Topology (H) Lecture 22 Lecturer: Zuoqin Wang Time: May 27, 2021 CLASSIFICATION of COVERING SPACES 1. the Universal Covering ¶

Topology (H) Lecture 22 Lecturer: Zuoqin Wang Time: May 27, 2021

CLASSIFICATION OF COVERING SPACES

1. The universal covering ¶ The universal covering. f In PSet11-1-2 we have seen that p∗ : π1(X, x˜0) → π1(X, x0) is injective. So the f f subgroup p∗(π1(X, x˜0)) in π1(X, x0) is isomorphic to π1(X, x˜0). The main theme today is to establish a “one-to-one correspondence” (in reverse order) between subgroups of π1(X, x0) and covering spaces over X. We ﬁrst start with the smallest subgroup of π1(X, x0), namely the subgroup {e}, which will give us the “largest” covering space. f f Deﬁnition 1.1. We say a covering space X is a universal covering if π1(X) '{e}. Example 1.2. • R is a universal covering space of S1, while S1 is not. • S2 is a universal covering space of S2, R2 is a universal covering space of T2, 2 2 1 the unit disc D is a universal covering space of Σ2 = T #T (shown below) . n • Sn is a universal covering space of RP (n ≥ 2). S3 is a universal covering space of L(p; q). SU(2) is a universal covering space of SO(3)2. • The universal covering of S1 ∨ S1 is the Cayley graph of ha, bi shown below.3

1 According to the uniformization theorem, D is a universal covering space of Σn for all n ≥ 2. 2In general the spin group Spin(n) is a universal covering of SO(n) for n ≥ 3. 3In general the Cayley graph of the free group on n generators is a universal covering of S1 ∨· · ·∨S1.

1 2 CLASSIFICATION OF COVERING SPACES

¶ The existence of universal covering space. Given a topological space X, we want to construct a universal covering. It turns out that it is not always possible to do so. To see this let’s suppose p : Xf → X is a universal covering. Then for any x ∈ X, there is a neighborhood U of x and an open Ü f Ü set U in X such that p|Ue : U → U is a homeomorphism. Now each loop γ in U can Ü −1 f be lifted to a loopγ ˜ in U by the map (p|Ue) . Since π1(X) = {e}, the lifted loopγ ˜ is null-homotopic in Xf. Composing the null-homotopy with the projection map p, we conclude that γ must be null-homotopic in X. In other words, for any x there is a neighborhood U of x such that any loop in U is null-homotopic in X. So the group homomorphism i∗ : π1(U, x) → π1(X, x) induced by the inclusion map is a trivial homomorphism. (Roughly speaking, X can’t have arbitrarily small “holes”.) Deﬁnition 1.3. We say a topological space X is semi-locally simply connected 4 if for each x ∈ X, there is a neighborhood U of x such that the group homomorphism

i∗ : π1(U, x) → π1(X, x) induced by the inclusion map i : U,→ X is trivial. Example 1.4. In PSet 3-2-3, we have seen the Hawai- ian earring X, i.e. a shrinking wedge of circles as showed in the picture. It is path-connected, locally path-connected, but NOT semi-locally simply connected, and thus admits no universal covering. On the other hand, the cone CX over the Hawaiian earring is contractible and thus is semi-locally simply connected, but it is not locally simply connected. Theorem 1.5. Suppose X is path-connected, locally path-connected. Then X admits a universal covering space Xf if and only if X is semi-locally simply connected. Proof. We have seen that the semi-locally simply connectedness is necessary. It remains to prove the converse. Idea. Given S1, how do you construct the topological space R? Any point in R is the end point of a lifted path (with fixed starting point) which corresponds to a path in S1 (with fixed starting point). Two lifted paths have the same end point if and only if the projected paths in S1 are path homotopic. So: points in R are in one-to-one correspondence with path homotopy classes of paths in S1 (with fixed starting point)! But, how do we define a topology on the space of homotopy classes of paths? We can simply take the compact-open topology on the space

