nLab
covering space

Contents

Definition

A covering space (or wrapping space) is a bundle p:EBp: E \to B in Top which is locally trivial and with discrete fiber. That is, a map p:EBp: E \to B is a covering space over BB if for each point xBx \in B, there exists an open neighborhood UU of xx evenly covered by pp: the pullback of pp over UU is isomorphic to a product bundle with discrete fiber E x=p 1(x)E_x = p^{-1}(x):

U×E x p 1(U) E π p U B\array{ U \times E_x & \cong & p^{-1}(U) & \to & E\\ & \pi \searrow & \downarrow & & \downarrow p\\ & & U & \hookrightarrow & B }

(the square is a pullback and the isomorphism maps (x,eE x)e(x, e \in E_x) \mapsto e).

Covering spaces over BB form an evident full subcategory Cov/BTop/BCov/B \hookrightarrow Top/B. These can be put together to form a replete, wide subcategory CovCov of TopTop, so that Cov/BCov/B is an over category, just as the notation suggests.

Remarks

  • Different points in BB may have non-isomorphic fibers. However, if open sets UU and VV are evenly covered by p:EBp: E \to B and have nonempty intersection, then there are canonical identifications

    E xE zE yE_x \cong E_z \cong E_y

    between typical fibers over xUx \in U, yVy \in V, and zUVz \in U \cap V. If BB is path-connected, then all the fibers match up to isomorphism (by the unique path-lifting lemma; see below).

  • Fibers may be empty. Some authors choose to forbid that, adding the condition that pp be a surjection, but the resulting category of covering projections over a space BB is not as nice as it would be without that condition.

  • The terms “covering space” and “covering projection”, while traditional, are certainly not optimal: they mislead by being too close to (open) “coverings”. James Dolan has suggested “wrapping space” as an alternative (as in the image of thread wrapping around a spool, to evoke the archetypal example of a covering projection, p:S 1:xexp(ix)p: \mathbb{R} \to S^1: x \mapsto \exp(i x)).

Relation to etale spaces

Every covering space (even in the more general sense not requiring any connectedness axiom) is an etale space, but not vice versa:

  • for a covering space the inverse image of some open set in the base BB needs to be, by the definition, a disjoint union of homeomorphic open sets in EE; however the ‘size’ of the neighborhoods over various ee in the same stalk required in the definition of étalé space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.

  • even if the the stalks of the etale space are finite, it need not be locally trivial. For instance the disjoint union iUi\coprod_i Ui of a collecton of open subsets of a space XX with the obvious projection ( iU i)X(\coprod_i U_i) \to X is etale, but does not have a typical fiber: the fiber over a given point has cardinality the number of open sets U iU_i that contain this particular point.

Fundamental theorem of covering spaces

The connection between covering spaces over BB and the fundamental group π 1(B)\pi_1(B) (for BB a connected space) is very old and runs very deep. An updated account involves shifting attention to representations of the fundamental groupoid Π 1(B)\Pi_1(B) (regardless of connectedness); we give a brief outline of the theory here.

Under some technical topological assumptions on the space BB, the fundamental theorem can be stated thus:

Theorem

The category of covering spaces Cov/BCov/B is equivalent to the category of functors Π 1(B)Set\Pi_1(B) \to Set.

More precisely, there is a functor

Fiber:Cov/BSet Π 1(B),Fiber: Cov/B \to Set^{\Pi_1(B)},

sending a covering space p:EBp: E \to B to the functor which maps the object bΠ 1(B)b \in \Pi_1(B) to the fiber E b=p 1(b)E_b = p^{-1}(b). Given a map [ϕ]:bc[\phi]: b \to c in Π 1(B)\Pi_1(B), where ϕ\phi is a path from bb to cc, the unique path-lifting lemma says that for any eE be \in E_b, there exists a unique fill-in ϕ˜:IE\tilde{\phi}: I \to E, that is the diagonal arrow making the following diagram commute:

{0} e E i p I ϕ B\array{ \{0\} & \stackrel{e}{\to} & E\\ i \downarrow & \nearrow & \downarrow p \\ I & \overset{\phi}{\to} & B }

and we define Fiber([ϕ]):E bE cFiber([\phi]): E_b \to E_c as the map sending eE be \in E_b to ϕ˜(1)E c\tilde{\phi}(1) \in E_c.

The precise statement of the theorem above is that if BB is locally path-connected and semi-locally simply-connected, then the functor FiberFiber is an equivalence of categories.

In fact, in the proof of this theorem one establishes an adjoint equivalence: one constructs a left adjoint to FiberFiber,

Set Π 1(B)Cov/B,Set^{\Pi_1(B)} \to Cov/B,

via a tensor product or weighted colimit construction, namely the one that extends (by left Kan extension along the Yoneda embedding on Π 1(B) op\Pi_1(B)^{op}) the functor

Π 1(B) opCov/B\Pi_1(B)^{op} \to Cov/B

that sends each object bb of Π 1(B)\Pi_1(B) to a universal covering space B˜ b\tilde{B}_b over the path-component of bb.

