The Hirzebruch formula: 45 years of history and the present state
by
Alexander Mishchenko
Lomonosov Moscow State University

The Hirzebruch formula: 45 years of history and the present state.

A.S.Mishchenko
(Moscow)

V.Rokhlin was the first who established the relation between signature of manifold with the Pontryagin classes in 4-dimensional case. A little bit later F.Hirzebruch wrote the common formula known now as the Hirzebruch formula.

The Hirzebruch formula says that for 4k-dimension oriented compact closed manifold X

signX = 22k á L(X), [X] ñ,

(1)

where

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

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

L(X)= Õ j   tj/2 th(tj/2)

is the Hirzebruch characteristic class.

There are different ways of generalizations of the Hirzebruch formulas mainly for non simly connected manifolds. Given closed oriented non simply connected manifold X, \pi = \pi₁(X), let

fX:X -->     B\pi

be the canonical mapping defined up to homotopy which induces the isomorphism

(fX)*:\pi1(X) -->     \pi.

Consider a finite dimensional representation

\rho:\pi -->     U(N).

Then one can consider cohomology groups with local system of coefficients generated by the representation \rho, H^2k(X), \rho. The \cup product induces a quadratic nondegenerated form on this group and let

sign\rhoX=sign(H2k(X, \rho), \cup ).

Then surely

sign\rho X = 22k á L(X)chfX*\xi\rho, [X] ñ,

(2)

where \xi\rho is the vector bundle over B\pi generated by the representation \rho. Inspite that both left hand and right hand sides of the formula (2) coinside with that of the formula (1) this generalization has further generalisations more useful. Namely one can construct at least right hand side of the formula(2) for more general notion of representation of the group \pi.

A natural universal generalization of the Hirzebruch formula associates with C^*-algebras. Let C^*[\pi] be the C^*-group algebra of the group \pi. Then each unitary representation of the group \pi can be uniquely extended to the representation [`(\rho)] of the algebra C<sup>*</sup>[\pi]. Let A=Im[`(\rho)] , [`(\rho)] :C^*[\pi] ([ || ( --> )])

Using different kinds of representations \rho one can obtain homotopy invariant characteristic classes, so called higher signaturs of the manifold X, sign_\rho(X). The crucial property of this formula is that the higher signature generated by a representatin \rho is not only invariant of homotopoy equivalence or smooth bordisms but it is invariant of topological bordisms. This observation allowed M.Gromov to present a short proof of topologocal invariance of rational Pontryagin classes.

The Hirzebruch formula can be generalized also to the case of combinatorial version of signature with coefficients in arbitrary vector bundle \xi over X. Due to M.Gromov the idea is to imitate the construction of the algebraic Poincare complex, generated by the combinatorial structure of the manifold X. Finally one can obtain a combinatorial invariant of the manifold X, sign(X, \xi), and a version of the Hirzebruch formula

sign(X, \xi) = 22k á L(X)ch\xi, [X] ñ,

Date received: May 28, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-16.