$S^1$-equivariant bordism, invariant metrics of positive scalar curvature, and rigidity of elliptic genera

TL;DR

This paper constructs generators for $S^1$-equivariant Spin-bordism groups, explores obstructions to positive scalar curvature metrics on $S^1$-manifolds, and provides bordism-based proofs of elliptic genus rigidity and $ ext{ extonehalf}$-genus vanishing.

Contribution

It introduces geometric generators for equivariant bordism groups with inverted 2, and applies these to analyze positive scalar curvature and elliptic genus rigidity.

Findings

Obstructions to positive scalar curvature are characterized by an $ ext{ extonehalf}$-genus of orbit spaces.

A bordism-theoretic proof of the vanishing of the $ ext{ extonehalf}$-genus for manifolds with non-trivial $S^1$-action.

A new bordism-based proof of the rigidity of elliptic genera.

Abstract

We construct geometric generators of the effective $S^1$-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which $S^1$-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 $\hat{A}$-genus of orbit spaces. This $\hat{A}$-genus generalizes a previous definition of Lott for orbit spaces of semi-free $S^1$-actions. As a further application of our results, we give a new proof of the vanishing of the $\hat{A}$-genus of a Spin manifold with non-trivial $S^1$-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.