Smooth Topology

The fundamental group detects loops. To detect higher-dimensional phenomena, such as a sphere wrapping around another sphere, or a vector field that cannot avoid vanishing, the cleanest tool is degree: for a smooth map between closed manifolds of the same dimension, count the preimages of a typical point, with signs. The count doesn't depend on the point or on deforming the map, and it proves Brouwer's fixed point theorem in every dimension, the hairy ball theorem (you can't comb a hairy ball flat), and the Poincaré–Hopf theorem: the zeros of a vector field on a closed manifold, counted with their indices, add up to the Euler characteristic.

What makes "a typical point" precise is Sard's theorem: the values at which a smooth map fails to be regular form a set of measure zero. This chapter works, following Milnor, with smooth submanifolds of Euclidean space, postponing abstract smooth manifolds to Book 8A.

By the end of this chapter you will be able to:

• define regular points and values, and use the preimage theorem;
• state Sard's theorem and use it to prove Brouwer's fixed point theorem in every dimension;
• define the degree of a smooth map between closed oriented manifolds of the same dimension, and compute it in examples;
• prove the hairy ball theorem for even-dimensional spheres;
• compute indices of zeros of vector fields and apply the Poincaré–Hopf theorem; state the Borsuk–Ulam theorem.

7.1There is always a calm point

7.2Smooth maps and regular values

Let $M \subseteq \mathbb{R}^k$ be a smooth submanifold of dimension $m$: near each point it is the graph of a smooth function, or equivalently the image of a smooth parametrisation with injective derivative (2B.9 The Inverse and Implicit Function Theorems). Its tangent space $T_xM \subseteq \mathbb{R}^k$ is the image of the derivative of a local parametrisation, an $m$-dimensional linear subspace. A map $f : M \to N$ between submanifolds is smooth if it extends locally to smooth maps of the ambient spaces, and its derivative $df_x : T_xM \to T_{f(x)}N$ is the restriction of the ambient derivative (2B.8 Calculus in Several Variables).

A point $x \in M$ is a regular point of $f$ if $df_x$ is surjective, and a critical point otherwise. A value $y \in N$ is a regular value if every point of $f^{-1}(y)$ is regular (in particular, if $f^{-1}(y)$ is empty), and a critical value otherwise (Figure 7.1).

The proof (Milnor, section 3) is an induction on dimension using Fubini's theorem (3A.5 Product Measures and Change of Variables) and Taylor estimates; for $m < n$ it is easy, since then every point is critical and the image of a smooth map from a lower-dimensional space has measure zero. Sard's theorem is used constantly in the form: regular values are dense, so one can always perturb a target point slightly to make it regular.

7.3Brouwer's fixed point theorem in every dimension

7.4Degree

Let $M$ and $N$ be closed (compact, without boundary) smooth manifolds of the same dimension, with $N$ connected, and $f : M \to N$ smooth. If $y$ is a regular value, $f^{-1}(y)$ is finite.

Degree mod 2. The number $\#f^{-1}(y)$ modulo $2$ is the same for all regular values $y$, and the same for homotopic maps (Milnor, section 4). It is the mod 2 degree $\deg_2f$. The identity has $\deg_2 = 1$ and a constant map $\deg_2 = 0$, so the identity of a closed manifold is not homotopic to a constant: a closed manifold is not contractible.

Oriented degree. If $M$ and $N$ are oriented, each regular preimage $x$ of $y$ gets a sign: $+1$ if $df_x$ preserves orientation (positive determinant in oriented charts), $-1$ if it reverses it. The degree is

an integer, independent of the regular value $y$ and unchanged by homotopy (Milnor, section 5). For maps of the circle it is the winding number of 7A.4 The Fundamental Group: $z \mapsto z^n$ has degree $n$. In Figure 7.1, a regular value of $x^3 - 3x$ between $-2$ and $2$ has three preimages with signs $+, -, +$: degree $1$, as for values with one preimage.

Properties (Exercise 7.8): $\deg(g\circ f) = \deg g\cdot\deg f$; a diffeomorphism has degree $\pm1$; the antipodal map $x \mapsto -x$ of $S^n$ has degree $(-1)^{n+1}$, since it is the composite of $n + 1$ reflections, each of degree $-1$. Hopf's degree theorem (1926) says that for maps $S^n \to S^n$ the degree is a complete invariant: two maps are homotopic if and only if they have the same degree.

7.5Vector fields

Index of a zero. Let $v$ be a vector field on an open set of $\mathbb{R}^n$ with an isolated zero at $z$. On a small sphere around $z$, the map $x \mapsto \frac{v(x)}{|v(x)|}$ goes from a sphere to a sphere; its degree is the index of $v$ at $z$. In the plane it is the number of times $v$ turns as you go once around $z$ (Figure 7.2):

• a source $v = (x, y)$, a sink $v = (-x, -y)$, a centre $v = (-y, x)$ and a spiral all have index $+1$;
• a saddle $v = (x, -y)$ has index $-1$;
• $v = \bar z^2$, in complex notation, has index $-2$ (a "monkey saddle").

