TAILIEUCHUNG - Đề tài " The homotopy type of the matroid grassmannian "

Characteristic cohomology classes, defined in modulo 2 coefficients by Stiefel [26] and Whitney [28] and with integral coefficients by Pontrjagin [24], make up the primary source of first-order invariants of smooth manifolds. When their utility was first recognized, it became an obvious goal to study the ways in which they admitted extensions to other categories, such as the categories of topological or PL manifolds; perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques. Modulo 2, this hope was realized rather quickly: it is not hard to see that the Stiefel-Whitney classes are. | Annals of Mathematics The homotopy type of the matroid grassmannian By Daniel K. Biss Annals of Mathematics 158 2003 929-952 The homotopy type of the matroid grassmannian By Daniel K. Biss 1. Introduction Characteristic cohomology classes defined in modulo 2 coefficients by Stiefel 26 and Whitney 28 and with integral coefficients by Pontrjagin 24 make up the primary source of first-order invariants of smooth manifolds. When their utility was first recognized it became an obvious goal to study the ways in which they admitted extensions to other categories such as the categories of topological or PL manifolds perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques. Modulo 2 this hope was realized rather quickly it is not hard to see that the Stiefel-Whitney classes are PL invariants. Moreover Whitney was able to produce a simple explicit formula for the class in codimension i in terms of the i-skeleton of the barycentric subdivision of a triangulated manifold for a proof of this result see 13 . One would like to find an analogue of these results for the Pontrjagin classes. However such a naive goal is entirely out of reach indeed Milnor s use of the Pontrjagin classes to construct an invariant which distinguishes between nondiffeomorphic manifolds which are homeomorphic and PL isomorphic to S7 suggested that they cannot possibly be topological or PL invariants 19 . Milnor was in fact later able to construct explicit examples of homeomorphic smooth 8-manifolds with distinct Pontrjagin classes 20 . On the other hand Thom 27 constructed rational characteristic classes for PL manifolds which agreed with the Pontrjagin classes and Novikov 23 was able to show that rationally the Pontrjagin classes of a smooth manifold were topological invariants. This led to a surge of effort to find an explicit combinatorial expression for the rational Pontrjagin classes analogous to Whitney s formula for the .

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