Communications in Analysis and Geometry

Volume 30 (2022)

Number 4

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

Pages: 869 – 890

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n4.a6

Authors

Demetre Kazaras (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Daniel Ruberman (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Nikolai Saveliev (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Abstract

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

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The first author was partially supported by NSF Grant DMS-1547145, the second author was partially supported by NSF Grants DMS-1506328 and DMS-1811111, and the third author was partially supported by a grant from the Simons Foundation #426269.

Received 27 March 2019

Accepted 28 October 2019

Published 30 January 2023