nLab
universal covering space

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

In point-set topology

The following is the classical discussion of universal covering spaces in point-set topology.

Proposition

(essential uniqueness of simply connected covering spaces)

Let XX be a topological space which is

  1. path-connected,

  2. locally path connected.

Then if E ip iXE_i \overset{p_i}{\to} X are two covering spaces over XX, i{1,2}i \in \{1,2\}, which are both path-connected and simply connected, then they are isomorphic as covering spaces.

Proof

Since both E 1E_1 and E 2E_2 are simply connected, the assumption of the lifting theorem for covering spaces is satisfied (this prop.). This says that there are horizontal continuous function making the following diagrams commute:

E 1 f E 2 p 1 p 2 XAAAAAAE 2 g E 1 p 1 p 2 X \array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X } \phantom{AAAAAA} \array{ E_2 && \overset{g}{\longrightarrow} && E_1 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X }
E i id E i p 1 p 2 X \array{ E_i && \overset{id}{\longrightarrow} && E_i \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X }

and that these are unique once we specify the image of a single point, which we may freely do (in the given fiber).

So if we pick any point xXx \in X and x^ 1E 1\hat x_1 \in E_1 with p(x^)=xp(\hat x) = x and x^ 2E 2\hat x_2 \in E_2 with p(x^ 2)=xp(\hat x_2) = x and specify that f(x^ 1)=x^ 2f(\hat x_1) = \hat x_2 and g(x^ 2)=x^ 1g(\hat x_2) = \hat x_1 then uniqueness applied to the composites implies fg=id E 2f \circ g = id_{E_{2}} and gf=id E 1g \circ f = id_{E_1}.

Definition

(universal covering space)

Let XX be a topological space which is

  1. path-connected,

  2. locally path connected.

Then a path-connected and simply connected covering space, is called the universal covering space of XX. This is well-defined, if it exists, up to isomorphism, by prop. 1.

Proposition

(universal covering space reconstructed from free and transitive fundamental group representation)

Let XX be a topological space which is well-connected in that it is

Then a universal covering space of XX (def. 1) exists.

Proof

By this prop. the covering space is connected and simply connected precisely if its monodromy representation is free and transitive. By the fundamental theorem of covering spaces every permutation representation of the fundamental groupoid Π 1(X)\Pi_1(X) arises as the monodromy of some covering space. Hence it remains to see that a free and transitive representation of Π 1(X)\Pi_1(X) exists. Let xXx \in X be any point, then Hom Π 1(X)(x,)Hom_{\Pi_1(X)}(x,-) is such a representation.

This is a functorial construction of a universal covering spaces

Top * wcCov * Top_*^{wc} \longrightarrow Cov_*

where Top * wcTop_*^{wc} denotes the full subcategory of pointed topological spaces Top *Top_* on the well-connected spaces and CovCov is the subcategory of Top * 2Top_*^2 of pointed maps of spaces with objects the covering space maps.

Specifically, if XX is a space with basepoint x 0x_0, we define X (1)X^{(1)} to be the space whose points are homotopy classes of paths in XX starting at x 0x_0, with the projection X (1)XX^{(1)}\to X projecting to the endpoint of a path. We can equip this set X (1)XX^{(1)}\to X with a topology coming from XX so that it becomes a universal covering space as above. As described at covering space, under the correspondence between covering spaces and Π 1(X)\Pi_1(X)-actions, the space X (1)X^{(1)} corresponds to the “regular representation” of Π 1(X)\Pi_1(X).

Remark

This regular representation can be seen to arise by taking a category of elements in the same way that the regular representation of a group is gotten by taking its action on itself: we can see that the universal covering groupoid Π(X) (1)\Pi(X)^{(1)} in the slice category Grpd/Π(X)\mathbf{Grpd}/\Pi(X) (see the universal covering \infty-groupoid below) is just the category of elements of the action of Π(X)\Pi(X) on itself, and can be topologized in a natural way by lifting the topology on XX along the canonical projection Π(X) (1)Π(X)\Pi(X)^{(1)} \to \Pi(X); decategorifying this yields X (1)X^{(1)}.

