Hutchings–Lee conjecture relating Seiberg–Witten and Morse theory invariants

Let $X$ be a closed $3$-manifold with $b_1(X)>0$, let $o$ be a homology orientation, and let $\operatorname{Spin}^c(X)$ denote the set of spin-c structures on $X$. Let $\operatorname{SW}_{X,o}\colon \operatorname{Spin}^c(X)\to\mathbb Z$ be the Seiberg–Witten invariant, and let $I_3\colon \operatorname{Spin}^c(X)\to\mathbb Z$ be the Morse theory invariant obtained from a Morse function $f\colon X\to S^1$ with no index $0$ or $3$ critical points. When $b_1(X)=1$, use the chamber determined by $r\,df$ for $r\gg0$. Hutchings–Lee conjecture. The Seiberg–Witten invariant agrees with the Morse theory invariant:

$$\operatorname{SW}_{X,o}=\pm I_3.$$

The conjecture connects Seiberg–Witten theory with a Morse-theoretic count of unions of closed orbits and flow lines. It is known when $f$ has no critical points, by applying Taubes’ theorem to a suitable symplectic structure on $X\times S^1$; Salamon proved an equivalent statement in that case. The general case remains open in the source.