A Remark on $\mathrm{Pin}(2)$-equivariant Floer homology

TL;DR

This paper explores the relationship between the monopole Fr{\

Contribution

It establishes connections between various Floer homology invariants and the $ ext{Pin}(2)$-equivariant Floer homology, clarifying which correction terms are significant.

Findings

Correction terms originate from specific subgroups: $bZ/4$, $S^1$, and $ ext{Pin}(2)$.

The monopole Fr{\

The analogues of the Involutive Heegaard Floer correction terms are related to $ ext{Pin}(2)$-equivariant Floer homology.

Abstract

In this remark, we show how the monopole Fr{\o}yshov invariant, as well as the analogues of the Involutive Heegaard Floer correction terms $\underline{d},\overline{d}$, are related to the $\mathrm{Pin}(2)$-equivariant Floer homology $\mathit{SWFH}^G_*$. We show that the only interesting correction terms of a $\mathrm{Pin}(2)$-space are those coming from the subgroups $\mathbb{Z}/4$, $S^1$, and $\mathrm{Pin}(2)$ itself.