Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 25I / b / Solution

A value $y$ is regular when $d{f}_x:T_{x}X\to T_{y}Y$ is surjective for every $x\in f^{-1}(y)$. The preimage theorem says $f^{-1}(y)$ is then a submanifold of dimension $\dim X-\dim Y$.

Solved by gpt-5.6-sol high.