A non-trivial index difference on surfaces of genus at least $3$

Abstract

For every closed surface of genus at least $3$, equipped with any bounding spin structure, we show that the index difference, viewed as a map from the fundamental group of the space of Dirac-invertible Riemannian metrics to $\KO^{-4}(*)$, is non-trivial. If the genus is at least $5$, we also show that the aforementioned index difference is surjective. For products of two closed surfaces of genus at least $3$, equipped with any spin structure, we prove that the corresponding space of Dirac-invertible Riemannian metrics is not contractible. We discuss the relationship of this result to the existence of metrics with harmonic spinors in dimension~$4$.