$\sp $ structure on a 4–manifold $W$ has a determinant line bundle whose first Chern class is a characteristic element in $H^{2}(W)$. For every characteristic element $c\in H^{2}(W)$ there exists a $\sp $ structure on $W$ with determinant line bundle whose first Chern class is equal to $c$ [@kirby Proposition 2.4.16]. If $H^{2}(W)$ contains no 2-torsion, then such a $\sp $ structure is unique. If $W$ is a 4–manifold with boundary ${Y_{m,n}}$, we will identify the restriction $\sp (W)\to \sp ({Y_{m,n}})$ with the corresponding restriction map on cohomology $H^{2}(W)\to H^{2}({Y_{m,n}})$ after making an appropriate choice of origins in the sets of $\sp $ structures.

Let $s_{1},s_{2}\in H_{2}({N_{m,n}})$ be the homology classes of the base spheres in the double plumbing. As we will see in Section \[App\], all the torsion $\sp $ structures on ${Y_{m,n}}$ extend to $\sp $ structures on ${N_{m,n}}$. For two integers $i$ and $j$, denote by $\mathfrak{t}_{i,j}$ the unique $\sp $ structure on ${N_{m,n}}$ for which $$\begin{aligned}
\langle c_{1}(\mathfrak{t}_{i,j}),s_{1}\rangle +m=2i\\
\langle c_{1}(\mathfrak{t}_{i,j}),s_{2}\rangle +n=2j\end{aligned}$$ and let $\mathfrak{s}_{i,j}=\mathfrak{t}_{i,j}|_{{Y_{m,n}}}$.

\[unique\] All the torsion $\sp $ structures on ${Y_{m,n}}$ are uniquely determined by

[1]{} & \_[i,j]{} 1im-1, 1jn-1\
& \_[0,j]{} 0jn-2\
& \_[i,0]{} 0im-2

with identifications $\mathfrak{s}_{m-2,0}=\mathfrak{s}_{0,n-2}$ and $\mathfrak{s}_{m-1,1}=\mathfrak{s}_{1,n-1}$.

Denote by 