Covering projections

Definition. A covering projection is a continuous map p : E → B with the property that B admits an open cover U such that for each U ∈ U there is a set F and a locally constant map q : p−1U → F such that  p  : p−1U → U × F q is a homeomorphism.

Commentary. A map from a space to a set is locally constant if it is continuous when the set is given the discrete topology. The top entry in the matrix is strictly speaking not p, but rather the map −1 pU : p U → U such that p−1U / E

pU p   U / B commutes. Giving such a map q is the same as giving a homeomorphism −1 h : p U → U × F such that pr1 ◦ h = pU . A map p : E → B is a trivial covering projection there exists a set F  p  and a locally constant map q : E → F such that : E → B × F is a q homeomorphism. Note that B × F is homeomorphic to the disjoint union of copies of B indexed by the elements of F . The trivial covering condition asserts that the map p just sends each copy of B to itself by the identity map. A covering projection is a map which is, locally in the base, a trivial covering projection.

Examples. e : R → S1 by t 7→ e2πit. 1 1 n pn : S → S by z 7→ z . p : Sn → RP n by v 7→ ±v.

1B : B → B and ∅ → B for any space B. Proposition. Let p : E → B be a covering projection. Let a, b ∈ B and let ω : I → B be a continuous map (a “path”) such that ω(0) = a and ω(1) = b. Let e ∈ E be such that p(e) = a. (a) There is a unique pathω ˜ : I → E such that pω˜ = ω andω ˜(0) = e. (b) Let Ω : I × I → B be such that Ω(0, 0) = b. There is a unique lift of Ω to a map Ω:˜ I × I → E such that Ω(0, 0) = e.

Suppose then that ω0 is a path in B with the same endpoints as ω and which is path-homotopic (i.e. homotopic through a homotopy which pre- serves the endpoints) to ω. Then the liftω ˜0 of ω0 withω ˜0(0) = e has ω˜0(1) =ω ˜(1), andω ˜ andω ˜0 are path-homotopic. We can conjoin paths if the end of the first is the start of the second: if α(1) = b = β(0), we define

 α(2t) if t ≤ 1/2 α ∗ β(t) = β(2t − 1) if t ≥ 1/2

This operation descends to an operation on path homotopy classes, and is associative and unital up to path homotopy: (α ∗ β) ∗ γ ' α ∗ (β ∗ γ) and 1a ∗ α ' α ∗ 1b, where 1a is the constant path at a = α(0). If we define −1 −1 −1 α (t) = α(1 − t), then α ∗ α ' 1b and α ∗ α ' 1a.

Definition. The fundamental group of the pointed space (B, b) is the set of paths in B beginning and ending at b, with the group structure given by juxtaposition. It is also called the Poincar´egroup, and is written π1(B, b).

Definition. A space B is simply connected if it is non-empty, and for any pair of points a, b ∈ B there is a unique path-class of paths from a to b. This is the same as requiring that B is nonempty, path connected, and for any (or even just some) point b ∈ B, π1(B, b) = 1. For example, any contractible space is simply connected. Under mild conditions, the fundamental group controls lifting across cov- ering projections.

Definition. A space B is locally path connected if for every point b and every open neighborhood U of b there exists a path-connected open neighborhood of b lying in U.

Proposition. Let X be connected and locally path connected space, and let p : E → B be a covering projection. Let x ∈ X and e ∈ E, and let f : X → B be a continuous map sending x to p(e). There is at most one continuous map g : X → E which sends x to e and covers f, and there is one if and only if

im(π1(X, x) → π1(B, b)) ⊆ im(π1(E, e) → π1(B, b))

Corollary. Let B be locally path connected space, and let p : E → B be a covering projection. Let e ∈ E and write b = p(e) ∈ B. Then π1(E, e) 7→ π1(B, b) is a monomorphism.

Monodromy. Let p : E → B be a covering projection, let b ∈ B, and let −1 e ∈ F = p (b). Let α ∈ π1(B, b). It lifts to a unique pathα ˜ : I → E withα ˜(0) = e. Let e · α =α ˜(1) ∈ p−1(b). This defines a right action—the −1 “monodromy action”—of π1(B, b) on p (b); that is,

(e · α) · β = e · (α ∗ β) , e · 1b = e .

Proposition. Assume that B is locally path connected, let b ∈ B, and let p : E → B be a covering projection. The isotropy group of e ∈ E under the action of π1(B, b) is

=∼ {σ ∈ π1(B, b): e · σ = e} ←− π1(E, e)

This construction defines a functor

Fb : CovB → Setπ1(B,b)

0 Here CovB has as objects the covering projections p : E → B, and CovB(p , p) = 0 0 {f : E → E : pf = p }. The target is the category of sets with right π1(B, b) action for objects, and equivariant maps for morphisms. The second proposition implies that if B is connected and locally path connected, then this functor is full and faithful.

Definition. A functor F : C → D is faithful if for every C,C0 ∈ obC, F : C(C,C0) → D(F (C),F (C0)) is injective; full if for every C,C0 ∈ obC, F : C(C,C0) → D(F (C),F (C0)) is surjective; representative if for every D ∈ obD there exists C ∈ obC such that F (C) is isomorphic to D; an equivalence of categories if it is full, faithful, and representative.

The functor Fb is also representative under a further mild condition. Definition. A space B is semi-locally simply connected if every point b ∈ B has an open neighborhood U such that π1(U, b) → π1(B, b) is the trivial group homomorphism.

Proposition. Let B be connected and locally path connected. There is a covering projection B˜ → B from a simply connected space if and only if B is semi-locally path simply connected.

A simply connected covering space of a connected locally path connected space is called a universal cover. It’s unique up to isomorphism.

Example. The covering projection e : R → S1 and (for n > 1) the double cover p : Sn → RP n are universal covers.

The right action of π1(B, b) on the fiber of a universal cover is free and transitive. This provides a computation of the fundamental group; for exam- 1 ple, π1(S , b) = Z. ˜ Fix e ∈ B lying over b ∈ B. Let σ ∈ π1(B, b). It determines an auto- morphism of the fiber Fb. Since the fiber functor is fully faithful, there is a ˜ ˜ unique map σ# : B → B over the identity of B such that σ#(e) = e · σ.I ˜ claim that this defines a left action of π1(B, b) on B. Here is the requisite calculation:

σ#(τ#e) = σ#(e · τ) = (σ#e) · τ = (e · σ) · τ = e · (στ) = (στ)#e using the fact that action of π1(B, b) is natural for covering maps.

Now if F is any right π1(B, b) set, we can form a covering space over B using the “balanced product” ˜ ˜ F ×π1(B,b) B = F × B/ ∼ , (xσ, e) ∼ (x, σ#e)

This defines a functor Setπ1(B,b) → CovB which is “quasi-inverse” to the fiber functor Fb: there are natural isomorphisms between composites and the appropriate identity functors. This completes the proof that the fiber functor is an equivalence of categories.