The rational symmetric signature of manifolds with finite fundamental group

Abstract

Let M be a closed, oriented manifold of dimension 2k. Poincart duality asserts that the rational intersection form (, )a : Hk@f; Q) x Hk(M; Q) -+ Q defined by (a, fl)o = (ctujl)[M] is a non-singular (- l)k-symmetric bilinear pairing. The isometry class of the rational intersection form is determined by the rank if k is odd and by the rank and signature if k is even. We wish to make a corresponding analysis of the equivariant intersection form in the case where M is the total space of a finite G-cover. Let G be a finite group and w: G + { f l} a homomorphism. A free (G, w)-manifold is a closed, oriented manifold M with a free G-action, so that for all g E G, g* [M] = w(g) [M]. If N is a closed manifold with finite fundamental group, its universal cover is a free (n,N, w,N)-manifold. The intersection form of a (G, w)-manifold satisfies the equivariance property (ga, g& = w(g)(m, &. The form has an invariant Lagrangian if there is a G-invariant subspace V c Hk(M; Q) so that I/ is equal to its orthogonal complement 1/l = {/?I (V, p>o = O}. S’ mce QG is semisimple, a form with an invariant Lagrangian admits a complementary Lagrangian, so is equivariantly hyperbolic. Our main result is: