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~$4$.

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BibTeX RIS

BibTeX

@misc{lockman2026nontrivial,
  title = {{A non-trivial index difference on surfaces of genus at least $3$}},
  author = {Samuel Lockman},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.12283},
  eprint = {2606.12283},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.12283},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - A non-trivial index difference on surfaces of genus at least $3$
AU  - Samuel Lockman
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2606.12283
UR  - https://arxiv.org/abs/2606.12283
AB  - 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~$4$.
ER  -

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