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UBC Theses and Dissertations

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UBC Theses and Dissertations

Homogeneity of combinatorial spheres Walker, Alexander Crawford 1968

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HOMOGENEITY OP COMBINATORIAL SPHERES by ALEXANDER CRAWFORD WALKER B . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS In the Department of * MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard • • THE UNIVERSITY OF BRITISH COLUMBIA November 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t udy . I f u r t h e r a g r ee t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Co lumb ia Vancouve r 8, Canada Date A b s t r a c t The o b j e c t of t h i s t h e s i s i s to cover the r e s u l t s of [1] from a piecewise l i n e a r p o i n t of view. The p r i n c i p a l r e -s u l t of [1] i s the theorem on the homogeneity of spheres, i . e . the complement of a c o m b i n a t o r i a l n - c e l l i n a c o m b i n a t o r i a l n-sphere i s a c o m b i n a t o r i a l n - c e l l . A piecewise l i n e a r p roof of t h i s theorem by a "long i n d u c t i o n " u s i n g r e g u l a r n e i g h -bourhoods and c o l l a p s i n g was gi v e n i n [4] . A d i r e c t p i ecewise l i n e a r p r o o f appeared r e c e n t l y i n [2] ; i t i s based on the e x i s t e n c e of a " c o l l a r " f o r the boundary of a c o m b i n a t o r i a l m a n i f o l d w i t h boundary. Our proof i s s i m i l a r t o the proof i n [ 2 ] . We proceed by-'induction on dimensions, p r o v i n g s i m u l -t a n e o u s l y the e x i s t e n c e of a c o l l a r f o r the boundary of a c o m b i n a t o r i a l m a n i f o l d with boundary and the homogeneity theorem. Prom [2] we adopted an argument which e l i m i n a t e s a~~ c e r t a i n c o m b i n a t o r i a l technique a p p l i e d i n [1] and i n v o l v i n g i n d u c t i o n on the l e n g t h of s t e l l a r s u b d i v i s i o n s . The r e s u l t s of [1] were p r e v i o u s l y i n t e r p r e t e d i n piecewise l i n e a r t opology by use,of a theorem i n [3]' s t a t i n g t h a t piecewise l i n e a r l y homeomorphic s i m p l i c i a l complexes have s u b d i v i s i o n s which are c o m b i n a t o r i a l l y e q u i v a l e n t i n the sense of. [ 1 ] . The t h e s i s i s d i v i d e d i n t o ' t h r e e p a r t s . The f i r s t g i v e s d e f i n i t i o n s and b a s i c p r o p e r t i e s r e l a t i n g t o s i m p l i c i a l complexes. The second concerns c o m b i n a t o r i a l m a n i f o l d s , and i n the t h i r d we p r e s e n t our proof of the piecewise l i n e a r homogeneity of spheres. Table of Contents Page 1. S i m p l i c l a l Complexes i n R n 1 2. Co m b i n a t o r i a l Manifolds 20 3. Homogeneity of Co m b i n a t o r i a l Spheres 31 Acknowledgements The t h e s i s was w r i t t e n under the s u p e r v i s i o n of Dr. E. L u f t whose constant help and encouragement was much a p p r e c i a t e d . I should a l s o l i k e t o thank the N a t i o n a l Research C o u n c i l of Canada f o r g i v i n g me f i n a n c i a l a s s i s t a n c e . - 1 -1. S i m p l i c i a l Complexes i n &n D e f i n i t i o n 1.1 Simplex i n ftn .; A k : A k = [y ... v^J the b a r y c e n t r i c . h u l l of l i n e a r l y Independent p o i n t s v^ e R n c a l l e d v e r t i c e s of A k . V n ' • k k ' i . e . A = (x : x e ft ,x = 2' ?.v, , §. > 0 and 2 = 1) i=0 1 1 1 1=0 x Dimension of A k ; dim ( A k ) dim (A k) = k-1 = one l e s s than /the number of v e r t i c e s of A k . Pace of A k ; A m < A k : ' I f f v ^ .... v^} c ( v 0 ... v k ) , then A m = Cv^ ... v£] .Is . '. " • ' a f a c e of A o f a p p r o p r i a t e dimension. .,, Boundary of A k ; 3Ak : 3Ak = ( A m : A m < A k , A m V A k ] = the s e t ' o f proper f a c e s of A k . ' . I n t e r i o r of A k , i n t ( A k ) : k n ; k ' '• k <• ~ I n t (A ) = (x : x.e ft" , x = 2 §,v, , f», > 0 and 2 g. = 1} 1=0 1 x . 1 . i=0 1 B a r y c e n t r e o f A k t bA k : -b A k = (v + .... + v v} . ' k+1 0 : K -- 2 -D e f i n i t i o n 1.2 ' S i m p l i c i a l Complex i n ftn ; K : A countable set of simplices C^)^ e L ' " A i 6 ^ ^ (i) A x e K and A g < A 1 => A g e K ( i i ) A X,A 2 e K => A x n A g < L and ^ n A g < A g . Dimension of K ; dim (K) : dim (K) = sup (dim (A) : A e K) . Realization of K ; |K| : |K| = (x : x e A , A e K) = the point set spanned by a l l simplices of K ' g a n ' Note: ( l ) I f A i s a simplex, the s i m p l i c i a l complex con-s i s t i n g of a l l faces of A including A i t s e l f ~ i s a complex and w i l l also be denoted, by A . The boundary of a simplex i s always a complex. (2) |A| = A by d e f i n i t i o n . Subcomplex L of K ; L c K : L c K i s a subcomplex of -K i f i t Is also a s i m p l i c i a l complex.. v F u l l Subcomplex L of K : - 3 -L c K i s a f u l l subcomplex of K i f : ( i ) L i s a subcomplex of K ( i i ) F o r A = [ v Q v k ] e K , i f ( v Q , v R ] c L then A e L P r i n c i p a l Simplex of K : A A e K i s a p r i n c i p a l simplex of K i f A i s not a f a c e of a h i g h e r dimensional simplex of K . Note: A e K i s p r i n c i p a l <=> K-(A] i s a complex. D e f i n i t i o