ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 19,57,58 Mishchenko Alexander S., Moscow State University, Russia Local combinatorial Hirzebruch formula The Hirzebruch formula signX = 2^{2k}⟨L(X), [X]⟩, dim X = 4k, can be generaized in different ways. In combinatorial case the Hirzebruch formula is known for different variants of representations ρ for fundamental group π_{1}(X): sign_{ρ}(X) = 2^{2k}⟨L(X)chξ_{ρ}, [X]⟩. We shall present similar formula for arbitrary vector bundle ξ. The main idea due to M. Gromov consits of considering an almost flat structure of the vector bundle ξ on the manifold X to immitate the Poincare duality in cohomology of the manifold X with coefficients in the bundle ξ. References: 1. A. S. Mishchenko, Homotopy invariants of nonsimply connected manifolds. I. Rational invariants. Math. USSR Izvestija 4 (1970), . No.3, 506–519.

2. M. Gromov, Positive curvature, Macroscopic dimension, Spectral Gaps and Higher signatures, Functional Anal. on the Eve og the 21st Century, v. II. Progress in Math., Basel–Boston: Birkhauser vol. 132, 1995.