Non-formality in $\mathrm{Pin}(2)$-monopole Floer homology

TL;DR

This paper explores the non-formality of the $ ext{Pin}(2)$-monopole Floer homology, providing explicit Massey products and methods for computing the homology of connected sums of three-manifolds.

Contribution

It introduces explicit descriptions of Massey products in $ ext{Pin}(2)$-monopole Floer homology and applies them to compute homology for connected sums.

Findings

Massey products can be explicitly described in this setting

Non-formality is demonstrated through these Massey products

Methods for computing homology of connected sums are developed

Abstract

In previous work, we introduced a natural $\mathcal{A}_{\infty}$-structure on the $\mathrm{Pin}(2)$-monopole Floer chain complex of a closed, oriented three-manifold $Y$, and showed that it is non-formal in the simplest case in which $Y$ is the three-sphere $S^3$. In this paper, we provide explicit descriptions of several Massey products induced on homology, and discuss how they can be used to compute the $\mathrm{Pin}(2)$-monopole Floer homology of connected sums in many concrete examples.