On a manifold, the index is computed in a chart and doesn't depend on the chart.

Milnor (section 6) proves this in two steps: the sum is the same for all vector fields (it equals the degree of a "Gauss map" of the boundary of a neighbourhood of $M$), and one example computes it. On a surface, take a triangulation and build a field with a source at each vertex, a saddle at the midpoint of each edge and a sink in each face: the sum is $V - E + F = \chi$ (7A.3 Manifolds and Surfaces). On $S^2$ a field flowing from the north pole to the south pole has a source and a sink, total $2$. On the torus, the field along the circles of one family has no zeros at all, total $0$. On a surface of genus $g \geq 2$, $\chi < 0$, so every vector field has zeros whose indices add up to a negative number: there must be saddles.

Transversality. The preimage theorem generalises: if $f : M \to N$ is transverse to a submanifold $Z \subseteq N$ (at each point of $f^{-1}(Z)$, the image of $df$ together with the tangent space of $Z$ spans the tangent space of $N$), then $f^{-1}(Z)$ is a submanifold of codimension equal to that of $Z$; and transversality can always be achieved by a small perturbation. This is the tool by which surfaces in three-manifolds are put in general position, used throughout three-manifold topology (10A.3 The Prime Decomposition).

For $n = 1$ this is the equator theorem of 2A.9 Continuous Functions. For $n = 2$: temperature and pressure (continuous functions on the sphere) agree at some antipodal pair. The proof shows that otherwise $g(x) = \frac{f(x) - f(-x)}{|f(x) - f(-x)|}$ would be an odd map $S^n \to S^{n-1}$, and odd maps between spheres have odd mod-2 degree in a suitable sense, which is impossible for maps from a higher-dimensional sphere (Matoušek, Using the Borsuk–Ulam Theorem, gives several proofs).

7.6History

Poincaré proved the index theorem for vector fields on surfaces in 1885, and the hairy ball theorem for $S^2$; Brouwer proved it for all even-dimensional spheres in 1912, the same year he introduced the degree of a map and proved his fixed point theorem in all dimensions. Heinz Hopf extended the index theorem to all dimensions and proved his degree theorem in 1926. Anthony Morse (1939) and Arthur Sard (1942) proved the measure-zero theorem for critical values; Karol Borsuk proved the Borsuk–Ulam theorem in 1933, answering a question of Stanisław Ulam. René Thom's transversality theorem dates from 1954. Milnor's Topology from the Differentiable Viewpoint (1965) is based on lectures of 1963.

7.7Exercises

Solution

(a) Critical point $0$, critical value $0$; regular preimages are circles. (b) The poles; critical values $\pm1$; regular preimages are circles of latitude. (c) Four critical points (top, bottom, and the top and bottom of the hole), critical values $3, 1, -1, -3$; regular preimages are one circle (near the top or bottom) or two circles (in between).

Solution

(a) In complex form $z^2$: index $2$. (b) $\bar z^2$: index $-2$. (c) Homotopic through fields with an isolated zero at $0$ to $(x, y)$: index $1$. (d) At $(0, 0)$, the linearisation $(y, -x)$ is a centre: $+1$. At $(\pi, 0)$, the linearisation $(y, x)$ is a saddle: $-1$.

Solution

(a) The constant field $(1, 0)$ on $\mathbb{R}^2$ is invariant under translations and descends; no zeros, sum $0 = \chi(T^2)$. (b) The sum must be $\chi(S^2) = 2$. Pushing the constant field $(1, 0)$ on $\mathbb{R}^2$ to $S^2$ by inverse stereographic projection gives a field on the sphere minus the north pole that extends continuously by $0$ there, with index $2$.

Solution

(a) $u^2 = -1$, so $\cos\theta + \sin\theta\,u$ multiplies like $e^{i\theta}$ in the plane spanned by $1$ and $u$. (b) We need $\sin k\theta\,u = \sin\alpha\,u_0$ and $\cos k\theta = \cos\alpha$. Either $u = u_0$ and $k\theta \equiv \alpha$, or $u = -u_0$ and $k\theta \equiv -\alpha \pmod{2\pi}$. As $\theta$ runs over $(0, \pi)$, $k\theta$ runs over $(0, k\pi)$, and the admissible values are $\alpha, 2\pi - \alpha, 2\pi + \alpha, 4\pi - \alpha, \dots$: exactly $k$ of them lie in $(0, k\pi)$, each giving one preimage. (c) $k$ preimages, each with sign $+1$.