By Norman Levitt

The booklet explores the potential of extending the notions of "Grassmannian" and "Gauss map" to the PL type. they're exceptional from "classifying area" and "classifying map" that are basically homotopy-theoretic notions. The analogs of Grassmannian and Gauss map outlined comprise geometric and combinatorial details. crucial purposes contain attribute category concept, smoothing concept, and the life of immersion satifying yes geometric standards, e.g. curvature stipulations. The e-book assumes wisdom of uncomplicated differential topology and package thought, together with Hirsch-Gromov-Phillips thought, in addition to the analogous theories for the PL classification. The paintings will be of curiosity to mathematicians desirous about geometric topology, PL and PD features of differential geometry and the geometry of polyhedra.

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**Extra resources for Grassmannians and Gauss Maps in Piecewise-Linear and Piecewise-Differential Topology**

**Sample text**

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.