Integral homology of a circle times a real projective plane ($H_0=\mathbb{Z},H_1=\mathbb{Z}\oplus \mathbb{Z}/2,H_2=\mathbb{Z}/2$)

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.