The sutured ECH–Heegaard Floer correspondence

Let $(M,\Gamma,\xi=\ker\alpha)$ be a sutured contact manifold, let $A\in H_1(M;\mathbb{Z})$, and let $\mathfrak{s}_\xi$ be the relative Spin$^c$ structure determined by $\xi$. Write $PD(A)$ for the Poincare dual of $A$. The sutured ECH–Heegaard Floer correspondence. There is an isomorphism

$$ECH(M,\Gamma,\xi,A)\simeq HF(-M,-\Gamma,\mathfrak{s}_\xi+PD(A)).$$

The authors expect to prove this extension of the closed-manifold correspondence between embedded contact homology and Heegaard Floer homology; it is presented as work in progress.