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Localization of a $\mathrm{KO}^{\ast}(\mathrm{pt})$-valued index and the orientability of the $\mathrm{Pin}^-(2)$ monopole moduli space

Jin Miyazawa

Algebraic & Geometric Topology 25 (2025) 887–918
Abstract

It is known that the Dirac index of a Spin c structure is localized to the characteristic submanifold. We introduce the notion of G±(n,s+,s) structure on a manifold as a common generalization of the Spin c structure and the Hn(s) structure defined by D Freed and M Hopkins, and formulate a version of characteristic submanifold for the G±(n,s+,s) structure. We show that the KO (pt )-valued index associated with the G±(n,s+,s) structure is localized to the characteristic submanifold. As an application, we give a topological sufficient condition for the moduli space of Pin (2) monopoles to be orientable.

Keywords
Witten deformation, localization of index, Seiberg–Witten theory, $K$-theory
Mathematical Subject Classification
Primary: 57K41, 58J20
References
Publication
Received: 17 September 2022
Revised: 22 September 2023
Accepted: 12 February 2024
Published: 16 May 2025
Authors
Jin Miyazawa
Graduate School of Mathematical Sciences
The University of Tokyo
Tokyo
Japan

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