$s_{1}^{*},s_{2}^{*}$ the basis for $H^{2}({N_{m,n}})$ which is Hom-dual to the basis $s_{1},s_{2}$. Then the restriction map $H^{2}({N_{m,n}})\to H^{2}({Y_{m,n}})$ maps $s_{i}^{*}\mapsto PD(\mu _{i})$ for $i=1,2$. Remember the presentation of the first homology group $$H_{1}({Y_{m,n}})=\left \langle \mu _{1},\mu _{2},\mu _{3}|\, m\mu _{1}+2\mu _{2}=0,\,2\mu _{1}+n\mu _{2}=0\right \rangle \;.$$ Its torsion subgroup is uniquely determined by the classes $i\mu _{1}+j\mu _{2}$ for $1\leq i\leq m-1$ and $1\leq j\leq n-1$, $i\mu _{1}$ for $0\leq i\leq m-2$ and $j\mu _{2}$ for $0\leq j\leq n-2$ with identifications $(m-2)\mu _{1}=(n-2)\mu _{2}$ and $(m-1)\mu _{1}+\mu _{2}=\mu _{1}+(n-1)\mu _{2}$.

The set ${\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }$ is decomposed into equivalence classes according to the $\epsilon $-relation [@OS1 Definition 2.11]. Furthermore, by the map $\mathfrak{s}_{z}\colon {\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }\to H^{2}({Y_{m,n}})$, every equivalence class of generators together with a fixed basepoint $z$ determines a $\sp $ structure and its corresponding cohomology class. By [@OS1 Lemma 2.19] for any two generators $\mathbf{x},\mathbf{y}\in {\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }$ the following holds: $$\mathfrak{s}_{z}(\mathbf{y})-\mathfrak{s}_{z}(\mathbf{x})=PD[\epsilon (\mathbf{x},\mathbf{y})]\;.$$

From the Heegaard diagram we obtain

$$\begin{aligned}
& \epsilon (\{x_{i+1},-\},\{x_{i},-\})=\mu _{1} \textrm { for $1\leq i\leq m-2$}\\
& \epsilon (\{x_{m},-\},\{x_{m-1},-\})=\epsilon (\{b_{n},-\},\{b_{n-1},-\})=\mu _{1}+\mu _{2}\\
& \epsilon (\{x_{m},-\},\{x_{1},-\})=(m-1)\mu _{1}+\mu _{2}=-\mu _{1}-\mu _{2}\\
& \epsilon (\{b_{i+1},-\},\{b_{i},-\})=\mu _{2}\textrm { for $1\leq 