first copy of ${\mathcal{T}^{+}}$ isomorphically to the second copy in each torsion $\sp $ structure $\mathfrak{s}$, dropping the absolute grading of the generator by one.

We use this result to determine whether the double plumbing ${N_{m,n}}$ can occur inside some 4–manifolds $X$ with $H_{2}^{+}(X)=2$. If it can, the complement $W=X\backslash \operatorname{Int}({N_{m,n}})$ is a negative semi-definite 4–manifold and [@OS4 Theorem 9.15] gives an obstruction depending on the correction terms of ${Y_{m,n}}=\partial W$.

In the manifold $X={\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2}$, every homology class $(x_{1},x_{2})\in H_{2}(X)$ with $(x_{1},x_{2})\in \{0,\pm 1,\pm 2\}^{2}\backslash \{(0,0)\}$ has a smooth representative of genus 0. Choosing two such representatives with algebraic intersection number 2, we check if they can have only 2 geometric intersections.

Next we consider the manifold $X=S^{2}\times S^{2}\# S^{2}\times S^{2}$. According to Wall [@WALL], every primitive homology class $(x_{1},x_{2},x_{3},x_{4})\in H_{2}(X)$ can be represented by an embedded sphere. We choose two such representatives with algebraic intersection number 2 and determine when the number of their geometric intersections has to be strictly greater than 2, thus not allowing the chosen homology classes to be represented by a double plumbing. We obtain the following estimates.

\[app\]\
a) Any two spheres representing classes $(2,2),(2,-1)\in H_{2}({\mathbb {C}}P^{2}\# {\mathbb {C}}P^{2})$ intersect with 