$\mathrm{Pin}(2)$-monopole Floer homology, higher compositions and connected sums

TL;DR

This paper investigates how $ ext{Pin}(2)$-monopole Floer homology behaves under connected sums, constructing an $ ext{A}_ ext{infty}$-module structure and spectral sequence to understand the algebraic and topological properties involved.

Contribution

It introduces a partially defined $ ext{A}_ ext{infty}$-module structure on the Floer chain complex and identifies the Floer complex of a connected sum with an $ ext{A}_ ext{infty}$-tensor product, advancing the algebraic understanding of Floer homology.

Findings

Constructed an $ ext{A}_ ext{infty}$-module structure on the Floer chain complex.

Identified the Floer complex of a connected sum with an $ ext{A}_ ext{infty}$-tensor product.

Showed that the Floer homology of $S^3$ has non-trivial Massey products.

Abstract

We study the behavior of $\mathrm{Pin}(2)$-monopole Floer homology under connected sums. After constructing a (partially defined) $\mathcal{A}_{\infty}$-module structure on the $\mathrm{Pin}(2)$-monopole Floer chain complex of a three manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to quasi-isomorphism the Floer chain complex of a connected sum with a version of the $\mathcal{A}_{\infty}$-tensor product of the modules of the summands. There is an associated Eilenberg-Moore spectral sequence converging to the Floer groups of the connected sum whose $E^2$ page is the $\mathrm{Tor}$ of the Floer groups of the summands. We discuss in detail a simple example, and use this computation to show that the $\mathrm{Pin}(2)$-monopole Floer homology of $S^3$ has non trivial Massey products