of the generators and finish the proof of Theorem \[th1\].

If $b_{1}(Y)>0$ then there is an action of the exterior algebra $\Lambda ^{*}(H_{1}(Y;{\mathbb {Z}})/\operatorname{Tors})$ on the groups $HF^{\infty }(Y,\mathfrak{s})$ and $\widehat{HF}(Y,\mathfrak{s})$ for every torsion $\sp $ structure $\mathfrak{s}$ on $Y$ [@OS1 Proposition 4.17, Remark 4.20]. Let $\gamma $ be a simple closed curve on the Heegaard surface $\Sigma $ in general position with respect to the $\alpha $ curves and let $[\gamma ]$ be its induced homology class in $H_{1}(Y,{\mathbb {Z}})$. Then the action is given by $$A_{[\gamma ]}([\mathbf{x},i])=\sum _{\mathbf{y}}\sum _{\{\phi \in \pi _{2}(\mathbf{x},\mathbf{y})|\, \mu (\phi )=1\}}a(\gamma ,\phi )\cdot [\mathbf{y},i-n_{z}(\phi )]\;,$$ where $$a(\gamma ,\phi )=\# \{u\in \mathcal{M}(\phi )|\, u(1\times 0)\in (\gamma \times \textrm{Sym}^{g-1}(\Sigma ))\cap \mathbb{T}_{\alpha }\}\;.$$ The value $d_{b}(Y,\mathfrak{s})$ is the least grading of an element of $HF^{\infty }(Y,\mathfrak{s})$ that is in the kernel of the action of $\Lambda ^{*}(H_{1}(Y;{\mathbb {Z}})/\operatorname{Tors})$ and whose image in $HF^{+}(Y,\mathfrak{s})$ is nonzero.

In the case of $Y={Y_{m,n}}$ and $\mathfrak{s}$ any torsion $\sp $ structure on ${Y_{m,n}}$, the image of $HF^{+}({Y_{m,n}},\mathfrak{s})$ in $d\widehat{HF}({Y_{m,n}},\mathfrak{s})$ is generated by two elements of the form $\{x_{r},b_{j},f_{2}\}$ and $\{x_{r},b_{j},f_{1}\}$. As we have shown in the beginning of this subsection, there are two homotopy classes of disks $\phi _{1}$ and $\phi 