F r ( h ( S t ' z~)) has a n e i g h b o r h o o d whose closure is a closed n-cell, b y 5.8 (using t h e fact t h a t no p o i n t of N belongs to X~

Points of iV in F r (h (St' z~)) require special consideration. The set F r (h (St' zi) ) is entirely c o n t a i n e d in h ( S t z~), so t h a t t h e problem is r e d u c e d to considering neighborhoods of points of F r ( S t ' zi) in St z i - St' z~. (This is t h e reason we " d e l e t e d " the neighborhoods h ( S t ' z~) f r o m X to f o r m N r a t h e r t h a n the neighborhoods h ( S t zJ.) B u t b y 9.5, Cl(St z~) is merely the cone over a n (n - 1)-manifold w i t h b o u n d a r y . I t is easy to show b y direct combinatorial means t h a t each p o i n t of F r ( S t ' z~) has a n e i g h b o r h o o d in St z ~ - St' z~

whose closure is a closed n-cell, a n d t h a t it has no n e i g h b o r h o o d which is itself an n-cell.

H e n c e N is an n-manifold with b o u n d a r y , a n d F r ( h ( S t ' z ~ ) ) c B d N . I f x E B d N - [.J i F r (h ( St' zi)), t h e n x E X ~-1. H e n c e B d N c Z.

F o r n = 1 a n d n = 2, it is also t r u e t h a t B d N = Z. (This follows f r o m t h e fact t h a t N n X "-1 lies in B d N , b y 5.8.) T h e n b y 9.2, there is a t r i a n g u l a t i o n of N which is com- patible with t h e given t r i a n g u l a t i o n of B d N = Y, a n d o u r t h e o r e m follows. (If n = 1, 9.2 does n o t apply, b u t this case is trivial.)

I f n = 3, there is m o r e work to do. The difficulty here is t h a t it need n o t be t r u e t h a t Z = B d N , for N fl X n-1 m a y contain points of l n t N . The points of (Int N ) 0 X n-1 h a v e index 1, b y 5.8. L e t A denote t h e set X 1 fl N; t h e n Z = A U B d N . There is a t r i a n g u l a t i o n of N, b y 9.4; we shall prove t h a t A is a locally t a m e subset of N. E v e r y point of A has index l, for it c a n n o t h a v e index 0 a n d lie in N. L e t x be in A, a n d let x n o t be in a n y set F r ( h ( S t ' z~)). L e t (St s, Ic, H) be a p o l y h e d r a l n e i g h b o r h o o d of x such t h a t dim s = 1.

B y the a r g u m e n t used in 5.8, t h e closed edges of H opposite/~ (s) m u s t f o r m a h o m e o m o r p h of a line segment or a circle, a n d A N St s is m e r e l y t h e set s. I t is easy to see t h a t St s is a n e i g h b o r h o o d of x satisfying t h e h y p o t h e s e s for A to be locally t a m e at x. On t h e o t h e r hand, suppose x is in F r ( h ( S t " zi) ). T h e n h - l ( x ) is a point of some 1-simplex of St z~, a n d h -1 ( X 1) N St zi consists of some of t h e 1-simplices of St ze Using these facts, one m a y readily show b y combinatorial means t h a t there is a n e i g h b o r h o o d of h -1 (x) in St zi - St' zi which satisfies the hypotheses for local tameness of A at x.

A is a subcomplex in the t r i a n g u l a t i o n of Z induced b y (L, h), since t h e original t r i a n g u l a t i o n (K, h) of X ~ was proper. T h e n b y 9.4, there is a triangulation of N which is compatible with t h e t r i a n g u l a t i o n of A U B d N = Z induced b y (L, h). This completes t h e p r o o f of t h e t h e o r e m for n = 3.

92 JAMES MUNKR~:S

9.7. COROLLARY. Let X be a locally polyhedral space o/ dimension n (n < 3). Given polyhedral neighborhoods (U~, k~, Ji) o/ each x i E X ~ there is a proper triangulation o/ X which is locally compatible with ( U , ki, J~) at xi, /or each i.

