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Write $A=S^2\times S^2$ and let $i:A\to X$ be its inclusion after the attachment. The cohomology ring of a product of two spheres iswhere $a,b$ come from the first and second factors and $ab$ is the chosen orientation class. The diagonal pulls both $a$ and $b$ back to the same generator of $H^2(S^2;\mathbb{Z})$.

There is one new three-cell. In the cellular chain complex, its boundary has coordinates $(1,1)$ in the two-dimensional cells. Thus the relevant differential isand the original four-cell still has zero boundary. This gives $H_2(X;\mathbb{Z})=\mathbb{Z}$, $H_3(X;\mathbb{Z})=0$, $H_4(X;\mathbb{Z})=\mathbb{Z}$, and no other positive homology groups. Equivalently, the relative cohomology sequence of $(X,A)$ gives an injective restriction in degree two with image $\mathbb{Z}(a-b)$, and an isomorphism in degree four.

Choose $u\in H^2(X;\mathbb{Z})$ and $v\in H^4(X;\mathbb{Z})$ by $i^{\ast }u=a-b$, $i^{\ast }v=ab$. Naturality of the cup product givesSince restriction is injective in degree four, $u^2=-2v$. All further positive-degree cup products vanish by dimension. ThereforeChanging the sign of $v$ would instead give $u^2=2v$; the displayed sign uses the product orientation $ab$. The factor $2$ is the characteristic feature of a cup square after a diagonal sphere attachment.

Let $x,y\in H^n(S^n\times S^n;\mathbb{Z})$ be the pullbacks of the two orientation classes. The cohomology ring of a product of two spheres hasand $xy$ generates $H^{2n}\cong \mathbb{Z}$. WriteThe matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ lies in $G{L}_2(\mathbb{Z})$ because a homeomorphism induces a cohomology ring automorphism.

If $n$ is even, graded commutativity givesThus $ab=cd=0$. Invertibility forces $A$ to be diagonal or anti-diagonal, and its two nonzero entries must each be $\pm 1$. Hence there are exactly eight possible actions: the signed permutation matrices.

Evenness is necessary. For $n=1$, the product is the torus, and every $A\in G{L}_2(\mathbb{Z})$ is induced by an integral linear automorphism of the torus. For example, the infinitely many matrices $\left(\begin{array}{cc}1& m\\ 0& 1\end{array}\right)$ give distinct actions on $H^1$.