n 1.3 L i n e a r Map, f : A - R n : A = [v, o k k f i s l i n e a r i f f (x) = f ( S §.v. ) = E §.f(v, ) V x e A i=0 1 1 1=0 1 • S i m p l i c i a l Map, f K - L A map f K| -* |L| , K,L s i m p l i c i a l complexes 5 (1) V A e K , f ( A ) i s a simplex of L (2) V A e K , f | A : A - |L| C fcn i s l i n e a r . Note: A s i m p l i c i a l map f : K -• L maps v e r t i c e s of K to v e r t i c e s of L S i r n p l i c i a l Isomorphism, f : K -• L : f : K -• L Is a s i r n p l i c i a l isomorphism i f i t i s a s i r n p l i c i a l map which i s l - l and onto on v e r t i c e s . Note: I f 3 a s i r n p l i c i a l - i s o m o r p h i s m f : K -* L we say-t h a t - K and L are s i m p l i c i a l l y isomorphic and w r i t e X « L . D e f i n i t i o n 1.4 .. ' Convex c e i l i n ftn ; C : C = (x :. x e & n ,. x '  s a t i s f i e s ^ ( x ) > b./ % I e I • - . . l ^ ( x ) = d j V J e J ' n where L., (x) = 2 a. x, _> b. i s a l i n e a r i n e q u a l i t y \J 1 e I 1 ...k=0 x k * 1 n i v ( x ) = S c . x, = i s a. l i n e a r e q u a t i o n \J 3 e J . 4•.' • • "k^O 3k K J • : " Note: (1) C I s convex, i . e . x,y e C => 'xy = d f ( ( i - t ) x + t y : 0 < t < 1} c C . (2).. A simplex i n ftn Is a convex c e l l i n & n # D e f i n i t i o n 1.5 Operations on Complexes: K a s i r n p l i c i a l complex • L^, Lg subcomplexes of K - 5 -Sum; + L 2 : •L-^ -rLg = (A : A e or A e L^} # I n t e r s e c t i o n ; I ^ A : ^ A L 2 = (i : A e and A e L 2 ) . Note: More g e n e r a l l y , i f L j and L 2 are s i m p l i c i a l " .-.complexes i n - & n 3> [A : A € or A e Lg) i s a • s i m p l i c i a l complex then •• ' L--j_-rL2 and 0 ' L 2 .are d e f i n e d as above. J o i n of S i m p l i c e s , A^-x-A1 : A k = [ v Q .... v k ] , A 1 = [v£ .... v[] where' v Q , v^, v^, ..., v£. are l i n e a r l y independent then A^-x-A1 =• [ v Q ... v^v^ ... v£] \ = a simplex of dimension ( k + l + l j , '.' Note: A simplex may be c o n s i d e r e d as the j o i n of'any two of i t s . f a c e s p r o v i d e d t h e s e / f a c e s have be-tween them as v e r t i c e s a l l the v e r t i c e s of the simplex. K i s the J o i n of sub-complexes and L 2 5 K = ^ v * ^ : A e K <=> A = A-^Ag , A 1 e 1^ A'2 e L 2 J o i n of Complexes •', K and L.; K*L : n. n 0 [K| c a , ' [L[ c a - * n 1 + n 2 + l c o n s i d e r |K| / |LJ C a 1 where |K| and. |L| are n l + n 2 + 1 c o n t a i n e d i n d i s j o i n t p l a n e s of ft- which t o g e t h e r n l + n 2 + 3 " . • •' span . ft . . • K*L = [ A 1 * A 2 : A x e K , A 2 e L} n n +n„+l lK*L| e f t - 1 ^ note t h a t t h i s j o i n i s ' d e f i n e d o n l y up to s i r n p l i c i a l i s o -morphism. The. f o l l o w i n g elementary p r o p e r t i e s h o l d : (1) L^iL^-L ) « L-j_*L2 + ^1*^3 w h e r e \> "S a r e subcomplexes of some complex K and ; the j o i n s are d e f i n e d . • • (2) 3 ( A 2 ) = SA^x-Ag -f A - * ^ , (3) I f f : Kg and g : ' -• L 2 are s i r n p l i c i a l maps ( - 7 -then f*g r'K-j^L^ - K-2*1'2 i s a w e l l - d e f i n € i d s i r n p l i c i a l map. D e f i n i t i o n 1 .6 K a s i r n p l i c i a l complex, A e K Star of A i n K ; St (A,K) : St (A;K) = (A' : A' < A", A" e K , A < A"} . • Link of A i n K ; Lk(A,K) : Lk (A,K) = {A' A' e St (A,K),Ar fl A = 0} . Residue of A i n K ; Res (A,K) : Res (A,K) = (A A : '"'A' e K and A { A',} . Note: ' (1) A' e Lk(A,K) <=> A*A' e K . (2) St (A,K) = A*Lk(A,K) . (3) : K = St (A,K) + Res (A,K) Elementary Properties: (1) v e K => Lk(v,K*L) = Lk(v,K)*L . (2) . Lk(A^*Ag,K) = Lk(A^,Lk(Ag,K)) . ; L c-.K a sub complex of K - 8 -Neighbourhood 'of L i n K « N ( L , K ) : , N(L,K) = {A' : A e K, A < A' , A'/! L / 0}'t ft(L,K) = CA.': A e N(L,K), A A L = 0 } . Note: L f u l l i n K <=> A -e N(L,K) .<=> A = A ^ ^ e L ^ " A 2 e N(L,K), Complement of L i n , K ; C ( L , K ) : K a s i m p l i c i a l complex, L c K a subcomplex . C(L,K) = [A : A e K, A < A', A e K, A' / L) . Note: (l.) I f K i s a s i m p l i c i a l complex and . v e K a v e r t e x . •• C(v,K) = Res (v,K). ' (2) |C(L,K)| = CL(|K| - JL|) /' ' ' " - ^ _ • where CL(X) = the t o p o l o g i c a l closure" of X . D e f i n i t i o n 1.7 S u b d i v i s i o n of a s i m p l i c i a l complex .. K j a(K) . i s a s u b d i v i s i o n of K i f : , ( i ) a(K) i s a s i m p l i c i a l complex, . :' ( i i ) A e a(K) => 3 A-'' .e K 3 A c.A'-. ( i i i ) |a(K)]'= iK| . . . - 9 -Elementary S u b d i v i s i o n : A e K , x e i n t A & ( x A ) ( K ) = X * 3 A * L k ( A , K ) + R e s ( A' K) S t e l l a r S u b d i v i s i o n ; a(K) : A composition of elementary s u b d i v i s i o n s B a r y c e n t r i c S u b d i v i s i o n of K r e l a t i v e to a subcomplex L of K ; B L ( K ) : B L ( K ) = ( A ' : A ' = A L*bA 1*...'*bAj c, A L'< ^ < ... < A keK A L e L , A 1 •{ L ^ Note: (1) I f a(K) i s a s u b d i v i s i o n of K and L i s -a subcomplex of K then a d e f i n e s a sub-d i v i s i o n a ( L ) of L . (2) I f a ( K ) i s a s u b d i v i s i o n of K and Y ( L ) i s a s u b d i v i s i o n of L , then CC(K)#Y(L) d e f i n e s a s u b d i v i s i o n of K-x-L . (3) I f L , K o are subcomplexes of K , L E K 0 E K > t n © n B L ( K ) d e f i n e s the bary-c e n t r i c s u b d i v i s i o n of K r e l a t i v e t o L 0 on K - 10 -Lemma 1.1 A a simplex C a convex c e l l S C O i E ^ Then C A A £ A-L , e 3A Proof; Choose A^ e SA 3 C f\ i n t A-j^  ^  0 and i s of maximum dimension i n BA wi t h t h i s p r o p e r t y . Suppose t h a t C f\ A £ A^ : i . e . 