= Integral homology of a circle times a real projective plane
{title2=$H_0=\mathbb Z,\ H_1=\mathbb Z\oplus\mathbb Z/2,\ H_2=\mathbb Z/2$}
Cover the circle by two arcs whose intersection has two components and take the product with the projective plane. The <Mayer–Vietoris sequence> has maps $(a,b)\mapsto(a+b,-a-b)$ on the two copies of the homology of the plane. Its kernels and cokernels, together with $H_1(\mathbb{RP}^2;\mathbb Z)=\mathbb Z/2$, give the displayed groups; all higher groups vanish. The product is a closed nonorientable three-manifold, but is homotopy equivalent to a noncompact oriented four-manifold.
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