Covering Spaces

The proof that $\pi_1(S^1) = \mathbb{Z}$ unwrapped the circle onto the line (7A.4 The Fundamental Group). The line is a covering space of the circle: locally it looks exactly like the circle, but loops that go around the circle unroll into paths that don't close up. This chapter develops covering spaces in general. They turn questions about the fundamental group into questions about symmetry: a simply connected space with a group $\Gamma$ acting on it freely and discontinuously covers its quotient, and the quotient has fundamental group $\Gamma$.

This is how the most important examples of three-manifolds arise. The 3-sphere is the group of unit quaternions, and it double covers the group of rotations $SO(3)$, so $\pi_1(SO(3)) = \mathbb{Z}/2$: a fact you can demonstrate with a belt. Quotients of the 3-sphere by finite groups of rotations are the spherical space forms, including the lens spaces and the Poincaré homology sphere, whose fundamental group has 120 elements but no abelian quotient. These are exactly the manifolds that the Ricci flow leaves behind when it shrinks a three-manifold with finite fundamental group to a round point (12C.1 Reading Off the Topology).

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

• define covering maps and use the path lifting, homotopy lifting and lifting criteria;
• describe the classification of coverings by subgroups of $\pi_1$, the universal cover and deck transformations;
• prove that a free, properly discontinuous action of $\Gamma$ on a simply connected manifold has quotient with fundamental group $\Gamma$;
• compute $\pi_1$ of $\mathbb{R}P^n$, tori and lens spaces, and construct the double cover $SU(2) = S^3 \to SO(3)$;
• describe the spherical space forms and the Poincaré homology sphere.

6.1Unwrapping phase

6.2Covering maps

Examples:

• $p : \mathbb{R} \to S^1$, $p(s) = e^{2\pi is}$, with infinitely many sheets (7A.4 The Fundamental Group).
• $p_n : S^1 \to S^1$, $z \mapsto z^n$, with $n$ sheets (Figure 6.2).
• $\mathbb{R}^n \to T^n = \mathbb{R}^n/\mathbb{Z}^n$.
• $S^n \to \mathbb{R}P^n$, identifying antipodal points, with $2$ sheets (Figure 6.2).
• $S^3 \to L(p, q)$, with $p$ sheets (7A.1 Topological Spaces and Quotients, last exercise).

The lifting lemma of 7A.4 The Fundamental Group holds for every covering map, with the same proof: paths and homotopies lift uniquely once a starting point is chosen. Three consequences follow (Exercise 6.4).

1. $p_* : \pi_1(\tilde X, \tilde x_0) \to \pi_1(X, x_0)$ is injective: a loop downstairs whose lift is a loop and is null-homotopic downstairs lifts to a null-homotopy upstairs.
2. A loop in $X$ lifts to a loop in $\tilde X$ exactly when its class lies in the subgroup $p_*\pi_1(\tilde X)$; the number of sheets equals the index of this subgroup.
3. (Lifting criterion) For a connected, locally path-connected $Y$, a map $f : Y \to X$ lifts to $\tilde X$ if and only if $f_*\pi_1(Y) \subseteq p_*\pi_1(\tilde X)$. In particular, every map from a simply connected space lifts.

6.3Universal covers and deck transformations

A covering $\tilde X \to X$ with $\tilde X$ simply connected is a universal cover. It exists for every connected, locally path-connected and "semi-locally simply connected" space, which includes every connected manifold, and it is unique up to isomorphism. It covers every other connected covering of $X$, and the connected coverings of $X$ are classified by subgroups of $\pi_1(X)$: each subgroup $H$ corresponds to the covering with $p_*\pi_1 = H$, and conjugate subgroups give isomorphic coverings (Hatcher, section 1.3).

A deck transformation of a covering is a homeomorphism $\phi : \tilde X \to \tilde X$ with $p\circ\phi = p$: it permutes the sheets over each point. For the universal cover, the deck transformations form a group isomorphic to $\pi_1(X)$, acting simply transitively on each fibre. For $\mathbb{R} \to S^1$ they are the translations $s \mapsto s + n$, and $\pi_1(S^1) = \mathbb{Z}$ again.

Turned around, this produces manifolds from group actions.

Such an action is free (no non-identity element has a fixed point). Conversely, a free action of a finite group on a Hausdorff space is a covering space action (Exercise 6.5); so is the action of $\mathbb{Z}^n$ on $\mathbb{R}^n$ by translations.

Examples.

• $\pi_1(T^n) = \pi_1(\mathbb{R}^n/\mathbb{Z}^n) = \mathbb{Z}^n$.
• $\pi_1(\mathbb{R}P^n) = \pi_1(S^n/\{\pm1\}) = \mathbb{Z}/2$ for $n \geq 2$, since $S^n$ is simply connected (7A.4 The Fundamental Group). The same answer as 7A.5 Computing π₁, by a different route.
• $\pi_1(L(p, q)) = \mathbb{Z}/p$, from the free action of $\mathbb{Z}/p$ on $S^3$ (7A.1 Topological Spaces and Quotients).
• A simply connected manifold has no non-trivial connected coverings: a covering of it is a covering by a space whose $p_*\pi_1$ has index equal to the number of sheets, and the index of a subgroup of the trivial group is $1$.