3 x e c H i 3 ^ / ^ now x e i n t A g , A g e SA c o n s i d e r the s m a l l e s t simplex A ^ 3 A-^  < A^ and Ag < A^ choose any y e C A i n t A-^  xy e C C i s convex xy e A^ A3 i s convex and m i n i m a l i t y of. A^ => A i n t A^ ^ $ => C H i n t A 3 ^ 0 c o n t r a d i c t i n g the m i n i m a l i t y of A^ * C A i £ A-, Theorem 1.1 K, L f i n i t e s i r n p l i c i a l complexes |L| £ |K| . Then "3 a s t e l l a r s u b d i v i s i o n a(K) and a s u b d i v i s i o n 'a(L) => a(L) £ a(K) . - 11 -By i n d u c t i o n on N , the number of s i m p l i c e s of N = 1 : L i s a v e r t e x , say v . |L| c |K| => v e i n t A , A e K . . . v , £ ^ v ^ j ( ^ ) i s the r e q u i r e d sub-d i v i s i o n of K. Assume the hypothesis to be t r u e i f L has <_ (N-l) s i m p l i c e s N : Consider A , a p r i n c i p a l simplex of L . L-{A] i s a s i m p l i c i a l complex wi t h <N s i m p l i c e s and }L - {A }| C |K| . Thus by the i n d u c t i o n assumption 3 a s t e l l a r s u b d i v i s i o n cf^(K) and a s u b d i v i s i o n a ( L - { A } ) => a(L-{A}) c o ^ K ) We order the s i m p l i c e s A e c£(K) with i n t A A i n t A ^ 0 ' by d e c r e a s i n g dimen-s i o n s : A ^ , A g .... A m , dim A ^ >^  dim A 2 _> .... >, dim. A m Choose x^,e i n t A^ A i n t A , i = 1,2,...m set a a ( K ) = ^ ^ ( ^ ( K ) ) 0 - 12 -and d e f i n e c(K) = ffm+i^K) * t h e r e s u l t a n t of these elementary s u b d i v i s i o n s of o-^(K) s e t a(L) = ( A / : A / e a ( K ) , i n t A'f\|L|^0'} C l e a r l y o(K) i s a s t e l l a r s u b d i v i s i o n of .K . a(L) E'^(K). by i t s d e f i n i t i o n . | a ( L ) j = JL|: ( i ) |L| c j a ( L ) j : • ' say x e |L| => x e i n t A, A e L i f i ^ A =>.x e I n t 2?, Z? e a(L-{A]) => % e a(K) — A^  u n a f f e c t e d by e ( x i , f f i ) ( o r ^ ( K ) ) , 1=1'-.... m => x e |a(L) [ — i n t 2" n JL| y V-i f A' = A , now x e jff(K)| => 3 A e cr(X) s x e i n t 2f => x e i n t A P\ i n t A => ,.' i n t A H |L| $ => A e a(L) ( i i ) | a ( L ) | c | L | : ' ' • s say x e |d(L)| => x. e i n t A' ,A' e a'(L) • V e a(L) "=> A' e a'(K) and • • i n t A'n JL| ^ 0 . - 13 -(a) i f i n t A ' A i n t A = 0 => i n t A ' r\ |L - { A } \ - / $ — i n t A ' f\|L| ^ $ now A' e a(K) => 3 a smallest A 1 e a 1 ( K ) d | A ' | C f A ^ => i n t A 1 0 | L - ( A } I ^ J2T = > - A 1 e a(L-{A}) — a ( L - ( A } ) E a 1 ( K ) => A 1 c |L| => A ' c |L| => x e |L| (b) i f i n t A ' O i n t A ^ 0 as cr(K) i s a s u b d i v i s i o n of cr^(K) 3 A x e a 1(K) => &' c A and i n t A-j^  A i n t L ^ $ thus A-, = A. and 1 % + l < K ) = e ( * . )(°i - W ) " i . e . A ' E * A p * A 0 e 3 A . x l - ^ . xl i f A 0 c A ' then 14 -x. * L c A => A' c A => xe A => x e |L| i f A 0 £ A (1) A 2 A i n t A = 0 => x ± . - * A 2 A i n t A = x ± • /. i n t A ' A i n t A ^ 0 => A' = 'x. c A 1 => x = A' = x. e |L| (2) i n t A 2 A i n t A ^ $ => A g = A^ and % + l W = £ ( x , A. ) ^ i 2 ^  2 (K)) 2 i . e . A c x. *x. *to » Ao e 9A. • 1 . 2 ~ > ^ ±2 i f A^ E A then x e [L| as above i f A ^ ^ A continue as i n case A 2 ^ A (3) A 2 A i n t A E 3A2=> A 2 A i n t A c A 3 , A 3 e SA g by lemma 1.2 => x. *A 0 A i n t A E x-,- *4o •1. 1 ^ => i n t A'A x, * f i , ^ 0 x l J => A' c x i *A ^ - 15 -a g a i n , i f - A then x e L i f A ^  2: A continue as i n case Ag ^ A Con t i n u i n g t h i s i n s p e c t i o n of the s u b d i v i s i o n of A^ i n form i n g a(K) from c-^K) we w i l l e v e n t u a l l y a r r i v e a t A ' <= x. *x. *. . . . *x. *A i A CT <=• A - i x i 2 . i p s s -or A 7 e x . *x. #x. *A , A 0 H i n t A = 0 1 1 . 12-.. X s s s which, i n e i t h e r case => A 7 c A => x e A => x e |L| C o r o l l a r y 1 .1 K a f i n i t e S i r n p l i c i a l Complex a^(K), a 2 ( K ) s u b d i v i s i o n s of K . ' . Then 3 a common s u b d i v i s i o n a(K) of a^(K) and cjg(K) . Proof Set a 2 ( K ) = K and a 2 ( K ) = L i n Thm. 1 . 1 . C o r o l l a r y 1 . 2 K a f i n i t e S i r n p l i c i a l Complex a(K) a s u b d i v i s i o n of K . Then 3 a s t e l l a r s u b d i v i s i o n o*(K) of K which i s a s u b d i v i s i o n of a(K) . - 16 -Proof: Set K = K , a(K) = L i n Thm. 1.1. D e f i n i t i o n 1.8 Piecewise l i n e a r Homeomorphism: K^, Kg s i m p l i c i a l complexes A piecewise l i n e a r homeomorphism, denoted PL homeomorphism, f : -» K^ i s a map f : [K . J -» |K2| 5> 3 s u b d i v i s i o n s £ ^ ( 1 ^ ) , a 2 ( K 2 ) of K^ and K 2 f o r which f : a^K^) -• a 2 ( K 2 ) Is a s i m p l i c i a l isomorphism. K^ and K g are s a i d to be PL homeomorphic. Note: (1) I f f : K x - K g and g : K g - K 3 are PL homeomorphisms, then gof : K^ K^ i s a PL homeomorphism by C o r o l . 1.1. (2) I f f : -* K 2 and g : L^ - L g are PL homeomorphisms, then g#f : K^*L^ -• K 2 * L 2 i s w e l l d e f i n e d and a P-L homeomorphism. D e f i n i t i o n 1.9 The C a r t e s i a n Product of a Complex with the U n i t i n t e r v a l : L e t I = [0,1] and K be a s i m p l i c i a l complex. I f |K| c ftn then ..Kxl |Kxl| c s n x i i c ' 8 n + 1 i s give n by: K x l = (A :A = (A o , 0 ) * ( b A 1 , l ) * * ( b A k , l ) , A 0<A 1< <A keK) Set KxO = (A :: A = (A 0 , 0 ) , i g . e K}.c K x l . • K x l = (A: : = ( b A 1 , l ) * * ( b A k , l ) , A]_< . ... .