= Rank-one intersection form and the projective-plane cohomology ring
{title2=$H^*(X;\mathbb Z)\cong\mathbb Z[u]/(u^3)$}
For a simply connected closed four-manifold with second <Betti number> one, <Poincare duality> and the <universal coefficient theorem for cohomology> make $H_2$ torsion-free of rank one. The integral <intersection form> is unimodular, so its one-by-one matrix is $[1]$ or $[-1]$. Hence the square of a generator $u\in H^2$ generates $H^4$, giving the displayed <cohomology ring>. This is a ring isomorphism; it need not preserve a preassigned orientation class.
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