Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 2

For $n>0$,The CW structure with one cell in every even dimension gives the additive groups, and the generator $x=c_1({\gamma }^{\ast })$ restricts compatibly along ${\mathbb{CP}}^{n-1}\subset {\mathbb{CP}}^n$; its powers generate each even group.

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The integral Künneth theorem gives a natural short exact sequence containing tensor and Tor terms. Here all cohomology groups are free abelian, so the Tor terms vanish and the external cup product is a ring isomorphism. Thus

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With the indicated points as basepoints, $X$ is the smash product ${\mathbb{CP}}^2\wedge {\mathbb{CP}}^2$. The quotient map identifies its positive-degree cohomology with the relative cohomology of the product modulo the wedge, which under the Künneth isomorphism is the ideal generated by $xy$. Thusin degrees $4,6,6,8$. The only nonzero product of positive-degree basis elements isTogether with the unit in degree zero, this determines the cohomology ring.

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Suppose such maps existed, and let $u$ generate $H^4(S^4;\mathbb{Z})$. Since $f^{\ast }{g}^{\ast }=1$, write $g^{\ast }u=ma$ and $f^{\ast }a=nu$; then $mn=1$, so $m=\pm 1$. Naturality of the cup product givescontradicting the nonzero degree-eight class computed in part c.

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