Vol. 5, No. 3, 2020

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Positive scalar curvature metrics via end-periodic manifolds

Michael Hallam and Varghese Mathai

Vol. 5 (2020), No. 3, 639–676
Abstract

We obtain two types of results on positive scalar curvature metrics for compact spin manifolds that are even-dimensional. The first type of result are obstructions to the existence of positive scalar curvature metrics on such manifolds, expressed in terms of end-periodic eta invariants that were defined by Mrowka, Ruberman and Saveliev (Mrowka et al. 2016). These results are the even-dimensional analogs of the results by Higson and Roe (2010). The second type of result studies the number of path components of the space of positive scalar curvature metrics modulo diffeomorphism for compact spin manifolds that are even-dimensional, whenever this space is nonempty. These extend and refine certain results in (Botvinnik and Gilkey 1995) and also (Mrowka et al. 2016). End-periodic analogs of K-homology and bordism theory are defined and are utilised to prove many of our results.

Keywords
positive scalar curvature metrics, maximal Baum–Connes conjecture, end-periodic manifolds, end-periodic K-homology, end-periodic eta invariant, vanishing theorems, end-periodic bordism
Mathematical Subject Classification 2010
Primary: 58J28
Secondary: 19K33, 19K56, 53C21
Milestones
Received: 18 February 2019
Revised: 8 January 2020
Accepted: 26 January 2020
Published: 28 July 2020
Authors
Michael Hallam
Mathematical Institute
Oxford University
Andrew Wiles Building
Woodstock Road
Oxford
United Kingdom
Varghese Mathai
School of Mathematical Sciences
University of Adelaide
North Terrace
Adelaide
Australia