<\k-e K} c K x l . C l e a r l y K x 0 « K and \K x l « B ( K ) . Note: |K x 11 = ' jK| x I . Theorem 1.2 to L k ( v , a 2 ( K ) ) . Proof: Since by C o r o l . 1.1 we may. take a common s u b d i v i s i o n a(K). of a-^(K) and a^(K) i t s u f f i c e s t o prove the theorem f o r • • K = - c ^ ( K ) and • a(K) =' a 2 ( K ) . We produce.a s u b d i v i s i o n ; Y ( L k ( v J K ) ) 3 Y(Lk(v,K)) « Lk(v,a(K)) -by " r a d i a l p r o j e c t i o n " " from v . „ . Define -r : |Lk ( v ; a ( K ) ) | -+ |Lk(v,K)|- on v e r t i c e s ' K a s i m p l i c i a l complex a^(K) , a 2 ( K ) subdivisions'' of K v e ct^(K) and v e.'a2>(K) • Then Lk ( v , a . ( K ) ) i s PL. homeomorphic - 18 -of Lk(v,a(K)) by, v 1 e Lk(v,a(K))' . . Choose a simplex A k of St (v,K) 3 v 1 £ A k . A k = v * A k - 1 , A k _ 1 e Lk(v,K) Set r(v^) = 'w^  e A k - 1 such that vv^ c vw-^  extend r to |Lk(v,a(K))[ f by defining, f o r A e Lk(v,a(K)) A = [v ... v k ] , A c some A k , A k e St(v,K) . r(A) = [ r ( v c ) ... r ( v k ) ] .. v o v k l i n e a r > l y independent => r ( v Q ) ... r ( v k ) l i n e a r l y independent => r(A) i s a simplex and r(A) c |Lk(v,K)| . . . Set Y(Lk(v,K)) = (r(A) : A e Lk(v,a(K))} . Y(Lk(v,K)) i s a subdivision of Lk(v,K): (a) (r(A) : A e Lk(y>a(K))) i s a simp. comp. since A X,A 2 e Lk(v,a(K)) => ^ h A 2 e Lk(v,a(K)) thus r ( A 1 ) n r ( A 2 ) = r ( A x D A 2) e O(A) : A e Lk(v,a(K))} - 19 -(b) |Lk(v,K)| = | { r ( A ) : A e Lk( v,a(K))} [ : by. defn \ (r<(A ) . : A e .Lk(v,a(K))} j c |Lk(v,K)| s a y x e |Lk(v,K)| => x e i A k _ 1 j , A k . = v-x-A 1^" 1 , A k e St (v,K) • ' thus x^ = va.- A J L k ( v , a ( K ) ) | has r ( x ^ ) = x => x e 1 2 ?(A) | , x-j. e i n t A , A 1 e..Lk(v,a(K)) By c o n s t r u c t i o n L k ( v , a ( K ) ) « y(Lk(v,K)) through the r a d i a l p r o j e c t i o n map r . C o r o l l a r y 1.3 K a s i r n p l i c i a l complex • a(X) a. s u b d i v i s i o n of K . v-•€ K s -Lk(v,a(K)) f\ -a(Lk(v,K)-)' ..=-' Then C(St (v,a(K))) ,a(,Si;(v,K)) i s PL homeomorphic to - Lk(v,K) x- I . • - 20 -Com b i n a t o r i a l M a n i f o l d s D e f i n i t i o n 2.1 Co m b i n a t o r i a l C e l l and Sphere £ n i s a combinatori n i f £>n Is PL homeomorphic to A n . c o m b i n a l c e l l of dimension y n i s a c o m b i n a t o r i a l sphere of dimension n i f Y n i s PL homeomorphic to 3 A n + 1 D e f i n i t i o n 2.2 Comb i n a t o r i a l M a n i f o l d of dimension n: JtJ1 i s a c o m b i n a t o r i a l m a n i f o l d of dimension; n I f V v e r t e x v of M^n , Lk(v, Jj£) i s a com-b i n a t o r i a l c e l l o r sphere of dimension ( n - l ) . Theorem 2.1 > £ n a comb, c e l l of dim n , £ m a comb, c e l l of dim m , yn a comb, sphere of ' dim n , ym a comb, sphere of dim m . Then, (a), £ n * £ m i s a comb. c e l l of dim (n+m+1) (b) , £ n * Y m i s a comb. c e l l of dim. (n+m+1) (c) Y n*Y m i s a comb. sphere, of dim (n+m+1) Proof : (a) We have PL homeomorphisms f : £ n A n «m A m : £ -• A - 21 -3 ' ?B homeomorphism • f*g'; £ n * £ m - A n * A m = A n + m + 1 • (b) We have PL homeomorphisms f : £ n -• A n . „ . ,,m g : Y 3A . thus f * g •: £ n * Y m - A n * S A 1 7 1 + 1 i s a PL homeomorphism. Now A°*3A m + ' ' ' » ^m+1) ( A m + 1 ) , let h : A ° * d A m + 1 -+ A 7 7 1 + 1 be the corresponding PL homeomorphism. ru, „n . An -1 7 Ab ^,m+lx An i Am+l An -1 Am+1' .n+m+1 Then i d : A *(A *3A ) = A *oA . -» A *A • = A A • . i s a PL homeomorphism .'. ( i d n*h).(f*g) : £ n * V m A n + m + 1 i s a "PL homeomorphism. • A n " x (c) Note that. oA 1 * 3 ' A k .« fi(x Ak+1) ( 3 A k + 2 ) # " . By induction.on n : n = 0 We have' PL homeomorphisms f : Y° -• BA1'' V ' ! a n d " g V Y m - o A r i + 1 . ' ' , ' ",' .*. f*g : Y°*Y m - 3A -1* 3 A 1 7 1 + 1 ; is a PL • ' homeomorphism.As noted 3 a PL. homeo- . morphim . h. : BA^SA 1 7 1 * 1 " 1 - 3A 1 7 1 + 2 -- . V h.'(f*g) : Y d * Y m - B A m + 2 13 a ; PL homeomorphism. Assume the hypothesis in dimensions <_ n • n+1 : We have PL homeomorphisms, V - f' : Y n + 1 "*' 3 A n + 2 ' and'" g : Y m - 3 A m + 1 , ; Consider the PL homeomorphism - 22 -f*g : Y n + 1 * Y m - 3 A n + 2 * 3 A m + 1 . now h * l m + 1 . : S A n + 2 * B A m + 1 - (3A W ^ V ^ 1 . 3A •= 3 A 1 * ( 3 A n + 1 * S A m + 1 ) i s a PL homeomorphism. By the I n d u c t i o n h y p o t h e s i s , 3 A n + 1 * 3 A m + 1 i s -PL homeomorphic. t o a A n + m + 1 ^ thus a A ' i * ( a A n + 1 * a A m + 1 ) * ; i s PL . homeomorphic ' n+m+2 " • to 3A by a g a i n a p p l y i n g . t h e above note. \ Theorem 2.2 . J ^ C ^ a comb. man. of dim n A k e JUP a k-simplex K a simp, comp., a(K) a s u b d i v i s i o n of K . £ n- a comb, n - c e l l , y n a comb. n-sphere. Then J '• ( i ) Lk(A , Ai ) i s a comb, (n-k-1) c e l l or sphere . • ( i i ) K Is a comb. man. of dim n <=>'; a(K) Is a comb. man. of- dim n , ( i i i ) «£n and y n are comb. mans. • of dim n . Proof: By i n d u c t i o n on n : n = 0 t r i v i a l • '' assume the theorem i s t r u e - f o r dimensions . < ( n - i ) . ••; •• n .: ( i ) By I n d u c t i o n on k = dim (A k) - 23 -k=0 A = v e -^ C /. Lk(bk, JC) i s a comb (n-1) . c e l l o r sphere s i n c e MJ1 i s a comb. man. of dim n . Assume ( i ) ' i s true f o r dlm(A) < k , A e MJ . k : A k = v * A k _ 1 , dim A k _ 1 = k-1 Lk(A k,X n) = Lk(v*A k' 1,X n) = L k ( v , L k ( ^ k - 1 , ^ n ) ) but L k ( A k - 1 , Jjp) i s a comb, (n-k) c e l l or sphere by the i n d u c t i o n hypothesis on k => L k ( A k 1,M^) i s a comb. man. of dim (n-k) by the i n d u c t i o n hypothesis on "\ n i n dimension (n-k) < n => L k ( A k , JU?) = L k ( v , L k ( A k " : L , ^ ) ) i s a comb.,(n-k-1) c e l l or sphere. ( i i ) <= Say a(K) . i s a comb. man. of. dim. n .. choose any v e K . Since v e a(K) , Lk(v,a(K)) i s a comb, (n-1) - c e l l or sphere and . Lk(v,a(K)) i s PL homeomorphic t o Lk(v,K). — — Thm. 1 . 