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A survey of the spherical space form problem by J. F. Davis

By J. F. Davis

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Were such a t r a n s f e r least coefficients locally the homomorphism depending o n l y on the embedding on the r e m a i n d e r of cohomology subdivision while the t r a n s f e r I plan might a chain-level t o be the c h a i n s on the f i r s t or on some f i n e r need m e r e l y be s i n g u l a r . e. m(e(K)). s u b d i v i s i o n of since f o r K. It an embedding f : the c o - c h a i n t*~(M) ordered c o n s i s t e n t with = m(M). 27 A slightly more cumbersome way of p r o v i n g the same t h i n g , (specifically for more i n s i g h t into the k th P o n t r j a g i n class pk ), one which o f f e r s G a b r i e l o v ' s approach is as f o l l o w s : Let Q be an arbitrarily-chosen n - p l a n e in V C G be d e f i n e d Q ~-n,n by V = {P E G Idim P N Q ) 2k} A l t e r n a t i v e l y V may be Q ~-n,n Q t h o u g h t as the set of n - p l a n e s RE G such t h a t o r t h o g o n a l n,~-n p r o j e c t i o n of R to Q (equivalently, Q to R) has n u l l i t y ) 2k.

Is an ( o r i e n t e d ) the o r i g i n a l Pk (p) simplex of triangulation, the f i r s t p ~ o , o = Z v ) Q E { t { p F) h ) to compute each summand on t h e r i g h t h a n d s i d e , c o r r e s p o n d i n g to a simplex T with o < 3, p l a c e d by the o b v i o u s c h a i n on a f i n e are a l l a real then we may compute {3) p ~ h a formula for Pontrjagin class. simplex n-k combinatorial structure barycentric real the v a l u e of some More e x p l i c i t l y , as in subcomplexes.

N U of TM at P On+k standard R , will r the Gauss map suffice X . o = n-dim o. a T~O C~)~ " takes ~ to 0 n bundle maPn TMo + Yn,k PL This q ~ clearly maps d e t e r m i n e s an a f f i n e plane X in U such t h a t P s e t c o n s t i t u t e s a PL the v e c t o r p-q of 0 the unique such p l a n e c o n t a i n i n g o. is Con- orthogo- W of d i m e n s i o n p enough, we c l a i m t h a t Gog U i s taken small P homeomorphically onto a PL takes g to c o n s t r u c t the s i m p l e x those p o i n t s to = ~ TM I~ ÷ TV o o L(o,M n ) Consider, t h e r e f o r e , a p o i n t p ~ ~ , and the f i b e r n o p.

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