The Fundamental Group

The hypothesis of the Poincaré conjecture is that every loop in the manifold can be shrunk to a point. This chapter turns that sentence into mathematics. Loops are continuous maps of a circle; "shrunk" means continuously deformed, a homotopy; and the loops at a point, up to homotopy, form a group, the fundamental group $\pi_1(X)$, which Poincaré introduced in 1895 precisely to tell spaces apart. A space whose fundamental group is trivial is simply connected, and the conjecture says that the 3-sphere is the only closed simply connected three-manifold.

The first real computation is for the circle: $\pi_1(S^1) = \mathbb{Z}$, the integer being the winding number, how many times a loop goes around. The proof lifts loops from the circle to the line, which is the first instance of a covering space (7A.6 Covering Spaces). The winding number then proves two classical theorems almost for free: Brouwer's fixed point theorem in the disc and the fundamental theorem of algebra.

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

• define homotopies of paths and maps, and the fundamental group, and prove it is a group;
• explain change of basepoint, induced homomorphisms, homotopy equivalence and contractibility;
• prove $\pi_1(S^1) \cong \mathbb{Z}$ by path lifting, and compute winding numbers;
• prove that the circle is not a retract of the disc, and deduce Brouwer's fixed point theorem in dimension $2$ and the fundamental theorem of algebra;
• prove that $S^n$ is simply connected for $n \geq 2$.

4.1A robot on a leash

4.2Homotopy

Let $X$ and $Y$ be topological spaces and $I = [0, 1]$.

Homotopy is an equivalence relation (Exercise 4.9). A homotopy is a continuous family of maps $f_s = H(\cdot, s)$ deforming $f_0$ into $f_1$. In a convex subset of $\mathbb{R}^n$, any two paths with the same endpoints are path homotopic by the straight-line homotopy $H(t, s) = (1 - s)\gamma_0(t) + s\gamma_1(t)$. In the plane minus the origin, the two paths around the two sides of the origin are not, as this chapter will prove.

4.3The fundamental group

A loop at $x_0$ is a path $\gamma : I \to X$ with $\gamma(0) = \gamma(1) = x_0$. Two loops can be concatenated: $\gamma\cdot\delta$ runs through $\gamma$ at double speed and then through $\delta$,

The group is usually not abelian (7A.5 Computing π₁): loops around two different holes in a twice-punctured plane don't commute.

Basepoints. If $\alpha$ is a path from $x_0$ to $x_1$, then $[\gamma] \mapsto [\bar\alpha\cdot\gamma\cdot\alpha]$ is an isomorphism $\pi_1(X, x_0) \to \pi_1(X, x_1)$. So for path-connected $X$ the group is independent of the basepoint up to isomorphism, and one writes $\pi_1(X)$. The isomorphism depends on $\alpha$, up to conjugation.

Induced homomorphisms. A continuous $f : X \to Y$ with $f(x_0) = y_0$ induces $f_* : \pi_1(X, x_0) \to \pi_1(Y, y_0)$, $f_*[\gamma] = [f\circ\gamma]$. It is a homomorphism, $(g\circ f)_* = g_*\circ f_*$ and $\operatorname{id}_* = \operatorname{id}$ (functoriality). So homeomorphic spaces have isomorphic fundamental groups: $\pi_1$ is a topological invariant.

Homotopy invariance. More is true: maps that are homotopic (through basepoint-preserving homotopies) induce the same homomorphism. Two spaces $X$ and $Y$ are homotopy equivalent if there are maps $f : X \to Y$ and $g : Y \to X$ with $g\circ f \simeq \operatorname{id}_X$ and $f\circ g \simeq \operatorname{id}_Y$; then $\pi_1(X) \cong \pi_1(Y)$. For instance the punctured plane $\mathbb{R}^2\setminus\{0\}$ is homotopy equivalent to the circle (retract each ray onto its point of the unit circle), and so is the annulus and the solid torus $D^2\times S^1$. A space homotopy equivalent to a point is contractible: $\mathbb{R}^n$, any convex set, any star-shaped set. Contractible spaces have trivial fundamental group.

