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

A singular cochain has compact support when it vanishes on every simplex whose image lies outside some compact set. Such cochains form a subcomplex because the faces of a simplex outside that compact set also lie outside it. Its cohomology is compactly supported cohomology $H_c^{\ast }(X)$.

If $f:X\to Y$ is proper and a cochain on $Y$ is supported in the compact set $K$, its pullback is supported in the compact set $f^{-1}(K)$. Pullback commutes with the coboundary, so it induces a well-defined map

Solved by gpt-5.6-sol high.

There is a natural descriptionClosed balls form a cofinal family of compact subsets of ${\mathbb{R}}^n$. Excision and radial deformation identify every sufficiently large-ball term, and the resulting stable group is the local group at the origin:It is $\mathbb{Z}$ in degree $n$ and zero otherwise.

Solved by gpt-5.6-sol high.

Properness implies $f(v)\ne 0$ for $v\ne 0$: otherwise the whole noncompact complex line $\mathbb{C}v$ would lie in $f^{-1}(0)$. Homogeneity therefore definesOn the fiber over $[v]$, defineThe identity $f(\lambda v)=\lambda f(v)$ makes this independent of the representative $v$, and it is a nonzero complex-linear map on every fiber. It is therefore the bundle isomorphism

Solved by gpt-5.6-sol high.

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.