6.4The 3-sphere and the rotation group

Let $\mathbb{H}$ be the quaternions, $q = a + bi + cj + dk$ with $i^2 = j^2 = k^2 = ijk = -1$, and $|q|^2 = a^2 + b^2 + c^2 + d^2$. The unit quaternions form the 3-sphere $S^3 \subset \mathbb{H} = \mathbb{R}^4$, and since $|qr| = |q||r|$ they form a group, isomorphic to $SU(2)$ (Exercise 6.6). Identify $\mathbb{R}^3$ with the pure quaternions $bi + cj + dk$. For a unit quaternion $q$, the map

preserves pure quaternions and their lengths, and is a rotation of $\mathbb{R}^3$: writing $q = \cos\frac\theta2 + \sin\frac\theta2\,u$ with $u$ a unit pure quaternion, $R_q$ is the rotation by angle $\theta$ about the axis $u$. Every rotation arises this way, and $R_q = R_{q'}$ exactly when $q' = \pm q$. So

is a 2-sheeted covering and a group homomorphism with kernel $\{\pm1\}$. Hence $SO(3) \cong S^3/\{\pm1\} = \mathbb{R}P^3$, and

A loop of rotations that turns once around an axis, $\theta$ from $0$ to $2\pi$, lifts to the path $q = \cos\frac\theta2 + \sin\frac\theta2u$ from $1$ to $-1$, which is not a loop: the loop is not contractible. Going around twice lifts to a loop in the simply connected $S^3$, and is contractible.

6.5Spherical space forms and the Poincaré homology sphere

A finite subgroup $\Gamma$ of $SO(4)$ that acts freely on $S^3$ gives a closed three-manifold $S^3/\Gamma$ with fundamental group $\Gamma$. Since $\Gamma$ acts by isometries of the round metric, $S^3/\Gamma$ inherits a metric of constant curvature $+1$: it is a spherical space form. Examples:

• lens spaces $L(p, q)$, $\Gamma = \mathbb{Z}/p$ acting by $(z, w) \mapsto (\zeta z, \zeta^qw)$ with $\gcd(p, q) = 1$;
• $\mathbb{R}P^3 = L(2, 1)$;
• quotients by finite subgroups $\Gamma \subset SU(2)$ acting by left multiplication on $S^3 = SU(2)$, which is always free (a group element other than $1$ fixes no point under left multiplication). The finite subgroups of $SU(2)$ are the preimages of the finite rotation groups of $\mathbb{R}^3$: cyclic, binary dihedral, and the binary tetrahedral, octahedral and icosahedral groups, of orders $24$, $48$ and $120$.

The Poincaré homology sphere is $S^3/I^*$, where $I^*$ is the binary icosahedral group: the $120$ unit quaternions that map to the $60$ rotational symmetries of a regular icosahedron. Poincaré found this manifold in 1904 (by a different construction) as a closed three-manifold with the same homology as $S^3$ but not homeomorphic to it. Its fundamental group $I^*$ has $120$ elements and is perfect: it equals its own commutator subgroup, so its abelianisation, which is $H_1$, is trivial (7A.8 Homology in Brief). It is the reason the Poincaré conjecture is stated in terms of $\pi_1$ and not homology (7A.9 The Poincaré Conjecture, Precisely).

6.6History

Covering spaces appeared in Riemann's work on multivalued functions (Riemann surfaces as coverings of the sphere) and were formalised by Poincaré and Hermann Weyl (Die Idee der Riemannschen Fläche, 1913). Poincaré's homology sphere appeared in the fifth supplement to Analysis Situs (1904). Heinrich Tietze introduced lens spaces in 1908, and Kurt Reidemeister classified them up to combinatorial equivalence in 1935. Heinz Hopf studied spherical space forms in the mid-1920s, and the classification of finite groups acting freely and linearly on $S^3$ was completed by Herbert Seifert and William Threlfall in 1930–33. William Rowan Hamilton discovered the quaternions in 1843. The neutron $4\pi$ experiments were published in 1975, Goldstein, Zebker and Werner's phase unwrapping method in 1988, and Shoemake's SLERP in 1985.

6.7Exercises

Solution

With $c = \cos\frac\theta2$, $s = \sin\frac\theta2$: $q\,i\,\bar q = (c + sk)i(c - sk) = (ci + sj)(c - sk) = c^2i - csik + scj - s^2jk$; using $ik = -j$ and $jk = i$: $= c^2i + csj + scj - s^2i = \cos\theta\,i + \sin\theta\,j$. Similarly $q\,j\,\bar q = -\sin\theta\,i + \cos\theta\,j$, and $k$ commutes with $q$, so it is fixed.

Solution

(a) The covering $S^3 \to S^3/\Gamma$ is a local isometry with $|\Gamma|$ sheets. (b) The Ricci flow is local and invariant under isometries, so a solution on $S^3$ that is invariant under $\Gamma$ descends; the round shrinking solution is invariant under all of $SO(4)$. (c) Volume scales by $(1 - 4t)^{3/2}$: $L(5, 1)$ has $\frac{2\pi^2}{5} \approx 3.95$, then $\frac{2\pi^2}{5}(\frac12)^{3/2} \approx 1.40$; the Poincaré sphere $\frac{2\pi^2}{120} \approx 0.164$, then $\approx 0.058$.