4As usual, X is called locally simply connected if any point has a simply connected neighborhood, i.e. a neighborhood U so that any loop in U is null-homotopic in U. This is stronger than semi-locally simply connected. CLASSIFICATION OF COVERING SPACES 3

of paths, and then take the quotient topology on the space of homotopy classes. But for our purpose below, we will take an alternative way which is more direct: we can show that “locally” the projection map is bijective. Thus by choosing a base of S1 consisting of small open sets (which form a base of the topology), we can deﬁne the “lifting” of these open sets to be open in the space of homotopy classes and thus get a base of a topology. That is what we need. Step 1. Construction of the set Xf. Let X be a path-connected, locally path-connected and semi-locally simply connected topological space with base point x0. We deﬁne f X = {[γ]p | γ is a path in X staring at x0}. By construction, there is a natural projection map from Xf to X, given by f p : X → X, [γ]p → γ(1). Since X is path-connected, p is surjective. Step 2. p is “locally” bijective.

For any path γ : [0, 1] → X with γ(0) = x0, let U ⊂ X be an open neighborhood of γ(1) which is path-connected and such that any loop in U is null-homotopic in X. Deﬁne

U[γ]p = {[γ∗λ]p | λ is a path in U with λ(0) = γ(1)}.

Note that U[γ]p is well-deﬁned, i.e. depends only on the path homotopy class [γ]p instead of on the path γ. Moreover, we have 0 0 (>) If [γ ]p ∈ U[γ]p , then U[γ]p = U[γ ]p . 0 0 0 To see this we suppose γ = γ ∗λ, then any element in U[γ ]p has the form [γ ∗λ∗λ ]p which is still an element in U[γ]p , and conversely any element in U[γ]p has the form 0 0 ¯ 0 [γ∗λ ]p = [γ∗λ∗λ∗λ ]p which lies in U[γ ]p . It is not hard to see that p = p| : U → U is bijective: U,[γ]p U[γ]p [γ]p

• Since U is path-connected, pU,[γ]p (U[γ]p ) = U. • Since any loop in U is null-homotopic in X, different choices of λ from γ(1) to a given point x ∈ U are homotopic in X and thus the map p is injective. Step 3. Define a topology on Xf. To define a topology on Xf, we first consider the following collection U = {U ⊂ X | U is open, path-connected and any loop in U is null-homotopic in X} of open sets in X. Since X is locally path-connected and semi-locally simply connected, U is a basis of the topology on X. 4 CLASSIFICATION OF COVERING SPACES

Claim: the collection Ü U := {U[γ]p | U ∈ U, γ is a path in X from x0 to a point in U} form a basis for a topology on Xf.

f 00 Reason: Obviously the union of such sets is X. Now suppose [γ ]p ∈ 00 0 U[γ]p ∩ V[γ ]p . Then γ (1) ∈ U ∩ V . Since U is a basis of the given topology on X, there is a set W ∈ U such that γ00(1) ∈ W ⊂ U ∩ V .

00 0 00 According to (>), we have U[γ]p = U[γ ]p and V[γ ]p = V[γ ]p . It follows

00 00 00 00 0 [γ ]p ∈ W[γ ]p ⊂ U[γ ]p ∩ V[γ ]p = U[γ]p ∩ V[γ ]p . Step 4. p is a covering map.

We have seen in Step 2 that pU,[γ]p : U[γ]p → U is bijective. In fact, pU,[γ]p also gives 0 a bijection between subsets V[γ ]p ⊂ U[γ]p and V ∈ U satisfying V ⊂ U: in one direction −1 we have p (V 0 ) = V , and in the other direction we have p (V ) ∩ U = V 0 , U,[γ]p [γ ]p U,[γ]p [γ]p [γ ]p 0 0 where [γ ]p ∈ U[γ]p is a path homotopy class such that γ has end point in V . It follows pU,[γ]p is a homeomorphism with respect to the topology constructed in Step 3. f Since p is continuous on each open set U[γ]p , p : XS→ X is continuous. It is a −1 covering map since for each U ∈ U, we have p (U) = [γ]p U[γ]p , which is a disjoint union in view of (>). Step 5. Xf is simply connected. f f Let’s take [cx0 ]p to be our base point in X. For any [γ]p ∈ X, if we let γt(s) = γ(st), f f f then [γt]p ∈ X and f(t) := [γt]p is a path in X connecting [cx0 ]p and [γ]p. So X is path-connected. [One need to check the continuity of f as a map from [0, 1] to Xf.] f Now let f : [0, 1] → X be any loop based at [cx0 ]p. Then p◦f is a loop in X based at x0 and f is a lifting of p◦f starting at [cx0 ]p. Let µt : [0, 1] → X be the path

µt(s) = p◦f(st). f Then g(t) := [µt]p deﬁnes a path in X starting at [cx0 ]p which is also a lifting of the loop p◦f. By the unique lifting property, we have [µt]p = f(t). In particular,

[p◦f]p = [µ1]p = [cx0 ]p. So p◦f is null-homotopic, i.e.

p∗([f]p) = e ∈ π1(X, [cx0 ]p).

