Homotopy equivalence of spaces of metrics with invertible Dirac operator

TL;DR

This paper proves that for cobordant closed spin manifolds of dimension at least 3, the spaces of metrics with invertible Dirac operators are homotopy equivalent, extending previous results on scalar curvature and surgery.

Contribution

It establishes a homotopy equivalence of metric spaces with invertible Dirac operators for cobordant spin manifolds, generalizing prior surgery results.

Findings

Homotopy equivalence of metric spaces for cobordant manifolds

Extension of surgery results to invertible Dirac operator metrics

Relative homotopy equivalence statement provided

Abstract

We prove that for cobordant closed spin manifolds of dimension $n\geq 3$ the associated spaces of metrics with invertible Dirac operator are homotopy equivalent. This is the spinorial counterpart of a similar result on positive scalar curvature of Chernysh/Walsh and generalizes the surgery result of Ammann-Dahl-Humbert on the existence of metrics with invertible Dirac operator under surgery. We also give a relative statement of this homotopy equivalence.