In cohesive homotopy theory

We describe now how the universal cover construction may be understood from the nPOV, using a fragment of cohesive homotopy theory.

The basic idea is that the universal cover of a space XX is the homotopy fiber of the canonical morphism

XΠ 1(X) X \longrightarrow \Pi_1(X)

from XX to its fundamental groupoid, which exists if both objects are regarded as “topological ∞-groupoids”. We may think of this as a precise way of the intuitive idea of “forcing Π 1(X)\Pi_1(X) to become trivial in a universal way”.

Example

(cohesive higher toposes of topological groupoids)

Let 𝒮\mathcal{S} be some ∞-cohesive site of spaces and consider

HSh (𝒮) \mathbf{H} \coloneqq Sh_\infty(\mathcal{S})

the corresponding cohesive (∞,1)-topos over this site. This is the (∞,1)-category of “topological ∞-groupoids” modeled on 𝒮\mathcal{S}.

A canonical choice relating to the traditional discussion would be 𝒮=Top lcont κ\mathcal{S} = Top^\kappa_{lcont} a small full subcategory of Top on locally contractible topological spaces, in which case the objects of H\mathbf{H} might be called “locally contractible topological ∞-groupoids”.

A more restrictive choice would be 𝒮=\mathcal{S} = CartSp, in which case the objects of H\mathbf{H} might be called Euclidean-topological ∞-groupoids.

In fact for the discussion of just universal covering spaces as opposed to the higher stages in the Whitehead tower it would be sufficient and more natural to take 𝒮\mathcal{S} the full subcategory of locally simply connected topological spaces and consider just the (2,1)-category H\mathbf{H} of stacks over this site.

But the following discussion is completely formal and applies globally to all such realizations. Namely all we need is that

Assumption

Assume that

  1. H\mathbf{H} is a cohesive (n,1)-topos

    HΠ Δ Γ B \mathbf{H} \array{ \overset{\Pi}{\longrightarrow} \\ \overset{\Delta}{\longleftarrow} \\ \overset{\Gamma}{\longrightarrow} \\ \overset{\nabla}{\longleftarrow} } \mathbf{B}

    over the given base (n,1)-topos of n-groupoids.

  2. such that its shape modality preserves homotopy fiber products over discrete objects (objects in the essential image of Δ\Delta).

We write

Unknown characterUnknown characterΔΠ ʃ \coloneqq \Delta \Pi

for the induced shape modality and

Xη XUnknown characterUnknown characterX X \overset{\eta_X}{\longrightarrow} ʃ X

for its unit morphism ona given object XX.

Example

Assumption 2 is satisfied for the case of (,1)(\infty,1)-toposes over an ∞-cohesive site and for (n,1)(n,1)-toposes by an nn-cohesive site as in example 1: by this prop..

Now consider XHX \in \mathbf{H} any object, for instance a topological space regarded as a 0-truncated topological infinity-groupoid.

Assume, just for ease of discussion, that XX is geometrically connected in that the 0-truncation of its shape is contractible:

τ 0(Unknown characterUnknown characterX)*. \tau_0(ʃ X) \simeq \ast \,.

Using the axiom of choice in the base topos H\mathbf{H} we choose a point

*τ 1(Unknown characterUnknown characterX). \ast \to \tau_1(ʃ X) \,.
Definition

The universal cover of XX is the homotopy pullback in

X^ * (pb) X L τ 1η X τ 1(Unknown characterUnknown characterX). \array{ \hat X &\longrightarrow& \ast \\ \downarrow &(pb)& \downarrow \\ X &\underset{L_{\tau_1} \circ \eta_X}{\longrightarrow}& \tau_1(ʃ X) } \,.

