The equivalence of two Pin(2)-equivariant Seiberg-Witten Floer homologies

Abstract

We show that for a rational homology 3-sphere $Y$ equipped with a self-conjugate spin$^c$-structure $\mathfrak s$, the $\operatorname{Pin}(2)$-equivariant monopole Floer homology of $(Y,\mathfrak s)$, as defined by Lin, is isomorphic to the $\operatorname{Pin}(2)$-equivariant Seiberg-Witten Floer homology of $(Y,\mathfrak s)$ defined by Manolescu.