2 Lk(v,K) i s a comb, (n-1) c e l l or sphere. - 2K -=> Say K Is a comb. man. of dim n . Choose any v e a(K) . Now v e i n t A , A e K . co n s i d e r S ( v A)^ K ) = v*3A.*Lk(A,K) + Res (A,K) . Lk(v,a(K)) i s PL homeomorphic to Lk(v,£^ v ~ Thm. 1.2 and Lk(v , e , ,(K)) = 3A*Lk ( A,K) . But SA i s a comb, sphere of dimension ( d i m - A - l ) and Lk(A,K) i s a comb, c e l l or sphere of dimension ( n - d i m & - l ) by ( i ) i n dimension n s i n c e K i s a comb. man. of dim n . .'. Lk(v,S£ v ^ j ( K ) ) i s a comb., c e l l or sphere of dim (n-1) Thm. 2.1 =>•Lk(v,a(K)) i s a comb (n-1) c e l l or sphere, ( i i i ) «£n tt'a-^(Ln) and A n i s a comb. man. of dim n <£n i s a comb. man. of dim n ( i i ) i n dim n . Y n « a 2 ( o A n + 1 ) and o A n + 1 i s a comb. man. of dim n L k ( v , o A n + 1 ) =. SA n y n i s a comb. man. of dim n — ( i i ) i n dim n . C o r o l l a r y 2.1 jdlj1 a comb. man. of dim n K a simp. comp. ^ K i s .PL homeomorphic to ^ C n . Then K ' i s a comb.1 man. of dim n . - 25 -Proof: Thm. 2.2 ( i i ) . Theorem 2.3 K , L s i r n p l i c i a l complexes . Then K*L i s a comb. man. <=> K and L are comb, c e l l s or spheres. Pro o f: <= Thm. 2.1 => By i n d u c t i o n on dim K*L , say n : n=l : Choose v e K , Lk(v,K*L) = Lk(v,K)*L L 4 0 => dim K = 0 => Lk(v,K) = $ .'. Lk(v,K*L) ,= L = comb O - c e l l o r sphere. Choose v e L to show K i s a comb. 0 c e l l or sphere. Assume the theorem i s tr u e f o r dimensions <. (n-1) n : Choose v e K , Lk(v,K*L) = Lk(v,K)*L - but Lk(v,K*L) = comb- (n-1) c e l l or sphere = comb man. of dim (n-1) Thm.. 2.2 ( i i i ) by the i n d . assumption Lk(v,K) and L _ are comb c e l l s o r spheres. Choose v e L to show K i s a comb, c e l l i or sphere. - 26 -D e f i n i t i o n 2.3 Boundary of a Co m b i n a t o r i a l M a n i f o l d : dX. n = [ A : A e X. n and L k ( ^ , A n ) i s a c o m b i n a t o r i a l c e l l } . Theorem 2.4 >>Cn a comb. man. of dim n a(Jin) a s u b d i v i s i o n of > L n £ n a comb, n - c e l l , Y n a comb, n-sphere Then ( i ) 3 X n i s a subcomplex of Mj1 ( i i ) 3(a(X n)) = a(9X n) ( I i i ) 9 £ n = Y 1 1" 1 , n >> 1 , n—1 Y a comb, ( n - l ) sphere ( i v ) * Y n = 0 . Proof: By i n d u c t i o n on n: n = 0 and 1 : t r i v i a l Assume.the theorem i s - t r u e i n dimensions < (n -1) n : . ( i ) a>Cn i s a subcomplex of Msn : say A e 3 > G N and A ' < A i . e . A = A ' * A " now Lk ( A,JLO n) = L k ( A " , L k ( A ' , > C N ) ) . I f L k ( A R , > C N ) comb, c e l l => Lk(A ' , > G n ) = a comb sphere of dim <_ (n -1) => L k ( A " , L k ( A / , ^ n ) ) = a comb, sphere ( i v ) i n ,dim'Lk(A',J^Cn) . - 2 7 -' C o n t r a d i c t i n g L k ( A J X / n ) = a comb, c e l l A.e 3 j i n /. Lk(A' ,Ji?) = a comb, c e l l . => A' e a J t f ( i i ) S a ( j t n ) " £ a(sXn) : ' ' ' , . Say A ' e ha(M,N) i . e . ' Lk(A 7 ,a(X, n)) = a comb, c e l l Choose x e i n t A 7 "• S ( x A / j ( a ( j L O n ) ) = 3."x * a A /*Lk ( A /,a(X n)) + Res ( A ' , a ( ^ n ) ) /. L k ( x , e ( x ^ / ) ( a ( X n ) ) = SA/*Lk(Av,a(v>On)) / = a comb, c e i l — — Thm. 2.1 Now A 7 e a^M?) => A ' E A , A e MJ1 9 A ' f | i n t A ^ 0 .*. x e i n t A . e x^ A)(^ n) = x*3A*Lk(AJ^t(51) + R e s ( A , ^ C n ) * .V Lk(x,£ ( x^ A )(J^ a)) = oA*Lk(A,Xn).. • • but Lk(x,£^ x ^(Mj1)). i s PL homeomorphic to L k ( x , & ( x ^ } ( a ( X n ) ) ) Thm. 1.2 .*. 3A*Lk(A,Jt n) i s a comb, c e l l and 3£ - a comb, sphere => L k ( A , ^ n ) i s a comb, 'v c e l l , Thm. 2.1 => A 7 e a(lM) - 28 -a( 0jUL n) c MMJ1) : Say A ' e aibMj1) => A' c 4 , A e 9 X n and i n t AHA' J4 $ } choose x e i n t A ' C\ i n t A e ( x A ) ( j ^ n ) = x*oA*Lk(A,jlL n) + Res (A,XC-n) .'. Lk(x,e^ x A)(^ n) = 3A*Lk(A,«A.n) = a comb., c e l l A e and Thm. 2 .1 A l s o e^x A / ^ ( a ( ^ n ) ) = x * B A / * L k ( A ' , a ( X n ) ) ' • ' + Res ( A ' , a ( X n ) ) L k ( x , e ( x #j(a(J(, n))) = oA'*Lk(A',a(J^ n)) but L k ( x , S ^ x ^ (a(v^C n))) i s PL homeomorphic to Lk(x,£^ x A)(X> n))' Thm. 1.2 .*. BA/*Lk(A/,a(^U1)) i s a comb. c e l l . and .SA' a comb, sphere => Lk(A' ,K(vX>n)) i s a comb, c e l l Thm. 2 .1 => A' e oa(J(, n) ( i i i ) d £ n = Y n _ 1 : <£nPL homeomorphic to A n => 3 s u b d i v i s i o n s a ( £ n ) » y ( A n ) .'. a ( a £ n ) = o(a(£ n)) « 3(Y(A n)) = YOA11) => d«£n i s a comb, sphere.of dim (n -1) ( i v ) oY n = 0 : Y n PL homeomorphic to 3 A n + 1 => - 29 -3 s u b d i v i s i o n s a ( y n ) « y ( o A n + 1 ) .'