Circle actions and scalar curvature
Abstract
We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$invariant metrics of positive scalar curvature on every $S^1$manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are noninvariant metrics of positive scalar curvature on many manifolds with circle actions. Results from equivariant bordism allow us to show that there is an invariant metric of positive scalar curvature on the connected sum of two copies of a simply connected semifree $S^1$manifold $M$ of dimension at least six provided that $M$ is not $\text{spin}$ or that $M$ is $\text{spin}$ and the $S^1$action is of odd type. If $M$ is spin and the $S^1$action of even type then there is a $k>0$ such that the equivariant connected sum of $2^k$ copies of $M$ admits an invariant metric of positive scalar curvature if and only if a generalized $\hat{A}$genus of $M/S^1$ vanishes.
 Publication:

arXiv eprints
 Pub Date:
 May 2013
 arXiv:
 arXiv:1305.2288
 Bibcode:
 2013arXiv1305.2288W
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Differential Geometry;
 53C21;
 57S15
 EPrint:
 25 pages