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

Wiemeler M.

Abstract

We construct geometric generators of the effective S1-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which S1-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 Â-genus of orbit spaces. This Â-genus generalizes a previous definition of Lott for orbit spaces of semi-free S1-actions. As a further application of our results, we give a new proof of the vanishing of the Â-genus of a Spin manifold with nontrivial S1-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.