A W(Z 2 ) Invariant for Orientation Preserving Involutions

Abstract

In this paper we calculate an invariant in W(Z2), the Witt ring of nonsingular, symmetric Z2-inner product spaces, for orientation-preserving involutions on compact, closed, connected 4n-dimensional manifolds M. This invariant with the Atiyah-Singer index theorem uniquely determines the orthogonal representation of Z on H2n(M; Z)/TOR. We also give an example to show that this invariant detects actions that the Atiyah-Singer theorem cannot. In this paper we calculate a torsion invariant for orientation-preserving involutions. This invariant and the Atiyah-Singer-Segal G-signature theorem allow one to compute precisely the element of W(Z; Z2) given by the orthogonal representation of Z2 on H2n(M; Z)/TOR. This cannot be done with the Atiyah-Singer-Segal theorem alone. We will return to this in the last section. In [4] Conner and Raymond define an invariant q(T, M) for orientationpreserving actions of Zp, p a prime, T a generator of Zp, on closed, compact, oriented manifolds M of dimension 4n. Briefly the action of Zp on M gives us an orthogonal representation of Z on H2n(M; Q). If we denote by w(T, M) and sgn(T, M) the rational Witt class and signature, respectively, of the inner product (x, y)p(x u y, [M]) on the subspace of fixed vectors then q(T, M) = w(T, M) sgn(T, M)1. In [1] the problem of expressing q(T, M) in terms of fixed point information was solved for P odd. In this paper we compute q(T, M) for p = 2 and give some applications. First some notation and background. W(R) will always denote the Witt group of nonsingular, symmetric, inner-product spaces over R, R a ring. W(Z2) is isomorphic to Z2 and the isomorphism is given by taking the rank of the inner product space mod 2. From now on we identify W(Z2) and Z2 by this isomorphism. If V is a disjoint union of a finite number of closed Received by the editors June 11, 1974. AMS (MOS) subject classifications (1970). Primary 57D85; Secondary IOC05.