4.4The fundamental group of the circle

Let $p : \mathbb{R} \to S^1$, $p(s) = (\cos2\pi s, \sin2\pi s)$, which wraps the line around the circle like a helix projected onto its axis (Figure 4.2). Each small arc of the circle has a preimage that is a disjoint union of copies of that arc, one in each interval $(k - \frac12, k + \frac12)$; this is what makes $p$ a covering map (7A.6 Covering Spaces).

For a loop in the punctured plane, $\gamma(t) \in \mathbb{C}\setminus\{0\}$, the winding number around $0$ is the degree of $\gamma/|\gamma|$, and for piecewise smooth loops it equals $\frac{1}{2\pi i}\oint_\gamma\frac{dz}{z}$, the integral of 5A.2 Cauchy’s Theorem and Its Consequences and the argument principle of 5A.3 Residues and Fourier Transforms: the integral measures the total change of the angle, which is $2\pi$ times the change of the lift.

4.5Consequences

The fundamental theorem of algebra, again. If $p(z) = z^n + a_{n-1}z^{n-1} + \dots + a_0$ had no root, then for each $r \geq 0$ the loop $\gamma_r(t) = p(re^{2\pi it})/|p(re^{2\pi it})|$ in $S^1$ would be defined, and the family would be a homotopy from $\gamma_0$ (constant, winding number $0$) to $\gamma_R$ for large $R$, which winds $n$ times because $p(z) \approx z^n$ on the large circle (Exercise 4.12). So $n = 0$. This is the topological core of the complex-analytic proofs in 5A.2 Cauchy’s Theorem and Its Consequences and 5A.3 Residues and Fourier Transforms.

Products. $\pi_1(X\times Y) \cong \pi_1(X)\times\pi_1(Y)$, since a loop in a product is a pair of loops and a homotopy is a pair of homotopies. So $\pi_1(T^2) = \pi_1(S^1\times S^1) \cong \mathbb{Z}^2$, and the torus is not simply connected: the 2-dimensional Poincaré theorem's first test case (7A.3 Manifolds and Surfaces).

In particular $S^3$ is simply connected: it satisfies the hypothesis of the Poincaré conjecture, which says it is the only closed three-manifold that does (7A.9 The Poincaré Conjecture, Precisely).

4.6History

Henri Poincaré defined the fundamental group in Analysis Situs (1895), as the group of loops up to deformation, and used it to distinguish three-manifolds with the same homology (7A.8 Homology in Brief). Luitzen Brouwer proved his fixed point theorem in 1910–12 (in dimension 3 it had been proved by Piers Bohl in 1904). The winding number goes back to Gauss and Cauchy, as the change of the argument along a closed curve. Camille Jordan studied closed curves and their deformations in the 1860s.

4.7Exercises

Solution

(a) $2$: the modulus stays positive and the argument increases by $4\pi$. (b) $0$: the loop stays in the half-plane $\operatorname{Re}z > 0$, where the argument has a continuous branch. (c) $1$. (d) $1 + 1 = 2$, since $e^{2\pi it} - \frac12$ winds once around $0$.

Solution

(a) $|p(z) - z^n| \leq \sum|a_k|R^k < R^n$ for $|z| = R$ under the condition, so $(1 - s)p(z) + sz^n = z^n + (1 - s)(p(z) - z^n) \neq 0$. (b) If $p$ has no zeros, $r \mapsto p(re^{2\pi it})$ for $0 \leq r \leq R$ is a homotopy in $\mathbb{C}\setminus\{0\}$ from a constant loop to a loop of winding number $n$, so $n = 0$, a contradiction.

Solution

$S^3$: simply connected (Theorem 4.8). $T^3$: $\pi_1 = \mathbb{Z}^3$. $S^2\times S^1$: $\pi_1 = \pi_1(S^2)\times\pi_1(S^1) = \mathbb{Z}$. $L(p, q)$: $\mathbb{Z}/p \neq 0$ for $p \geq 2$. Only $S^3$ is simply connected, and it is the sphere, so none is a counterexample.