Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 25F / Solution

A point $y$ is a regular value of $f$ when every derivative $D_{x}f$ for $x$ in $f^{-1}(y)$ is surjective. Sard theorem says that the critical values of a smooth map have measure zero. The stack-of-records theorem says that the inverse image of a regular value is a smooth submanifold of codimension $\dim Y$.

Maps $f$ and $g$ are smoothly homotopic when a smooth $F:X\times [0,1]\to Y$ has endpoint maps $f$ and $g$. Perturb such a homotopy relative to its ends so that it is transverse to $y$. Then $F^{-1}(y)$ is a compact one-dimensional manifold whose boundary is $f^{-1}(y)\bigsqcup g^{-1}(y)$. Every compact one-dimensional manifold has an even number of boundary points, proving the parity formula.

The hypothesis on $U$ says that the degree modulo two of $h$ is one. Suppose no $h(x)$ and $h(-x)$ were antipodal. Thenwould be defined, even, and homotopic to $h$ by normalized straight-line interpolation. For a regular value, the fibres of the even map $k$ occur in antipodal pairs, so its degree modulo two is zero. Homotopy invariance gives a contradiction. Hence some antipodal pair is mapped to an antipodal pair.

Solved by gpt-5.6-sol high.