The universal property of X^\hat X is immediate from the abstract setup:

  1. X^\hat X is simply connected, in that τ 1Unknown characterUnknown characterX^*\tau_1 ʃ \hat X \simeq \ast

    This is because Unknown characterUnknown characterXʃ X is discrete by construction, and hence so is τ 1(Unknown characterUnknown characterX)\tau_1(ʃ X). So by assumpotion 2 applying Unknown characterUnknown characterʃ to the above square yields another homotopy pullback of the form

    Unknown characterUnknown characterX^ * (pb) Unknown characterUnknown characterX Unknown characterUnknown characterη X τ 1(Unknown characterUnknown characterX). \array{ ʃ \hat X &\longrightarrow& \ast \\ \downarrow &(pb)& \downarrow \\ ʃ X &\underset{ʃ \eta_X}{\longrightarrow}& \tau_1(ʃ X) } \,.

    Now the long exact sequence of homotopy groups (in B\mathbf{B}) applied to this homotopy fiber sequence is of the form

    0π 1(X^)π 1(X)=π 1(X) 0 \to \pi_1(\hat X) \longrightarrow \pi_1(X) \overset{=}{\longrightarrow} \pi_1(X) \to \cdots

    which implies that π 1(Unknown characterUnknown characterX)\pi_1( ʃ X ) and hence τ 1Unknown characterUnknown character\tau_1 ʃ is indeed trivial.

  2. Let EXE \longrightarrow X be any other object of H /X\mathbf{H}_{/X} such that τ 1(Unknown characterUnknown characterE)*\tau_1(ʃ E) \simeq \ast, then there is morphism

    E X^ X \array{ E && \longrightarrow && \hat X \\ & \searrow && \swarrow \\ && X }

    This is because the naturality of the shape unit and of the truncation unit gives a homotopy commuting square of the form

    E η E Unknown characterUnknown characterE τ 1(Unknown characterUnknown characterE)* X η X Unknown characterUnknown characterX τ 1Unknown characterUnknown characterX \array{ E &\overset{\eta_E}{\longrightarrow}& ʃ E &\overset{}{\longrightarrow}& \tau_1(ʃ E ) \simeq \ast \\ \downarrow && && \downarrow \\ X &\underset{\eta_X}{\longrightarrow}& ʃ X &\longrightarrow& \tau_1 ʃ X }

    and thus a cone over the diagram which defines X^\hat X via its universal property.

This shows that X^\hat X is the universal cover on abstract grounds.

We may also check explicitly that X^\hat X is given as the space of homotopy classes of paths in XX from the given basepoint. To that end we use that, as least over the site CartSp, the shape of XX is represented by the topological path ∞-groupoid. See at shape via cohesive path ∞-groupoid.

The following is old material that deserves to be harmonized a bit more with the above stuff.

Proposition

Let XX be a suitably well behaved pointed space. The universal cover X (1)X^{(1)} of XX is (equivalent to) the homotopy fiber of XΠ(X)X \to \Pi(X) in the (∞,1)-category H=Sh (,1)(Top cg)\mathbf{H} = Sh_{(\infty,1)}(Top_{cg}) of topological ∞-groupoids.

In other words, the principal ∞-bundle classified by the cocycle XΠ 1(X)X \to \Pi_1(X) is the universal cover X (1)X^{(1)}: we have a homotopy pullback square

X (1) * X Π(X). \array{ X^{(1)} &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \Pi(X) } \,.

Urs Schreiber: may need polishing.

Proof

We place ourselves in the context of topological ∞-groupoids and regard both the space XX as well as its homotopy ∞-groupoid? Π(X)\Pi(X) and its truncation to the fundamental groupoid Π 1(X)\Pi_1(X) as objects in there.

The canonical morphism XΠ(X)X \to \Pi(X) hence XΠ 1(X)X \to \Pi_1(X) given by the inclusion of constant paths may be regarded as a cocycle for a Π(X)\Pi(X)-principal ∞-bundle, respectively for a Π 1(X)\Pi_1(X)-principal bundle.

Let π 0(X)\pi_0(X) be the set of connected components of XX, regarded as a topological \infty-groupoid, and choose any section π 0(X)Π(X)\pi_0(X) \to \Pi(X) of the projection Π(X)π 0(X)\Pi(X) \to \pi_0(X).