. a ( o y n ) = B ( a ( y n ) ) « a ( y ( B A n + 1 ) ) = y ( o ( 3 4 . n + 1 ) ) ' = J2T" => d y n = 0 C o r o l l a r y 2.2 -^Cy 1 1 a comb. man. of dim n " bM? = 0 <=> V v e r t e x v e ^ n , Lk(v,jAP) i s a comb, sphere. Proof: => by defn of bMP <= by Thm. 2 A ( i i i ) and ( i v ) a c o m b i n a t o r i a l c e l l cannot be a sphere. Hence i f \J v e L k ( v , J ^ ) i s a sphere, bMj1 cannot c o n t a i n v e r t i c e s . But bjLn i s a subcomplex by Thm. 2.4 ( i ) so i t must be empty. Theorem 2.5 J-C n a comb. man...of dim n 3 3 X n ^ 0 Then bMj1 i s a comb. man. of dim (n-1) S 3 ( o J ^ n ) = 0 . Proof: (a) L k ( v , 3 X n ) = o L k ( v , X P ) : A e L k ( v , 3 j H n ) <=> v*Ae3X n <=> L k ( v = L k ( A , L k ( v , 0 ^ ) ) i s a comb, c e l l <=> A e 3Lk(v, jLCn) - 30 -bM? i s a comb. man. of dim ( n - l ) d 3(3 Jl?)=0: v e => Lk(v,3> H , ) = 3Lk(v, Jt(?) L k ( v , j t C n ) i s a comb, c e l l of dim (n-1) .'. 3Lk(v , J U J 1 ) i s a comb, sphere of dim (n-2) Thm. 2.4 ( i i i ) => 3-Wi1 i s a comb. man. of dim (n-1) 3 3 ( 3 X ? ) = 0 . - 31 -Homogeneity of Combinatorial Spheres Lemma 3.1 £^ £ 2 c o m b i n a t o r i a l c e l l s h : 3-S^  -* a P L homeomorphism Then 3 a PL homeomorphism A, A «n «n _ t < Proof: 3 P L homeomorphisms : -» A n , g 2 : tfj - A n , s e t h ' = S s U ^ ' ^ g ^ l ^ n ; h' : 3 A n -• 3 A n i s a PL homeomorphism. Consider i t s u n d e r l y i n g s i r n p l i c i a l isomorphism h' : 0^(3 A n ) - a 2 ( 3 A n ) choose v e i n t A n , set h' = h ' * l v : 0^(3 A n ) * v = o^ ( A n ) - a 2 ( 3 A n ) * v = 2 2(A n) Ii = g 2 ^ 0 ^ / o S ] _ : "* ^ 2 * s a homeomorphism 3 ^ 3 £ ? = h ' Lemma 3.2 ^ a c o m b i n a t o r i a l c e l l of dim n C n PL homeomorphic to 3 X nxI with C n A £ n = 3 £ n . Then £ n + C n i s " a c o m b i n a t o r i a l c e l l of dimension n. .' • • " . Proof.: 3 -PL homeomorphisms .  h-^. : <Sn -» A.n , h 2 : C n -* 3 £ n x l ' take h 3 = ( h - J ^ X 1 ; E ) ° h2 : ° n "* S A n x i . c o n s i d e r h o i « ' *, : 9 £ n - s A n by lemma 3.1 i t may be •3 3& n c C A, n r A extended t o h^ : & - A . thus h ^ , and i-h^ : d e f i n e a PL homeomorphism. -•' h : C n + £ n - 3 A n x I + A n . which i s PL homeomorphic to Lemma 3.3 • S^, c o m b i n a t o r i a l n - c e l l s 3 3<£^ = a £ 2 = £ i ^ £ 2 * T h e n £ 1 + £ 2 i s a c o m b i n a t o r i a l n-sphere. Proof: .. By lemma 3.1 3 PL homeomorphisms \ •• £ i - 3 S i * v i - « \ * *l+i2 9 h I - V 1 1 h 2 , : * g - ^ * v 2 ; v 2 ^  v x v 2 / ^ + 3 ,.h 2 l a £n - 1,^ s i n c e £ ^ . n £g = a£j = a£g '. / 3 a PL homeomorphism h : + £ 2 S £ j * V l + a ^ * v 2 but 3£^*v^ + 3£ 2*v 2. = •'3£^ -x-(v^  + v 2 ) which i s a comb, n-sphere by Thm. 2.1. ' - 33 -Theorem 3»1 yn a comb, n-sphere £ n c Y n a comb, n - c e l l Then C ( £ N , Y N ) i s a comb, n - c e l l . P r oof: By i n d u c t i o n on n : n = 0 : Y ° = 2 O - s i m p l i c e s , £° = 1 O-simplex .*. C ( . £ 0 , Y ° ) = 1 O-simplex . = a comb. O - c e l l . Assume the theorem i s tr u e i n dimensions <_ ( n - l ) . T h i s assumption i m p l i e s r e s u l t (A), the e x i s t e n c e of a c o l l a r , of the boundary of a c o m b i n a t o r i a l 1 m a n i f o l d of dimension n which w i l l be used t o prove the main theorem i n t h a t dimension. (A). jLn a comb, m a n i f o l d of dim n 3 bJHj1 i s f u l l i n , J ^ n . Then N(bJLLnjY(M.n)) i s PL homeomorphic to bJLLn x I where y(Jln) = , B a ^ n ( ^ ( , n ) = 1 s t b a r y c e n t r i c s u b d i v i s i o n of JJ^ r e l a t i v e t o bjUj1 Proof: By i n d u c t i o n on n : Case n = 0 is. t r i v i a l . We assume (A) i s true f o r dimension <_ (n-1) as i n the main theorem, case n : ( l ) Defn of .0(A) f o r A e hA* : 0 ( A ) = [ L ± : A 1 ' e L k ( A , Y ( X / ) ) and . - 34 -A 1 n ajX/= si n c e bjL? i s f u l l i n y(Jl*) we have t h a t N(aJt,Y(v/0)) = S A *o(A, ) AeaX a l s o Lk(A,Y(JO)) = N(Lk(A,aX),Lk(A,Y(X)))+0(A ) s i n c e Lk(A,aJt) = 3Lk( A , Y ( X ) ) = Lk( A , y(JC)) fl 3>L (2) P r o p e r t i e s of 0 ( A ) : I. I f A e hMj i s an l - s i m p l e x , then 0 ( A ) i s a comb. (n-t-1) c e l l : ( i ) Choose a ve r t e x v not co n s i d e r e d so f a r . Since Lk(A,aX/) Is a comb, ( n - l - 2 ) sphere, v*Lk(A,a>C)) i s a comb, ( n - l - l ) c e l l . A l s o Lk(A,v(«>6j) i s a comb, ( n - l - l ) c e l l 3 L k ( 4 , Y ( J J b ) ) H v-x-Lk(A,aX) i s Lk(A,aX) = aLk(A,Y(^)) = a(v*Lk(A,3X)) ^Mj i s f u l l i n y(Jt). Thus v*Lk(A,a>C)+Lk(A,YOC)) i s a comb. (n-E--l) sphere — - Lemma 3«3« ( i i ) N(Lk(A,aJ^),Lk(A,Y(^)) « N(Lk(A,aX) , y(Lk(A,A)) where Y(Lk(A,^C)) = ^ L K ( 4 J 3 ^ ) L k ( A , > 0 ) but N(Lk(A,aX) , ' Y ( L k ( A , ^ ) ) ) i s PL homeomor-p h i c t o L k ( A , a ^ ) x I by (A) i n -dimension ( n - l - l ) - 35 -( i i i ) N(Lk(A,5>L),Lk(A,Y(X))) i s a " c o l l a r " of v#Lk(A,a>t) i n the sense of Lemma 3 - 2 , hence v*Lk(A ,aX,)+N(Lk(A ,aX),Lk(A ,Y (>U)) i s a comb, ( n - l - l ) c e l l say SP'*"1 . ( i v ) Now 0(A)+N(Lk(A/a>C),Lk(A,Y(jtC')))+v*Lk(A,a'X) = L k ( A , Y ( ^ ) ) + v*Lk(A,a^C) = a comb, ( n - l - l ) sphere — — ( i ) say Y m-t-1 And .so, by the i n d u c t i o n h y p othesis t o the main theorem i n "dim ( n - l - l ) , C ( £ N " ^ " 1 , Y N " ^ " 1 ) i s a comb, ( n - l - l ) c e l l (v) But 0(A) 0 [N(Lk(A,ail/),Lk(A,Y(jtC)))+v*Lk(A,a>tC)] = N(Lk(A , a j l L>,Lk(A , Y(Jt))) which can on l y c o n t a i n s i m p l i c e s of dimension < (n Thus C(&n'l~1,yn~l~1) = 0(A) . And 0(A) i s a "' comb, ( n - i - l ) c e l l . I I . f o r A r A 2 e a-**/, A 1 < A2 => 0 ( A . 