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Extra resources for Using Borsuk-Ulam theorem. Lect. on topological methods in combinatorics and geom

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M } ] : L (M) E (M;R). 19 defined for M bounds o r i e n t a b l y , i . e . q, in M = @W. ) some Under t h i s neighborhood of M. the s i g n a t u r e of the o r i e n t e d m a n i f o l d There are then two is merely the usual a l g e b r a i c - t o p o l o g i c a l s i g n a t u r e of a m a n i f o l d - w i t h - b o u n d a r y i . e . form in (If We assume W be equipped w i t h a Riemannian m e t r i c i s o m e t r i c obvious " c a n d i d a t e s " f o r The f i r s t 4i-i M o r i e n t e d Riemannian ( 4 i - 1 ) - m a n i f o l d s 2J-dimensional r a t i o n a l the s i g n a t u r e of homology.

6 to the l a r g e r c o n t e x t of A-homology m a n i f o l d s . 14 Corollary. Given an i - d i m e n s i o n a l c h a r a c t e r i s t i c class A-homology n - m a n i f o l d s , t h e r e e x i s t s a l o c a l representing for o r d e r e d formula c. We c l a i m t h a t t h i s extension is e a s i l y achieved. It r e a d i l y be seen by the reader t h a t the basic d e f i n i t i o n s s-cell c complex, s - b l o c k b u n d l e , s - c e l l , etc. will s-ball, - may be mimicked in c c n t e x t a p p r o p r i a t e to the study of A-homology m a n i f o l d s .

21 than a l o c a l - o r d e r e d formula. However, d e s p i t e i t s attractive c a n o n i c i t y , t h i s formula has not been made " c o m b i n a t o r i a l " in the sense t h a t a d i r e c t a l g o r i t h m f o r computing i t purely from the combi4i -I n a t o r i a l data of is not y e t known. Indeed, the t r a d i t i o n a l difficulties in a c t u a l l y computing the ~ - i n v a r i a n t of a smooth Riemannian m a n i f o l d c e r t a i n l y seem to p e r s i s t in t h i s c o n t e x t as well as f a r as is c u r r e n t l y known.

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