560 at 368 552 at 372 460 at 430 470 at 420 520 at 441 520 at 294 335 at 290 628 at 320 295 at 320 596 at 253 318 at 252 606 at 256 472 at 270 443

![The Heegaard diagram of ${Y_{m,n}}$](zacetek "fig:") \[fig:heeg\]

based on which we can determine the torsion $\sp $ structures. A generator $\mathbf{x}\in {\mathbb {T}}_{\alpha }\cap {\mathbb {T}}_{\beta }$ belongs to a torsion $\sp $ structure if and only if $\left \langle c_{1}(\mathfrak{s}_{z}(\mathbf{x})),\mathcal {P}\right \rangle =0$, which happens exactly when $\sum _{x_{i}\in \mathbf{x}}\overline{n}_{x_{i}}(\mathcal{P})=1$. Thus, the torsion $\sp $ structures of our chain complex contain the following generators: $$\{\{x_{i},b_{j},f_{k}\},\{x_{i},c_{k},e_{r}\},\{y_{k},a_{k},f_{r}\},\{y_{1},c_{k},d_{r}\},\{u_{k},a_{2},e_{r}\}\}$$ for $k,r\in \{1,2\}$ and $i=1,\ldots ,m$ and $j=1,\ldots n$. There are $2(mn+2m+6)$ generators of the torsion $\sp $ structures on ${Y_{m,n}}$.

[Notation for $\sp $ structures]{} \[notspin\] There is a one-to-one correspondence $$\delta ^{\tau }\colon \sp ({Y_{m,n}})\to H^{2}({Y_{m,n}})$$ [@OS1 Subsection 2.6]. Thus, we may identify the $\sp $ structures on ${Y_{m,n}}$ with cohomology classes, or even with their Poincaré dual homology classes in $H_{1}({Y_{m,n}})$. Then the natural map $c_{1}\colon \sp ({Y_{m,n}})\to H^{2}({Y_{m,n}})$ assigning to any $\sp $ structure its first Chern class is connected to $\delta ^{\tau }$ by $c_{1}(\mathfrak{s})=2\delta ^{\tau }(\mathfrak{s})$.

Similarly, a 