Kronheimer–Manolescu's Seiberg–Witten–Ozsváth–Szabó Floer homology conjecture

Let $Y$ be a three-manifold with $b_1(Y)=1$ and a nontorsion $spin^c$ structure ${\mathfrak{s}}$. Let ${\text{SWF}}$ and ${\text{SWF}}_0$ be the associated periodic Seiberg–Witten pro-spectra, and let $H_*$, $H_*^{S^1}$, $cH_*^{S^1}$, and $tH_*^{S^1}$ denote ordinary, Borel, co-Borel, and Tate homology theories. Kronheimer–Manolescu's conjecture.

$$\widehat{HF}_n=H_n({\text{SWF}}_0)=H_n({\text{SWF}}),\qquad HF^+_n=H^{S^1}_{n+1}({\text{SWF}}_0)=H^{S^1}_{n+1}({\text{SWF}}),$$ $$HF^-_n=cH^{S^1}_{n+2}({\text{SWF}}_0),\qquad HF^{\infty}_n=tH^{S^1}_n({\text{SWF}}_0).$$

The same should hold for manifolds with $b_1(Y)=0$ after replacing ${\text{SWF}}_0$ by ${\text{SWF}}$. This conjecture predicts explicit identifications between the Ozsváth–Szabó and Seiberg–Witten Floer theories; the paper presents them as expected relationships rather than proving them.