Positive scalar curvature metrics via end-periodic manifolds

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.