Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

TL;DR

This paper introduces a relative $L^2$-$ ho$-invariant for Dirac operators on spin manifolds with boundary, demonstrating its role in classifying positive scalar curvature metrics and revealing infinite moduli space components under certain conditions.

Contribution

It defines a new bordism-invariant for positive scalar curvature metrics on manifolds with boundary and explores its implications for the structure of the moduli space of such metrics.

Findings

Existence of infinitely many bordism classes of psc metrics under torsion conditions.

The moduli space of psc metrics has infinitely many path components.

Potential extension to delocalised $ ho$-invariants for further results.

Abstract

In this paper, we define a relative $L^2$-$\rho$-invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of positive scalar curvature metrics which are collared near the boundary. As an application, we show that if a $4k+3$-dimensional spin manifold with boundary admits such a metric and if, roughly speaking, there exists a torsion element in the difference of the fundamental groups of the manifold and its boundary, then there are infinitely many bordism classes of such psc metrics on the given manifold. This result in turn implies that the moduli-space of psc metrics on such manifolds has infinitely many path components. We also indicate how to define delocalised $\eta$-invariants for odd-dimensional spin manifolds with boundary, which could then be used to obtain similar results for $4k+1$-dimensional manifolds.