2 ) E 0(4^) say A e 0 ( A 2 ) => A e Uc(& ',y(M,)) and A r\ bjXy~ 0 . = > A * A 2 e = > A * A 1 -e YUC) => A e L k ( A 1 , Y ( X ) ) and & C\ bJl= $ = > A e 0(4^)' - 36 -I I I . f o r A e bJXs , 30(A) = U 0(A.') A<A* ( i ) say A ± e 30(A) = > A X £ Lk(A,y(-U,)) , A 1 H 3 X = 0, and Lk(A-^,0(A)) = a comb. c e l l . now L k ( A ^ , L k ( A , Y ( X ) ) ) = a comb-sphere — A e 3J-G => Lk(A,Y(jA,)) i s a comb, sphere , thus "Lk(A 0(A)) E Lk(A 1,Lk(A,v(X/))) — 0(A) £ L k ( A , Y ( X ) ) - But L k ( A 1 , 0 ( A ) ) ^ L k ( A 1 , L k ( A , Y ( ^ t ) ) ) .*. 3 A 2 e L k ( A 1 , L k ( A , Y ( X ) ) ) , A 2 / L k ^ O ^ ) ) => A 2 * A X e L k ( A , Y U C ) ) as A 2 / L k ( A 1 , 0 ( A ) ) we must have A 2 * \ ^ ^ # but A 1 A 3 > t n = V A 3 = A 2 * A x n S A 2 < A Consider A 3 # A * c l e a r l y A < A 3 * A we show^ A-j_' e 0 ( A 3 * - A ) : • A g-x-A^ e Lk ( A , Y ( - M ^ ) ) = > A 2 - ) F A 1 * A e y(jAj) = > A 3 * A ] _ * A e Y ( A C ) — A 3 < A 2 = > A 1 e Lk ( A 3 * A , Y ( j > C )) and _ A X A = 0 " => A X e 0 ( A 3 * A ) => A-, e U o ( A ' ) ~ A < A . * A . 2 - 37. -( i i ) say A . e U O(A') i . e . A E O ( A ' ) f o r x some A < A 7 . By I I A € 0 ( A ) we must show Lk( A-j_ , 0 ( A )) i s a comb. c e l l . Now A - L ft SLk(A,y ( > C ) ) , — 9Lk(A,Y(>L)) = aX, A Lk(A,v(X)) L k ( A 1 , L k ( A , Y (X) )) i s a comb, sphere of the same dimension as L k ( A ^ , 0 ( A ) ) but . L k ( A 1 , 0 ( A ) ) c Lk(A 1,Lk(A,Y(X))) And as a comb, k-sphere cannot c o n t a i n a comb, k-sphere as a proper subcomplex L k ( A ^ , 0 ( A ) ) must be a comb, c e l l — — 0 ( A ) i s a comb. man. A 1 e SO ( A ) IV. A X , A 2 € then 0 ( A V ) A 0 ( A O ) = / a s i m P l e x o f Y ( X ) # i f A ^ a n d " A 2 are not f a c e s of O ( A ^ ) where A 3 i s the s m a l l e s t simplex of y{jL>) c o n t a i n i n g A ]_ and •£_2 a s f a c e s ( i j i f A e 0(A 3) then A e 0(A- L ) f\ 0 ( A 2 ) , . A T L , A 2 < A^' by I I ( i i ) say A e 0(4^) f\ 0 ( A 2 ) , now A = bA'-j*...'.'*DA,k , A l ^ ' ^ A ' K e - 38 -I. i n the c a n o n i c a l rep. of y(JL) A e U c ^ Y p ^ ) ) * A e L k ( A 2 , Y ( X ) ) => A-x^ and A * A 2 6 Y(^ 1) but A -j^Ag € aX-=>A*A1 = A^gj^^A'-^*'• •'*bA' k A *A 2 = A aj^^A • . .-x-bA k i n c a n o n i c a l r e p r e s e n t a t i o n =>A 1,A 2 < A ^ => A 1 * 4 2 e >XC =>A 1*A 2 e 3>>{/ 3 f u l l i n JiL A 3 = A 1 * A 2 n a s A\eO(A3) <=>AeO(A 1)no(A 2) . (3) Consider now the s i r n p l i c i a l complex bMjx.1 : f o r A e 3>Cx 0 , s e t 'D(L) = [&1 3 ^ x 0 = gi, A x * A e ay<xi) c BXCx 1 BJbtxl « ,6(3 jtC)y C l e a r l y 3 X * 0 i s f u l l i n S^tx 1 and BJ/LxI = s A*D(A) aicxo « 3>tL (4) P r o p e r t i e s of D(A)": I f A. ^  e sX/O i s an t-simplex, then ^(A^) i s a co m b i n a t o r i a l •(n-i'-l) c e l l : • " oJt(,xl - 39 -D(A 1) = ( ( b A ^ l ) * . . . * ( b A m , l ) : A1<-...<Am e aJO = (bA 1,l)*{(bA 2,l)*...*(bA m,l):A l<...<A m e*X2 « (bA 1,l)*/3(Lk(A 1,3X)) = a comb, (n-^-l) c e l l — - Lk(A^,a^C/) i s a comb. (n-L-1) sphere I I . I f A X , A 2 e 3 ^ x 0 , A x < A g then D(A 2) <£ D(A X) : ' say A e D(A 0) = > A * A o e a X x I , A O 3 X * 0 = 0 = > A * A 1 e 3>{,x I , A x < A 2 => A e D ( A X ) I I I . I f A e SX X° , ^ D ( A ) = U D ( A ' ) A<A' (I) say A - a e 3D(A)^ => LkCA-^lKA)) = a comb, c e l l but LkCA-pLkCAjaXxI)) ^Lk(A 1,D(A)) and Lk(A 1,Lk(A,2LXxl)) 5/ Lk ( A 1,D ( A ) ) /. 3 A 2 e Lk(A 1,Lk(A,a JL(/I)) , A 2 t Lk ( A^,D ( A ) ) , and h 1 e D ( A 3 ) , ,. A 3 = A 2 * A - L 0 ay^ xO ( I I ) say A ! e D ( A ' ) A < A ' - 40 -Lk(A- L,Lk(A,b j { / . I )) i s a comb, sphere of the same dim as Lk(A^,D(A)) => Lk(AT_,D(A)) cannot be a combinatorial sphere since L k ^ . D ^ ) ) £-Lk(&1,Lk(A.,bjlxl)) => Lk(A1,D(A)) i s a comb, c e l l => A1 e 3D(A) . IV. I f & l f L 2 e oX/0 then D(A 1)nD (4 2 ) = 4 r 0 i f A-j_ and A 2 are not faces of a simplex of S-XCxl^ D(A 3) where A ^ i s the smallest simplex of 3 ^ x 1 ' containing A ]_ and A 2 as faces (i ) i f A e D(A 3) then A e D(A-L) f\ D(A 2) by I I ( i i ) i f A e D ^ ) A D(A 2) =>A= (bA 1 , l ) * ( b A 2 , l ) * . . . * ( b A ^ , l ) A 1<A 2<.. .<A^  e 9 ^ and A = ( b A 2 , l ) * ( b A 2 , l ) * . . .*(bA^,l) ^2<A2<...<Am e 3J<C but we must have m = A m .". A 2*^2 <A^ =>A 1 *4 2 = A 3 e 3>C has A e D(A 3) <=> A e D(A 1) H D(A 2) . - 41 -( 5 ) We now e x h i b i t a PL homeomorphism h : N(O(.X C),YUO) -.3/6<I note t h a t N(O>(,Y( ) = s A*0(A) and A e 3 / 6 3X0^1 = S A*D(A) Ae3>C><0 we d e f i n e h i n d u c t i v e l y on s i m p l i c e s of N(3X(,, Y(XC) ) i n order of d e c r e a s i n g dimension. Choose any A . n _ 1 e 9 x 0 . 0 ( A n ~ 1 ) = v x and D(A n _ 1) = v 2 d e f i n e h| , :A n" 1*0(A n~ 1) -d A *0(A ) ^ n* 1*D(A n" 1) by h = ^ n - l * 6 w h e r e eCv^) = v Q Assume we have d e f i n e d h on A *0 ( A . ) V l > k Choose A k e 3X(,as 30(A k) = U 6 (A**) and k<^ 3D(A k) = U D(A^) we have d e f i n e d k<l-hj , : 30(A k) - 3D(A k) a l r e a d y . - h| may be 30(A k) 30(A K) extended t o hi by lemma 3«1« . 