Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 313 / 3 / Solution

For a smooth map $\varphi :M_1\to M_2$ between connected oriented closed manifolds of equal dimension and a volume form $\omega$ on $M_2$, the topological degree is defined byThe standard area form on the unit sphere isSince ${\int }_{S^2}\omega =\mathrm{vol}(S^2)$, this gives