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

Normalize on the unit sphere: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 givesSince 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 $\mathrm{deg}F=1$.

The original homogeneous map is properly homotopic to the cone on $F$ by radial normalization; the norms $\Vert f(v)\Vert$ 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 $f^{\ast }:H_c^{\ast }({\mathbb{C}}^n)\to H_c^{\ast }({\mathbb{C}}^n)$ is the identity.

Solved by gpt-5.6-sol high.