Then the Π(X)\Pi(X)-principal \infty-bundle classified by the cocycle XΠ(X)X \to \Pi(X) is its homotopy fiber, i.e. the homotopy pullback

UCov(X) π 0(X) X Π(X). \array{ UCov(X) &\to& \pi_0(X) \\ \downarrow && \downarrow \\ X &\to& \Pi(X) } \,.

We think of this topological \infty-groupoid UCov(X)UCov(X) as the universal covering \infty-groupoid of XX. To break this down, we check that its decategorification gives the ordinary universal covering space:

for this we compute the homotopy pullback

UCov 1(X) * x X Π 1(X), \array{ UCov_1(X) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{x}} \\ X &\to& \Pi_1(X) } \,,

where we assume XX to be connected with chosen baspoint xx just to shorten the exposition a little. By the laws of homotopy pullbacks in general and homotopy fibers in particular, we may compute this as the ordinary pullback of a weakly equivalent diagram, where the point ** is resolved to the universal Π 1(X)\Pi_1(X)-principal bundle

E xΠ 1(X)=T xΠ 1(X). \mathbf{E}_x \Pi_1(X) = T_x \Pi_1(X) \,.

(More in detail, what we do behind the scenes is this: we regard the diagram as a diagram in the global model structure on simplicial presheaves on Top. In there all our topological groupoids are fibrant, hence all we have to ensure is that one of the morphisms of the diagram becomes a fibration, which is what the passage to E xΠ 1(X)\mathbf{E}_x \Pi_1(X) achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in \infty-prestacks. Then by left exactness of \infty-stackification, the image of that in \infty-stacks is still a homotopy pullback. )

The topological groupoid E xΠ 1(X)\mathbf{E}_x \Pi_1(X) has as objects homotopy classes rel endpoints of paths in XX starting at xx and as morphisms κ:γγ\kappa : \gamma \to \gamma' it has commuting triangles

x γ γ y κ y \array{ && x \\ &{}^{\mathllap{\gamma}}\swarrow && \searrow^{\mathrlap{\gamma'}} \\ y &&\stackrel{\kappa}{\to}&& y' }

in Π 1(X)\Pi_1(X). The topology on this can be deduced from thinking of this as the pullback

E xΠ 1(X) * x Π 1(X) I d 0 Π 1(X) \array{ \mathbf{E}_x \Pi_1(X) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Pi_1(X)^I &\stackrel{d_0}{\to}& \Pi_1(X) }

in simplicial presheaves on Top. Unwinding what this means we find that the open sets in Mor(E xΠ 1(X))Mor(\mathbf{E}_x \Pi_1(X)) are those where the endpoint evaluation produces an open set in XX.

Now it is immediate to read off the homotopy pullback as the ordinary pullback

UCov 1(X) E xΠ 1(X) X Π 1(X). \array{ UCov_1(X) &\to& \mathbf{E}_x \Pi_1(X) \\ \downarrow && \downarrow \\ X &\to& \Pi_1(X) \,. }

Since XX is categorically discrete, this simply produces the space of objects of E xΠ 1(X)\mathbf{E}_x \Pi_1(X) over the points of XX, which is just the space of all paths in XX starting at xx with the projection UCov 1(X)XUCov_1(X) \to X being endpoint evaluation.

This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in

In the petit \infty-toposes over the space

On the other hand, we can view a space XX as the little (,1)(\infty,1)-topos Sh (,1)(X)Sh_{(\infty,1)}(X) of (,1)(\infty,1)-sheaves on XX. If XX is locally connected and locally simply connected in the “coverings” sense, then Sh(X)Sh(X) is locally 1-connected.

In fact, for the construction of the universal cover we require only the (2,1)-topos Sh (2,1)(X)Sh_{(2,1)}(X) of sheaves (stacks) of groupoids on XX, so we will work in that context because it is simpler. The construction can be adapted, however, to produce a “universal cover” of any locally 1-connected (,1)(\infty,1)-topos.

Let EE be any (2,1)-topos which is locally 1-connected. This means that in the unique global sections geometric morphism (E *,E *):EGpd(E^*,E_*)\colon E\to Gpd, the functor E *E^* has a left adjoint E !:EGpdE_!\colon E \to Gpd, which is automatically GpdGpd-indexed. The fundamental groupoid of EE is defined to be Π 1(E)E !(*)\Pi_1(E)\coloneqq E_!(*), where ** is the terminal object of EE.

