Normalize on the unit sphere:
$$
F(v)=\frac{f(v)}{\|f(v)\|}:S^{2n-1}\longrightarrow S^{2n-1}.
$$
This is an $S^1$-equivariant map of the <Hopf fibration> covering $\bar f$, and it has degree one on every circle fiber. The bundle isomorphism in part c gives
$$
\bar f^*c_1(\gamma_{1,n})=c_1(\gamma_{1,n}).
$$
Since this first Chern class generates the cohomology ring of $\mathbb{CP}^{n-1}$, $\bar f$ acts identically on its cohomology. Naturality of the oriented <Gysin sequence of a sphere bundle> then gives $\deg F=1$.

The original homogeneous map is properly homotopic to the cone on $F$ by radial normalization; the norms $\|f(v)\|$ are bounded above and away from zero on the unit sphere, so this homotopy is proper. Thus $f$ has proper degree one. It fixes the generator of $H_c^{2n}(\mathbb C^n)$, and all other compactly supported groups vanish. Therefore $f^*:H_c^*(\mathbb C^n)\to H_c^*(\mathbb C^n)$ is the identity.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 3 / d / Solution

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The complex Hopf fibration is the principal circle bundle
$$
S^1\longrightarrow S^{2n-1}\longrightarrow\mathbb{CP}^{n-1}.
$$
It is the unit-sphere bundle of the tautological complex line bundle.


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