Topological obstructions to certain Lie group actions on manifolds

Abstract

Given a smooth closed S 1 -manifold M, this article studies the extent to which certain numbers of the form (f* (x) P · C) [M] are determined by the fixed-point set M S1 , where f: M → K(π 1 (M), 1) classifies the universal cover of M, x ∈ H* (π 1 (M); Q), P is a polynomial in the Pontrjagin classes of M, and C is in the subalgebra of H* (M; Q) generated by H 2 (M;Q). When M S1 = O, various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free S 1 -action extends to an action by some semisimple compact Lie group G, then (f(x) · P · C) [M] = 0. Similar vanishing results are obtained for spin manifolds admitting certain S 1 -actions.