Topological proof of theG-signature theorem forGfinite

Abstract

Part One. By considering the cyclic subgroup generated by a given element, we may restrict our attention to cyclic group actions. By a Zd manifold (g, M) we will mean a smooth, oriented, compact manifold (without boundary unless otherwise stated or obvious) together with an orientation preserving diffeomorphism of order d. Frequently, we will omit the g in referring to Zd manifolds M. We denote the disjoint union by +, the disjoint union of r copies of M by rM, and the disjoint difference (one reverses the orientation of the second manifold) by —. A Z d manifold If bounds (resp. rationally bounds) if M (resp. r M) is the boundary of a Zd manifold. Two Zd manifolds are bordant (resp. rationally bordant) if their disjoint difference bounds (resp. rationally bounds). The collection of bordism classes of Zd manifolds forms a graded ring O*(Zd) in the usual manner. Let (g, B) denote the d-fold branched cover of S2 along d points together with the deck translation which rotates neighborhoods of the fixed points through an angle 2π/d. Let (g, PJ denote the action on ^-dimensional complex projective space given by g[z0, , zn_lf zn] — [zQ, , zn_lf <ozn]. Here and throughout part one ω = exp(2πi/d). The fix point set has two components: the point [0, , 0, 1] where the action on the trivial normal bundle is multiplicatio n by ώy and