A Remark on $\mathrm{Pin}(2)$-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.