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

TL;DR

This paper demonstrates that periodic eta-invariants serve as obstructions to positive scalar curvature cobordisms between certain manifolds, using advanced index theory and minimal surface techniques.

Contribution

It introduces new obstructions based on periodic eta-invariants for positive scalar curvature cobordisms in dimensions 4 to 6, linking index theory with geometric topology.

Findings

Periodic eta-invariants obstruct positive scalar curvature cobordisms.

Bordism groups are infinite for certain fundamental groups.

Combines minimal surface techniques with end-periodic index theorems.

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 $\Omega^{spin,+}_{n+1}(S^1 \times BG)$ are infinite for any non-trivial group $G$ which is the fundamental group of a spin spherical space form of dimension $n=3$ or $5$.