= Gysin ring splitting for an odd-dimensional sphere bundle
{c}
{title2=$H^*(S(E);\mathbb Z)\cong H^*(B;\mathbb Z)\otimes\Lambda_{\mathbb Z}(a)$}
Let $E$ be an oriented real <vector bundle> of positive even rank $r$, with zero <Euler class>, and suppose the integral <cohomology> of $B$ is torsion-free. The <Gysin sequence> gives $0\to H^j(B)\to H^j(S(E))\to H^{j-r+1}(B)\to0$. Choose an odd-degree class $a\in H^{r-1}(S(E))$ with $\pi_!a=1$. The <projection formula for sphere bundle integration> identifies the cohomology additively with $\pi^*H^*(B)\oplus\pi^*H^*(B)a$. It is torsion-free, so <graded commutativity of the cup product> gives $a^2=0$ from $2a^2=0$. These facts establish the displayed ring isomorphism, not just a decomposition of groups.
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