is that Step $(4)$ fails when the fiber surface $S$ has genus one; the relevant summand of $\underline{HF}^+(-Y_0(K),{{\mathbb F}}[C_n])$ is infinite dimensional in this case. To account for this, we alter our coefficients through the discussion, replacing ${{\mathbb F}}$ with a certain Novikov field $\Lambda$, which is the coefficient module for Floer homology perturbed by a $2$-form. Using Floer homology perturbed by a $2$-form Poincar[é]{} dual to a meridian of the binding, the case of genus one proceeds exactly as above.

Essentials of the Proof {#subsec:proof}
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With the general outline of our proof in place, we turn to the details of the argument. Suppose $W: Z_1\to Z_2$ is a compact oriented cobordism between closed connected oriented 3-manifolds $Z_1$ and $Z_2$. For each [[ ]{}]{}structure ${\mathfrak{s}}$ on $W$ there is an induced homomorphism between the Heegaard Floer homology groups of $Z_1$ and $Z_2$. More generally, if $\mathbb{A}$ is a module for the group algebra ${{\mathbb F}}[H^1(Z_1; {\mathbb{Z}})]$, there is a homomorphism in Floer homology with twisted coefficients, $$F_{W,{\mathfrak{s}}}^\mathbb{A}: {\underline}{HF}^+(Z_1, {\mathfrak{s}}_1; \mathbb{A}) \to {\underline}{HF}^+(Z_2, {\mathfrak{s}}_2; \mathbb{A}\otimes_{{{\mathbb F}}[H^1(Z_1)]} \mathbb{K}(W)),$$ where ${\mathfrak{s}}_i = {\mathfrak{s}}|_{Z_i}$, and $\mathbb{K}(W) = {{\mathbb F}}[{\mbox{Im}}(H^1(\partial W)\to H^2(W, \partial W))]$ (c.f. [@HolDiskFour Theorem 3.8]). In the case that $W$ consists 