As discussed here in the (,1)(\infty,1)-case, the construction of Π 1(E)\Pi_1(E) is a left adjoint to the inclusion of groupoids into locally 1-connected (2,1)-toposes (which sends GGpd/GGpd GG\mapsto Gpd/G \simeq Gpd^G). Thus we have a geometric morphism EΠ 1(E)E\to \Pi_1(E) (where we regard Π 1(E)\Pi_1(E) as the (2,1)-topos Gpd Π 1(E)Gpd^{\Pi_1(E)}).

Suppose, for simplicity, that EE is connected. Then Π 1(E)\Pi_1(E) is also connected, and so we have an essentially unique functor *Π 1(E)*\to \Pi_1(E). We define the universal cover of EE to be the pullback (2,1)-topos:

E (1) E * Π 1(E)\array{ E^{(1)} & \to & E \\ \downarrow & & \downarrow \\ * & \to & \Pi_1(E)}

(where of course ** denotes the terminal (2,1)-topos GpdGpd).

Now observe that *Π 1(E)*\to \Pi_1(E) is a local homeomorphism of toposes, since we have GpdGpd/Π 1(E)/(*Π 1(E))Gpd \simeq Gpd/\Pi_1(E)/(*\to \Pi_1(E)). Since local homeomorphisms of toposes are stable under pullback, E (1)EE^{(1)}\to E is also a local homeomorphism, i.e. there exists an object E˜E\widetilde{E}\in E and an equivalence E (1)E/E˜E^{(1)} \simeq E/\widetilde{E} over EE. Moreover, it is not hard to see that E˜\widetilde{E} can be identified with the pullback

E˜ * η * E *(Π 1(E))=E *(E !(*))\array{ \widetilde{E} & \to & * \\ \downarrow & & \downarrow \mathrlap{\eta} \\ * & \to & E^*(\Pi_1(E)) \mathrlap{= E^*(E_!(*))}}

in EE. Note that the bottom map *E *(E !(*))*\to E^*(E_!(*)) is E *E^* applied to the unique map *E !(*)*\to E_!(*), while the right-hand map is the unit of the adjunction E !E *E_!\dashv E^*.

In order to see that this is a sensible definition, we first observe that E (1)E^{(1)} is itself locally 1-connected (since it is etale over EE). Moreover, it is actually 1-connected, which is equivalent to saying that E !(E˜)=*E_!(\widetilde{E}) = *. This is because the “Frobenius reciprocity” condition for the adjunction E !E *E_!\dashv E^* (which is equivalent to saying that E !E_! is GpdGpd-indexed) applied to the defining pullback of E˜\widetilde{E} implies that we also have a pullback

E !(E˜) E !(*) id * E !(*)\array{ E_!(\widetilde{E}) & \to & E_!(*) \\ \downarrow & & \downarrow \mathrlap{id} \\ * & \to & E_!(*)}

which clearly implies that E !(E˜)=*E_!(\widetilde{E}) = *.

Thus, E (1)E^{(1)} is a connected and simply connected space with a local homeomorphism to EE, but is it a covering space? In other words, is it locally trivial? Since we have supposed that EE is locally 1-connected, as a (2,1)-category it can be generated by 1-connected objects, i.e. objects UU such that E !(U)*E_!(U)\simeq *. In particular, we have a 1-connected object UU and a regular 1-epic? U*U\to *.

