Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 114 / 1 / 1 / 3

The standard cellular chain complex of the Klein bottle givesThe universal coefficient theorem for cohomology therefore yields $H^0=\mathbb{Z}$, $H^1=\mathbb{Z}$, and $H^2=\mathbb{Z}/2$. The degree-one generator is pulled back from the base circle in the circle-bundle description of $K$, so its square is zero. Thus every positive-degree product vanishes. The same calculation for ${\mathbb{RP}}^2\vee S^1$ gives the integral cohomology ring of the wedge of the real projective plane and a circle, and hence

There is no map $Y\to K$ inducing this isomorphism. The Klein bottle is an aspherical space, and its fundamental group is torsion-free. Any map ${\mathbb{RP}}^2\to K$ therefore induces the trivial homomorphism $\mathbb{Z}/2\to {\pi }_1(K)$ and is null-homotopic. Its pullback on $H^2$ is zero, while the circle summand has no degree-two cohomology. Consequently every map $Y\to K$ is zero on $H^2(K;\mathbb{Z})$ and cannot be a cohomology isomorphism.