In PSet 11-1-2 we have seen that p∗ is injective. So [f]p = e. It follows f π1(X, [cx0 ]p) = {e}, f i.e. X is simply connected. CLASSIFICATION OF COVERING SPACES 5

2. Classification of covering spaces ¶ Existence of covering spaces with general fundamental group. Having constructed universal covering, now we can set up a correspondence between 5 subgroups H ⊂ π1(X, x0) and covering spaces over X. First we prove the existence of a covering space for any H: Theorem 2.1 (Existence of covering spaces). Let X be path-connected, locally path- connected and semi-locally simply connected. Then for any subgroup H ⊂ π1(X, x0), f −1 there is a covering space p : XH → X and a basepoint x˜0 ∈ p (x0) such that f p∗(π1(XH , x˜0)) = H.

0 f f Proof. Let H ⊂ π1(X, x0) be a subgroup. For two points [γ]p, [γ ]p ∈ X, where X is the universal covering space of X that we constructed above, we deﬁne

0 0 0 [γ]p ∼ [γ ]p ⇐⇒ γ(1) = γ (1) and [γ∗γ¯ ]p ∈ H. H This is an equivalence relation since H is a subgroup. [Check this.] f Now we deﬁne XH to be the quotient space f f XH = X/ ∼, H f and letx ˜0 ∈ XH to be the equivalence class containing [cx0 ]p. It remains to check f • The natural projection p : XH → X induced by [γ]p 7→ γ(1) is a covering map. 0 Reason: Suppose [γ]p ∼ [γ ]p, and suppose U ⊂ X is open, path- H connected, and such that i∗ : π1(U, γ(1)) → π1(X, γ(1)) is trivial. Then for any path λ in U with λ(0) = γ(1), λ∗λ¯ is null-homotopic in X and thus

0 0 0 [γ∗λ∗γ ∗λ]p = [γ∗λ∗λ∗γ ]p = [γ∗γ ]p ∈ H, 0 i.e. [γ∗λ]p ∼ [γ ∗λ]p. In other words, the “basic” neighborhood U[γ] H p f 0 and U[γ ]p are the same. It follows that p is a projection map (since X is a covering of X). f • p∗(π1(XH , x˜0)) = H. f Reason: Let γ be a loop in X based at x0. Then its lifting to X

starting atx ˜0 = [cx0 ]p ends at [γ]p. So the image of this lifted path is f a loop in XH if and only if [γ]p ∼ [cx ]p, i.e. if and only if [γ]p ∈ H. H 0 The conclusion follows from PSet 11-1-2(b). 5One should compare this with the fundamental theorem of Galois theory: given a ﬁeld extension (with some additional assumptions), there is a one-to-one correspondence with intermediate ﬁelds and subgroups of its Galois group. 6 CLASSIFICATION OF COVERING SPACES

¶ Isomorphic covering spaces.

Given the existence of a correspondence from subgroups H ⊂ π1(X, x0) to covering spaces over X, next we study the uniqueness in this correspondence. First we need a definition to identify different covering spaces over X. f Definition 2.2. We say two covering spaces p1 : X1 → X f and p2 : X2 → X over the same base space X are iso- f h f f f X1 ' / X2 morphic if there exists a homeomorphism h : X1 → X2 s.t. p1 p2 ~ p1 = p2 ◦ h. X The map h is called a covering space isomorphism.