We claim that if UU is any 1-connected object of EE, then E˜\widetilde{E} is trivialized (or split) over UU, in that U×E˜U\times \widetilde{E} is equivalent, over UU, to U×E *SU\times E^*S for some SGpdS\in Gpd. For pulling back the defining pullback to UU, we obtain

U×E˜ U U×η U U×E *(Π 1(E)).\array{ U\times \widetilde{E} & \to & U \\ \downarrow & & \downarrow \mathrlap{U\times \eta} \\ U & \to & U\times E^*(\Pi_1(E)).}

But U×E *(Π 1(E))(E/U) *(Π 1(E))U\times E^*(\Pi_1(E)) \cong (E/U)^* (\Pi_1(E)), so to give a map U(E/U) *(Π 1(E))U \to (E/U)^* (\Pi_1(E)) over UU is the same as to give a map (E/U) !(*)Π 1(E)(E/U)_!(*) \to \Pi_1(E) in GpdGpd. But (E/U) !(*)*(E/U)_!(*)\simeq *, since UU is 1-connected, and Π 1(E)\Pi_1(E) is connected, so there is only one such morphism. Therefore, the two maps UU×E *(Π 1(E))U\to U\times E^*(\Pi_1(E)) in the pullback above are in fact the same, and in particular both are the pullback to E/UE/U of the map *Π 1(E)*\to \Pi_1(E). Thus, U×E˜U\times \widetilde{E} is equivalent to (E/U) *(S)U×E *(S)(E/U)^*(S) \cong U\times E^*(S), where S=Ω(Π 1(E))S= \Omega(\Pi_1(E)) is the loop object of Π 1(E)\Pi_1(E), i.e. what we might call the fundamental group of the connected (2,1)-topos EE.

Therefore, since E˜\widetilde{E} is trivialized over any 1-connected object, and EE is generated by 1-connected objects, E˜\widetilde{E} is locally trivial. Moreover, since ** is a discrete object of EE, so is E˜\widetilde{E}. Thus, if we specialize all this to the case E=Sh (2,1)(X)E=Sh_{(2,1)}(X) of (2,1)-sheaves on a topological space, then we conclude that E˜\widetilde{E} is an honest 1-sheaf on XX which, when regarded as a local homeomorphism over XX, is locally trivial (hence a covering space), connected, and 1-connected—i.e. a universal cover of XX.

Examples

Example

(real line is universal covering of circle)

Let

  1. 1\mathbb{R}^1 be the real line with its Euclidean metric topology;

  2. S 1{x 2|x=1} 2S^1 \coloneqq \left\{ x\in \mathbb{R}^2 \;\vert\; {\Vert x\Vert} = 1 \right\} \subset \mathbb{R}^2 be the circle with its subspace topology induced from the Euclidean plane.

Consider the continuous function

1 p S 1 2 t (cos(2πt),sin(2πt)). \array{ \mathbb{R}^1 &\overset{p}{\longrightarrow}& S^1 \subset \mathbb{R}^2 \\ t &\mapsto& (\cos(2\pi t), \sin(2\pi t)) } \,.

This exhibits the universal covering space (def. 1) of the circle.

Proof

Let pS 1p \in S^1 be any point. It is clear that we have a homeomorphism of the form

S 1p (0,1). \array{ S^1 \setminus p &\overset{\simeq}{\longrightarrow}& (0, 1) } \,.

and hence a homeomorphism of the form

S 1×Disc() (0,1)×Disc() p 1(S 1{p}) (t,n) (cos(2πnt),sin(2πnt)). \array{ S^1 \times Disc(\mathbb{Z}) &\simeq& (0,1) \times Disc(\mathbb{Z}) &\overset{\simeq}{\longrightarrow}& p^{-1}(S^1 \setminus \{p\}) \\ && (t,n) &\mapsto& (cos(2\pi n t), \sin(2\pi n t)) } \,.

Now for p 1p 2p_1 \neq p_2 two distinct point in S 1S^1, their complements constitute an open cover

{S 1p iS 1} i{1,2} \left\{ S^1 \setminus p_i \subset S^1 \right\}_{i \in \{1,2\}}

and so this exhibits p: 1S 1p \colon \mathbb{R}^1 \to S^1 as being covering spaces.

Now

  1. S 1S^1 is path-connected and locally path connected (this example);

  2. 1\mathbb{R}^1 is simply connected (this example).

Therefore pp exhibits 1\mathbb{R}^1 as a universal covering space of S 1S^1, by def. 1.

References

See the references at covering space.

Revised on July 20, 2017 03:36:58 by Urs Schreiber (46.183.103.8)