Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 23H (Differential Geometry)

A point $x\in X$ is critical if $d{f}_x:T_{x}X\to T_{f(x)}Y$ is not surjective. A critical value is an image of a critical point; a regular value is a point $y\in Y$ for which every point of $f^{-1}(y)$ is regular, including vacuously values with empty preimage. Sard theorem says that for a smooth map of finite-dimensional manifolds, the critical values have measure zero in every coordinate chart of the target. In particular regular values are dense.

Assume $X$ is a compact smooth $n$-manifold without boundary and $Y$ a connected smooth $n$-manifold without boundary. For a regular value $y$, the inverse function theorem makes $f^{-1}(y)$ discrete, and compactness makes it finite. Define ${\mathrm{deg}}_{2}f=|f^{-1}(y)|(\mathrm{mod}2)$. No orientations are required.

The parity is independent of the regular value: join two regular values by a path and perturb the path in its interior to be transverse to $f$. Its inverse image is a compact one-dimensional manifold with boundary consisting of the two endpoint fibres. Every compact one-manifold has an even number of boundary points, so the two counts have the same parity. The same argument proves homotopy invariance. For noncompact $X$ one instead needs a proper map to retain compactness of the relevant inverse images.

If $f$ omitted a point of $Y$, that point would be a regular value with empty fibre and hence degree zero. Independence on the connected target gives the contradiction. Therefore a nonzero degree modulo two forces surjectivity.

Parametrize the torus by $F(u,v)=((2+\cos u)\cos v,(2+\cos u)\sin v,\sin u)$. Its outward Gauss map isThe derivative $N_u$ has unit length, while $N_v$ has length $|\cos u|$ and is perpendicular to $N_u$. Thus rank drops precisely at $u=\pi /2$ or $3\pi /2$. These are the two circles of height $z=1$ and $z=-1$ and radius two, and their images are the north and south poles. Therefore the critical values are $(0,0,\pm 1)$.

For any nonpolar target point, its third coordinate determines two values of $u$, one with positive and one with negative $\cos u$. For each, the longitude fixes a unique $v$ modulo $2\pi$, with a shift of $\pi$ when $\cos u$ changes sign. Thus every regular value has exactly two preimages, one on the outer and one on the inner side of the torus. ConsequentlyThis explicit two-sheet description justifies the count without relying on a sketch.