On positive scalar curvature bordism

Using standard results from higher (secondary) index theory, we prove that the positive scalar curvature bordism groups $\mathrm{Pos}^\operatorname{spin}_{4n} (G \times \mathbb{Z})$ are infinite for any $n \geq 1$ and $G$ a group with nontrivial torsion. We construct representatives of each of these classes which are connected and with fundamental group $G \times \mathbb{Z}$. We get the same result for $\mathrm{Pos}^\operatorname{spin}_{4n+2} (G \times \mathbb{Z})$ if $G$ is finite and contains an element which is not conjugate to its inverse. This generalizes the main result of Kazaras, Ruberman, Saveliev, “On positive scalar curvature cobordism and the conformal Laplacian on end-periodic manifolds” to arbitrary even dimensions and arbitrary groups with torsion.

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P.P. thanks Ministero Istruzione Universit`a Ricerca for partial support through the PRIN 2015 Spazi di Moduli e Teoria di Lie.

T.S. and V.F.Z. thank the German Science Foundation and its priority program “Geometry at Infinity” for partial support.

Received 22 December 2019

Accepted 7 April 2020

Published 17 August 2023