$\mathrm{Pin}(2)$-Monopole Floer homology and the Rokhlin invariant

TL;DR

This paper establishes that the bar version of $ ext{Pin}(2)$-monopole Floer homology for a 3-manifold with a self-conjugate spin$^c$ structure is determined by the triple cup product and Rokhlin invariants, linking index theory and topology.

Contribution

It provides a new topological characterization of $ ext{Pin}(2)$-monopole Floer homology using classical invariants and cup products, extending Atiyah's results to three dimensions.

Findings

Floer homology is determined by triple cup product and Rokhlin invariants.

Connects mod 2 index theory with Floer homology.

Provides a 3D analogue of Atiyah's results on spin structures.

Abstract

We show that the bar version of the $\mathrm{Pin}(2)$-monopole Floer homology of a three-manifold $Y$ equipped with a self-conjugate spin$^c$ structure $\mathfrak{s}$ is determined by the triple cup product of $Y$ together with the Rokhlin invariants of the spin structures inducing $\mathfrak{s}$. This is a manifestation of mod $2$ index theory, and can be interpreted as a three-dimensional counterpart of Atiyah's classic results regarding spin structures on Riemann surfaces.