Heegaard–Floer homology of $\partial {N_{m,n}}$ [@OS2] and use an obstruction theorem [@OS4 Theorem 9.15] to see when ${N_{m,n}}$ can indeed be realized inside a given 4–manifold $X$ with $b_{2}^{+}(X)=2$.

Denote by ${Y_{m,n}}$ the boundary of ${N_{m,n}}$. For two integers $i$ and $j$, denote by $\mathfrak{t}_{i,j}$ the unique $\sp $ structure on ${N_{m,n}}$ for which $$\begin{aligned}
\label{eqsp}
\langle c_{1}(\mathfrak{t}_{i,j}),s_{1}\rangle +m=2i\end{aligned}$$ $$\begin{aligned}
\label{eqsp1}
\langle c_{1}(\mathfrak{t}_{i,j}),s_{2}\rangle +n=2j\;,\end{aligned}$$ where $s_{1},s_{2}\in H_{2}({N_{m,n}})$ are the homology classes of the base spheres in the double plumbing. Let $\mathfrak{s}_{i,j}=\mathfrak{t}_{i,j}|_{{Y_{m,n}}}$.

Throughout the paper, we denote by ${\mathbb{F}}$ the field ${\mathbb {Z}}_{2}$ and by $\mathcal{T}^{+}$ the quotient module ${\mathbb{F}}[U,U^{-1}]/U{\mathbb{F}}[U]$. Our main result is the following:

\[th1\] Let $Y={Y_{m,n}}$ be the boundary of a double plumbing of two disk bundles over spheres with Euler numbers $m$ and $n$, where $m,n\geq 4$. The Heegaard–Floer homology $HF^{+}(Y,\mathfrak{s})$ with ${\mathbb{F}}$ coefficients is given by $$\begin{aligned}
& HF^{+}(Y,\mathfrak{s}_{m-1,1})={\mathcal{T}^{+}}_{(d(m-1,1))}\oplus {\mathcal{T}^{+}}_{(d(m-1,1)-1)}\oplus {\mathbb{F}}_{(d(m-1,1)-1)}\\
& HF^{+}(Y,\mathfrak{s}_{i,j})={\mathcal{T}^{+}}_{(d(i,j))}\oplus {\mathcal{T}^{+}}_{(d(i,j)-1)}\\
& HF^{+}(Y,\mathfrak{s}_{0,k})={\mathcal{T}^{+}}_{(d_{1}(m,k+1))}\oplus {\mathcal{T}^{+}}_{(d_{1}(m,k+1)-1)}\\
& HF^{+}(Y,\mathfrak{s}_{l,0})={\mathcal{T}^{+}}_{(d_{1}(n,l+1))}\oplus {\mathcal{T}^{+}}_{(d_{1}(n,l+1)-1)}\end{aligned}$$ for $1\leq i\leq m-1$, $1\leq j\leq n-1$, $0\leq k\leq n-2$, $0\leq l\leq m-2$ and $(i,j)\notin \{(m-1,1),(1,n-1)\}$, where the subscripts denote the absolute gradings of the bottom elements and

[1]{} & d(i,j)=,\
& d\_[1]{}(t,i)=.

The action of the exterior algebra $\Lambda ^{*}(H_{1}(Y,{\mathbb {Z}})/\operatorname{Tors})$ on $HF^{+}(Y,\mathfrak{s})$ maps the 