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

Research paper by Michael Wiemeler

Indexed on: 11 Jul '16Published on: 11 Jul '16Published in: Mathematics - Geometric Topology



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. 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.