Proo/. W e p r o c e e d b y i n d u c t i o n o n n. X n-1 is a l o c a l l y p o l y h e d r a l space of d i m e n s i o n less t h a n n, so b y t h e i n d u c t i o n h y p o t h e s i s t h e r e is a p r o p e r t r i a n g u l a t i o n (K, h) of X n-1 w h i c h is l o c a l l y c o m p a t i b l e w i t h t h e p o l y h e d r a l n e i g h b o r h o o d (U~ N X n 1, k~, Li) a t x~, for e a c h x~ E ( X ~ - I ) ~ = X ~ (L~ is a s u b c o m p l e x of J i ) . T h e n (K, h) is also l o c a l l y c o m p a t i b l e w i t h (Ui, k~, J~) a t x~, so t h a t 9.6 applies.

9.8. THEOREM. I / X is a locally polyhedral space o/ dimension n (n < 3), X m a y be triangulated.

9.9. LEMMA. Let X be a locally polyhedral space, and let Y be a closed admissible sub- space o / X . A n y proper triangulation o/ X induces a triangulation o/ Y.

Proo/. L e t ~ (x) - 1 if x E Y, a n d :r (x) = 0 otherwise. Since Y is a d m i s s i b l e , ~ is c o n s t a n t on e a c h s i m p l e x of e v e r y p o l y h e d r a l n e i g h b o r h o o d on X . B y 4.4 a n d 4.5, g i v e n a p r o p e r t r i a n g u l a t i o n of X , ~ is c o n s t a n t on e a c h s i m p l e x of X u n d e r t h i s t r i a n g u l a t i o n . Since Y is closed, i t is t h e p o l y t o p e of a s u b c o m p l e x of X.

9.10. THEOREM. Let X be a locally triangulable space o / d i m e n s i o n n (n ~ 3), and let Y be a weakly locally tame closed subset o / X . Then there is a triangulation o / X under which Y is a polyhedron.

Proo/. This generalizes T h e o r e m 8 of [2]. L e t B be t h e c o l l e c t i o n of all p o l y h e d r a l n e i g h b o r h o o d s ( U, h, L) on X such t h a t Y c o n t a i n s e v e r y s i m p l e x of U w h i c h i t i n t e r s e c t s . F r o m t h e d e f i n i t i o n of w e a k local t a m e n e s s , i t is clear t h a t B covers X . To s h o w t h a t IB is c o n s t a n t on e v e r y s i m p l e x of a n e l e m e n t of B, one uses t h e a r g u m e n t of 3.8 (with t h e a d d i t i o n a l f a c t t h a t t h e h o m e o m o r p h i s m g t h e r e d e f i n e d m a p s Y i n t o itself). T h e n B is a basis for a s t r u c t u r e . L e t ~ (x) = 1 if x E Y, o t h e r w i s e l e t cr (x) = 0. T h e n ~ satisfies t h e h y p o t h e s e s of 4.3, so t h a t b y 4.4, :r is c o n s t a n t on e v e r y s i m p l e x of e a c h p o l y h e d r a l n e i g h - b o r h o o d of t h e s t r u c t u r e on X . H e n c e Y is a n a d m i s s i b l e s u b s p a c e of X , a n d 9.9 applies.

9.11. R E M A R K . T h e s e r e s u l t s d o n o t i m p l y t h a t t h e t r i a n g u l a b i l i t y of t h e g e n e r a l l o c a l l y p o l y h e d r a l s p a c e of d i m e n s i o n n w o u l d follow f r o m a t r i a n g u l a t i o n t h e o r e m for t h e g e n e r a l n - m a n i f o l d w i t h b o u n d a r y , or f r o m a s t r o n g e r t h e o r e m i n v o l v i n g local tameness~

like 9.4. R e m o v i n g t h e s i n g u l a r p o i n t s f r o m X m a k e s X l o c a l l y a m a n i f o l d w i t h b o u n d a r y o n l y a t p o i n t s of i n d e x n, n - 1, a n d n - 2. I f n = 3, t h e p o i n t s a t w h i c h X fails t o be a m a n i f o l d w i t h b o u n d a r y a r e i s o l a t e d , a n d one can use p o l y h e d r a l n e i g h b o r h o o d s t o t r i a n -

T H E T R I A N G U L A T I O N O F L O C A L L Y T I ~ I A N G U L A B L E S P A C E S 93

g u l a t e a n e i g h b o r h o o d of this set. A f t e r d e l e t i n g these n e i g h b o r h o o d s , one has left a m a n i f o l d w i t h b o u n d a r y to t r i a n g u l a t e . I f n > 3, t h e set of p o i n t s where X fails t o be a m a n i f o l d w i t h b o u n d a r y has positive d i m e n s i o n , a n d it is n o t clear how one could proceed i n this case.

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