Combinatorial fiber bundles
Talk by Prof. Dr. Nikolai Mnev
Date: 18.05.11 Time: 16.00 - 17.00 Room:
There is a long story of local combinatorial formulas for the characteristic classes of combinatorial manifolds. Here ”local” means that the characteristic class should be represented by a simplicial cochain which is a sum of contributions depending only on local com- binatorics (combinatorics of stars). The existence of such formulas is a simple result, but it is a challenge to find them. There are Whitney for- mulas for Z2 Stiefel-Whitney classes (proved by Halperin and Toledo), there are formulas for the first rational Pontryagin class (Gabrielov- Gelfand-Losik, Cheeger, Gaifullin). This story heavily depends on the explicit combinatorial cohomology models of Grassmanians. I will describe a new purely combinatorial model for the PL-Grassmanian BPLn. This allows us to obtain a clear notion of combinatorial fiber bundle, obtain a canonical Gauss map for a combinatorial manifold. New combinatorics leads to new ideas on formulas. One can see the Thom class of a bundle and get a general formula for the Euler class. One can cohomologically identify a combinatorial fiber bundle with a special homotopy local system of cochain complexes and thus realize cohomology trivializations, twisting cochain (in the sense of O’Brain- Toledo-Tong) and its traces, which leads to a kind of Chern classes. This is an informal report on the work in progress.Loading Document
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