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arXiv · 2606.12283

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

Samuel Lockman
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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~$4$.

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Samuel Lockman. 2026-06-10. A non-trivial index difference on surfaces of genus at least $3$. https://arxiv.org/abs/2606.12283

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