Combinatorial Hirzebruch formula
by
Alexander S. Mishchenko
Moscow State Unoversity

The well know Hirzebruch formula says that for 4k-dimensioanal orientable compact closed manifold X the following equation holds

sign X = 22k <L(X), [X] >,

where

sign X = sign (H2k(X, C), \cup )

is the signature of nondegenerated quadratic form in the cohomology groups H^2k(X, C), defined by \cup -product,

L(X) = Õ j  \fractj2/2th (tj2/2)

is the Hirzebruch characteristic class defined by formal generators t_j by

\sigmak(t1, ..., tn) = ck (cTX).

V.A. Rokhlin was the first who esatblished the simplest case of the Hirzebruch formula in 4-dimensional case. In 1952 Rokhlin expressed the signature of 4-dimensional manifold in terms of the first Pontryagin class. A year later F. Hirzebruch wrote his formula for expressing the signature of arbitrary oriented manifold in terms of characteristic Pontryagin classes, so called Hirzebruch formula.

During last 45 years many different generalisations of the Hirzebruch formula were opened. Among them: algebraic setting for symmetric signature of non simply connected manifolds, functional setting for C^*-group algebras of fundamental groups, the Hirzebruch formula for Fredholm and asymptotic representations, smooth version due to Atiyah-Singer index formula.

One the most interesting conjecture was formulating by S. Novikov on so called higher signatures, being open till now.

The last consequences of Hirzebruch formula are: new short proof of topological invariance of rational Pontryagin classes due to M. Gromov (1995), combinatorial local Hirzebruch formula for combinatorial manifolds due to M. Gromov and author (1998).

Partly supported by RFFS No.9901-01201 and INTAS No. 96-1099

Date received: July 5, 1999