0(A k) Define hj v ^A k*0(A k) - A k*D(A k) by A *0(A ) h'= 1 ,*h| . n o t i n g t h a t 0(A k) f l 0(A~) = 0(A L) l/> A 0 ( A ) by i n d u c t i o n we d e f i n e h on A k*0(A k)^A k e 3>C , k = 0, 1, .... (n-1) . . . • Thus we have a PL homeomorphism - 2f2 -h : N(ai£,Y(i(,)) ~* ZJI* I • (B) Proof of the Main Theorem in dimension n : (i) Set K = C(£ n,Y n) , clearly yn = K + £ n . (a) K i s a comb. man. of dim n : Choose, any v e K , LkCv,vn) = Lk(v,£n)"'+ Lk(v,K) v / £n'=> Lk(v,K) = Lk(v,Y n) = a comb, (n-l) sphere * . v e £ n=>Lk(v,Y n) = a comb, (n-1) sphere i f Lk(v,£n) is a sphere, => Lk(v,£ n) = Lk(v,Yn),' ' < => Lk(v,K) = 0 ' contradicting v e K .*. Lk(v,£ n) i s a comb, (n-l) c e l l - thus Lk(v,K) = C(Lk(v,£n)'', Lk(v,Y n)) = a comb, (n-1) c e l l by the induction assumption to the main" theorem. Lk(v,K) i s a comb, (n-1) c e l l or sphere => K is a comb. man. of-dim n . (b) oK = 3 £ n = K n £ n ; A e oK <=> Lk(A,K) is a comb, c e l l <=> Lk(A,£ n) is a comb, ce l l — (a) '<=> A e 3 £ n - ^3 -a l s o A € K fl £ n => Lk(.A,£ n) and Lk(A,K) are comb, c e l l s — a s i n (a) => A e SK.'and A e 3 £ n  v \ 3K = d £ n = K 0. £ n ( i i ) Given &n £ Y N c o n s i d e r Jl = C(£ n,-Y-) We show Jib i s a c o m b i n a t o r i a l n - c e l l Now, ^ C i s a comb. man. of dim n &'bAs= 3 £ n = JL D £ n — ( i ) As 3 = Jif\ £ n , by Lemma 3*1 we may co n s i d e r yn as A,+ bJL*v .. i ( i i i ) Consider a 'PL homeomorphism h : y n = Jl + *jl*v - 3 A n + 1 . L e t ; A n + 1 = w*An . Then 3 A n + 1 = w*3An + A n . We note t h a t i f x , y e 3 A , then there i s a PL homeomor-phism of 3 A n + ^ onto i t s e l f which maps x onto y . Hence we may assume t h a t the PL homeomorphism h s a t i s f i e s the p r o p e r t y 1 . h(v) = w . We choose now s u b d i v i s i o n s a^(yn) of yn and a 2 ( 3 A n + 1 ) of 3 A n + 1 such t h a t , h za^y*1) - » . a 2 ( 3 A n + 1 ) i s a s i r n p l i c i a l ; ;. isomorphism and such t h a t L I ^ V ^ ^ Y 1 1 ) ) n a 1 ( L k ( v , Y n ) ) = Lk(v^(y* 1 ) ) fl ^ (BJK/) = 0 - 44 -and L k ( w , a 2 ( o A n + 1 ) ) n a 2 ( L k ( w , 9 A n + 1 ) ) L k ( w , a 2 ( 3 A n + 1 ) ) n a 2 ( 3 A n ) = 0 . By C o r o l l a r y 1.3, C ( S t ( w , a ( a A n + 1 ) ) , a ( S t ( w , 9 A n + 1 ) ) ) Is PL homeomorphic to . 3A n x I . By Lemma 3.2,-C ( S t ( w , a 2 ( 3 A n + 1 ) ) , a 2 O A n + 1 ) ) = C ( S t ( w , a 2 ( aA n + 1 ) ) , a 2 ( S t ( w , a A n + 1 ) ) ) + a 2 ( A n ) i s a c o m b i n a t o r i a l c e l l of dimension n . Since h ( S t ( v , a 1 ( Y n ) ) ) = S t ( w , a 2 ( a A n + 1 ) ) , ~ , we conclude t h a t C ( S t ( v , a 1 ( Y n ) ) , a 1 ( Y n ) ) i s " a c o m b i n a t o r i a l n - c e l l . To s a t i s f y the hypothesis of (A) with r e s p e c t to a^(JjL) and a^(bMs) , we d e f i n e . - K5 -°<v n ) - B 2 a i ( a ^ v ) ( a 1 ( v n ) ) • ( i v ) By ( i i i ) , C (St(v,a ( v n ) ) , ct( y n ) ) i s a comb, n - c e l l . We w i l l produce a PL homeomorphism of ^AL onto C ( S t ( v , a ( v n ) ) , a ( Y n ) ) . ^ Observe t h a t 3 PL homeomor- / , ^ ^ ^ " " ^ phisms . ' ^^M*«U)MJl)) h 1:N(aa(X),aU))^Xxl - (A) / ? > ^ ^ < ^ h 2 : C ( S t ( v , a ( Y n ) ) , S t ( v , Y n ) ) -* S j l i x l — C o r o l . 1.3 X ~ ~ " ^ " v v ^ ~ S t ( v ' Y n ) Thus 3 a PL homeomorphism .• • . s t ( v * a ( Y )) g : N('oa(Jt),a(X)J-- N(3a(iC)MM?) )+C(St (v,a( y n ) ),St (v,y n)) • $ S , N ( a a ( ^ ) , a ( X ) ) = l d N ( c a ( ^ ) , a ( X ) ) Using ,g and id C( N(a a(y6),a(iC) ),a(it)) ' s i n c e they agree, on i n t e r s e c t i o n s of domains, we may d e f i n e a PL homeomorphism h :.X- C { S t ( v , a ( Y n ) ) , a ( Y n ) ) ..' Thus JL i s a c o m b i n a t o r i a l n - c e l l . -C o r o l l a r y 3.1 JttJ1 a comb. man. of dim-n 9 bjij1 i s - 46 -f u l l i n 4 n . B ( ^ n ) the 1 s t b a r y c e n t r i c s u b d i v i s i o n of jLn r e l a t i v e t o sXJ1 . Then N(aX n,B(^(5 1)) i s PL homeomorphic to bMj1 X I . Proof: c f . (A) of Thm. 3.1. C o r o l l a r y 3«2  rtn ~ „n ^ „n jc!\ c o m b i n a t o r i a l n - c e l l s ^ n -1 J fi £g r = 0 - £ " n bZ* = £ n _ 1 a comb, (n-1) c e l l . Then §^ + £ 2 i s a c o m b i n a t o r i a l n - c e l l . Proof: , n - l 2 a PL homeomorphism h : £' n-1 C ( £ n , 3 ^ ) i s a comb, (n-1) c e l l by theorem 3»1« By lemma 3»1, 3 a PL homeomorphism h 1:C(X n " 1 , a J C j ) - v 1 * 3 A : n-1 An-1 A which extends the PL homeomorphism h. S i m i l a r l y j | a PL homeomorphism h ^ C ^ " 1 , . ^ ) - v ^ S A 1 1 " 1 which extends the PL homeomorphism h . Set AN = An"1*v1 ,-and A£ = AN v 2 we have • PL homeomorphism' h[ : afij = f £ n ~ 1 + C ( £ n " 1 , a £ ] n ) - 3 A j Hence n-1 - 47 -and h 2 :' = £ n - 1 + C ( < 2 n - : L , S ^ ) - oA* w i t h ^ l ^ n - l = 4^-1 ' By lemma 3*1 these may be extended t o PL homeomorphisms - V': x i - A I Thus g i v i n g a PL homeomorphism ' h :. *J + ^ - A* + A* = A n " 1 * ( v 1 +! v 2 ) and A11"*1 # (v-j^ + v 2 ) i s a comb, n - c e l l — Thm. 2.1. ^ \ - 48 - ' • B i b l i o g r a p h y [ l ] Alexander, J . W., The c o m b i n a t o r i a l t h e o r y of complexes, . Ann. of Math., 31 .(1930), 292-320. ' [2] ' Hudson, J . F. P., Piecewise l i n e a r topology, l e c t u r e n otes, U n i v e r s i t y of Chicago (1966/67). [3] Newman, M. H. A., Combinatory t o p o l o g y and e u c l i d e a n n-space, Proc. London Math. Soc. 30 (1930), 339-3^6. [4] Zeeman, E. C , Seminar on c o m b i n a t o r i a l topology, I n s t i t u t des Hautes Etudes S c i e n t i f i q u e s (1963), mimeographed. 

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