On the Chern numbers for pseudo-free circle actions

Abstract

Let (M, psi) be a (2n + 1)-dimensional oriented closed manifold with a pseudo-free S-1-action psi : S-1 x M -> M. We first define a local data L (M, psi) of the action which consists of pairs (C, (p (C); (q) over right arrow (C))) where C is an exceptional orbit, p (C) is the order of isotropy subgroup of C, and (q) over right arrow (C) is an element of (Z(p(C)) (x))(n) is a vector whose entries are the weights of the slice representation of C. In this paper, we give an explicit formula of the Chern number < c(1)(E)(n), [M/S-1]> modulo Z in terms of the local data, where E = M x (S1) C is the associated complex line orbibundle over M/S-1. Also, we illustrate several applications to various problems arising in equivariant symplectic topology. c.International Press of Boston, Inc.