On positive scalar curvature cobordisms and the conformal Laplacian on end-periodic manifolds

Abstract

We show that the periodic $\eta$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of cobordisms with positive scalar curvature metrics between manifolds of dimensions $4$ and $6$. The proof combines a relative version of the Schoen--Yau minimal surface technique with an end-periodic index theorem for the Dirac operator. As a result, we show that the bordism groups ${\Omega }_{n+1}^{spin,+}(S^1\times BG)$ are infinite for any non-trivial group $G$ which is the fundamental group of a spin spherical space form of dimension $n=3$ or $5$.

arXiv commentFinal version, to appear in Communications in Analysis and Geometry. Some clarifications and revisions in response to referee's comments