Equivariant Seiberg–Witten Floer and Heegaard Floer homology correspondence

Let $Y$ be a rational homology $3$-sphere and let $\mathfrak{s}\in \operatorname{Spin}^c(Y)$ be a $\operatorname{Spin}^c$ structure. The groups $HF^{SW,+}_{*,U(1)}(Y,\mathfrak{s})$, $HF^{SW,-}_{*,U(1)}(Y,\mathfrak{s})$, $\widehat{HF}^{SW}_*(Y,\mathfrak{s})$, and $HF^{SW}_{\mathrm{red},*}(Y,\mathfrak{s})$ are the equivariant Seiberg–Witten Floer groups, while $HF^+_*(Y,\mathfrak{s})$, $HF^-_*(Y,\mathfrak{s})$, $\widehat{HF}_*(Y,\mathfrak{s})$, and $HF_{\mathrm{red},*}(Y,\mathfrak{s})$ are the corresponding Heegaard Floer groups.

Equivariant Seiberg–Witten Floer and Heegaard Floer correspondence. For every such $Y$ and $\mathfrak{s}$, there are isomorphisms

$$\begin{array}{ccc} HF^{SW,+}_{*,U(1)}(Y,\mathfrak{s}) \cong HF^+_*(Y,\mathfrak{s}), &\qquad & HF^{SW,-}_{*,U(1)}(Y,\mathfrak{s}) \cong HF^-_*(Y,\mathfrak{s});\\[3mm] \widehat{HF}^{SW}_*(Y,\mathfrak{s}) \cong \widehat{HF}_*(Y,\mathfrak{s}), &\qquad & HF^{SW}_{\mathrm{red},*}(Y,\mathfrak{s}) \cong HF_{\mathrm{red},*}(Y,\mathfrak{s}). \end{array}$$

This conjecture proposes that the equivariant Seiberg–Witten Floer invariants agree with the Heegaard Floer invariants introduced by Ozsváth and Szabó. The preceding result identifies the corresponding infinity versions, but the asserted isomorphisms for the plus, minus, hat, and reduced theories are not resolved in the supplied source context.