Remark 2.3. (1) One can easily check that this deﬁnes an equivalence relation on the set of all covering spaces over X. (2) Let p : Xf → X be a covering space. Then Aut(p) := {h : Xf → Xf | h is a covering space isomorphism} is a group under composition. Deﬁnition 2.4. The group Aut(p) is called the Deck transformation group, or covering transformation group of the covering p : Xf → X. [Some authors also call this the Galois group of the covering.] f f Now suppose h : X1 → X2 is a covering space isomorphism. Then for any x0 ∈ X, h −1 −1 is a one-to-one map from the set p1 (x0) to the set p2 (x0). In particular, if we choose −1 −1 x˜1 ∈ p1 (x0) and letx ˜2 = h(˜x1) ∈ p2 (x0), then the following “pointed-diagram” commutes: f h f (X1, x˜1) ' / (X2, x˜2)

p1 p2 % y (X, x0) Note that from this diagram we must have (by using both h and h−1) f f (p1)∗(π1(X1, x˜1)) = (p2)∗(π1(X2, x˜2)), i.e. isomorphic covering spaces gives the same subgroup in π1(X, x0).

¶ Uniqueness of covering spaces. Now we are ready to prove Proposition 2.5 (Uniqueness of covering space). Let X be path-connected and locally f f path-connected. Suppose p1 :(X1, x˜1) → (X, x0) and p2 :(X2, x˜2) → (X, x0) are CLASSIFICATION OF COVERING SPACES 7 path-connected covering space. Then there exists a covering space isomorphism h : f f (X1, x˜1) → (X2, x˜2) if and only if f f (p1)∗(π1(X1, x˜1)) = (p2)∗(π1(X2, x˜2)). Proof. It remains to prove that if the condition holds, then there exists a covering space f f isomorphism h :(X1, x˜1) → (X2, x˜2). By the existence criterion of lifting, we may lift f f p1 to a continuous mapp ˜1 :(X1, x˜1) → (X2, x˜2) such that

p1 = p2 ◦ p˜1. f f Similarly we may lift p2 top ˜2 :(X2, x˜2) → (X1, x˜1) such that

p2 = p1 ◦ p˜2.

It follows by the unique lifting property thatp ˜1 ◦ p˜2 = Id e andp ˜2 ◦ p˜1 = Id e . Sop ˜1 X2 X1 is the covering space isomorphism we are looking for. In particular we see that given any path-connected, locally path-connected and semi-locally simply connected space, the universal covering must be unique.

¶ Classification of covering spaces. Combining Theorem 2.1 and Proposition 2.5 we get Theorem 2.6 (The classification of covering spaces, with basepoint). Let X be path- connected, locally path-connected and semi-locally simply connected. Then there is a bijection between the set of basepoint-preserving isomorphism classes of path-connected f covering spaces p :(X, x˜0) → (X, x0) and the set of subgroups of π1(X, x0), given by f f (X, x˜0) ! p∗(π1(X, x˜0)). We can also ignore the basepoint in the correspondence. In this case we have Theorem 2.7 (The classification of covering spaces, without basepoint). Let X be path-connected, locally path-connected and semi-locally simply connected. Then there is a bijection between isomorphism classes of path-connected covering spaces p : Xf → X and conjugacy classes of subgroups of π1(X, x0). f Proof. We have seen in PSet 11-1-2(e) that for a covering space p :(X, x˜0) → (X, x0), −1 f if we change the basepointx ˜0 tox ˜1 ∈ p (x0), the two subgroups p∗(π1(X, x˜0)) and f p∗(π1(X, x˜1)) are conjugate to each other. f 0 Conversely suppose we are given any subgroup H = p∗(π1(X, x˜0)) and any H = −1 g Hg, where g = [γ]p ∈ π1(X, x0). We letγ ˜ be the lifting of γ with start pointx ˜0, and −1 f −1 denotex ˜1 =γ ˜(1) ∈ p (x0). Then one can easily prove p∗(π1(X, x˜1)) = g Hg. As a consequence of the classification theorem, we get 8 CLASSIFICATION OF COVERING SPACES

Corollary 2.8. Let X be path-connected, locally path-connected and semi-locally simply f f connected. Suppose p1 :(X1, x˜1) → (X, x0) and p2 :(X2, x˜2) → (X, x0) are two path- connected covering spaces. If f f (p1)∗(π1(X1, x˜1)) ⊂ (p2)∗(π1(X2, x˜2)), f f then there exists a covering map p3 :(X1, x˜1) → (X2, x˜2) such that p1 = p2 ◦ p3. In particular we see why a universal covering space is “universal”: Corollary 2.9. If Xc is a universal covering of X, and Xf is any connected covering of X, then Xc is a (universal) covering of Xf.