S1-Equivariant bordism, invariant metrics of positive scalar curvature, and rigidity of elliptic genera

Abstract

We construct geometric generators of the effective [Formula: see text]-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which [Formula: see text]-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the same manifold, the only obstruction to the existence of such a metric is an [Formula: see text]-genus of orbit spaces. This [Formula: see text]-genus generalizes a previous definition of Lott for orbit spaces of semi-free [Formula: see text]-actions. As a further application of our results, we give a new proof of the vanishing of the [Formula: see text]-genus of a Spin manifold with nontrivial [Formula: see text]-action originally proven by Atiyah and Hirzebruch. Moreover, based on our computations we can give a bordism-theoretic proof for the rigidity of elliptic genera originally proven by Taubes and Bott–Taubes.