We now spell out the details.

In terms of a bar-construction

Given a space BB, let B|B| be BB retopologized with the discrete topology, and consider the pullback in TopTop

Path(B) B I ev 0,ev 1 B×B id×id B×B \array{ Path(B) & \to & B^I & \\ \downarrow & & \downarrow & \langle ev_0, ev_1 \rangle\\ |B| \times B & \overset{id \times id}{\to} & B \times B & }

Let Path¯(B)\overline{Path}(B) be the quotient of Path(B)Path(B) by the equivalence relation “homotopy rel boundary”. We can think of Path¯(B)\overline{Path}(B) as a sum of spaces

bBB˜ b,\sum_{b \in B} \tilde{B}_b,

fibered in the obvious way over B|B| (the set of all basepoints bb), where B˜ b\tilde{B}_b is the space of paths in BB which begin at bb, modulo homotopy-rel-boundary. The space B˜ b\tilde{B}_b can be thought of the universal covering space over the connected component of a point bBb \in B, considered as a space based at bb.

We have a span

Path¯(B) B B\array{ & & \overline{Path}(B) & & \\ & \swarrow & & \searrow & \\ |B| & & & & B }

with an obvious (contravariant) composition action compcomp of the fundamental groupoid Π 1(B)\Pi_1(B), itself regarded as a span

Π 1(B) B B\array{ & & \Pi_1(B) & & \\ & \swarrow & & \searrow & \\ |B| & & & & |B| }

with a monad structure in the bicategory of spans. The action gives a map

comp:Π 1(B)× BPath¯(B)Path¯(B),comp: \Pi_1(B) \times_{|B|} \overline{Path}(B) \to \overline{Path}(B),

of spans from B|B| to BB.

Now suppose given an object FF of Set Π 1(B)Set^{\Pi_1(B)}, i.e., a covariant action of the fundamental groupoid, that is to say a span F:1BF: 1 \to |B| equipped with an action α\alpha of the monad Π 1(B):BB\Pi_1(B): |B| \to |B| in Span(Top)Span(Top). The data of a right-handed action compcomp on Path¯(B)\overline{Path}(B) and the left-handed action α\alpha on FF gives rise to a two-sided bar construction

B(Path¯(B),Π 1(B),F),B(\overline{Path}(B), \Pi_1(B), F),

which here is a simplicial object in the category of spans from 11 to BB, whose two face maps from degree 1 to degree 0 take the form:

F× BΠ 1(B)× BPath¯(B) F× Bcomp α× BPath¯(B) F× BPath¯(B) \array{ & F \times_{|B|} \Pi_1(B) \times_{|B|} \overline{Path}(B) & \\ & {}^{\mathllap{F \times_{|B|} comp}}\downarrow \downarrow^{\mathrlap{\alpha \times_{|B|} \overline{Path}(B)}} & \\ & F \times_{|B|} \overline{Path}(B) & }

The coequalizer of this pair provides a canonical augmentation of the two-sided bar construction, and may be called the tensor product

Path¯(B) Π 1(B)F\overline{Path}(B) \otimes_{\Pi_1(B)} F

(the seemingly opposite placement of the two tensor factors, as compared against the span constructions above, is simply an artifact of the discrepancy between diagrammatic order of composition, and the traditional order in which right actions are covariant and left actions contravariant).

As a span from 11 to BB, that is as a bundle over BB, this tensor product is indeed a covering space over BB, assuming that BB is locally connected and semi-locally simply connected. Finally, the functor

Path¯(B) Π 1(B):Set Π 1(B)Cov/B\overline{Path}(B) \otimes_{\Pi_1(B)} -: Set^{\Pi_1(B)} \to Cov/B

is under these conditions quasi-inverse to the fiber functor

Fiber:Cov/BSet Π 1(B)Fiber: Cov/B \to Set^{\Pi_1(B)}
Remark

An abstract way of considering the functor FiberFiber is that it is obtained by homming:

Fiber(p:EB)(b)=(Cov/B)(B˜ b,p)Fiber(p: E \to B)(b) = (Cov/B)(\tilde{B}_b, p)

and this forces its left adjoint to be given by the tensor product construction described above.

Special case: the universal covering space

As a special case, consider the permutation representation Π 1(B)Set\Pi_1(B) \to Set given by the discrete fibration

cod:Π 1(B)Bcod: \Pi_1(B) \to |B|

David Roberts: shouldn’t such a discrete fibration then give rise to a functor BSet|B| \to Set? If you mean Mor(Π 1(B))Mor(\Pi_1(B)), then this could probably be described as the total tangent groupoid, which is the action groupoid for the action of Π 1(B)\Pi_1(B) on itself.

Todd Trimble: I didn’t make myself clear then. Recall that if CC is an internal category in a category EE (with E=SetE = Set in this discussion), then one defines E CE^C by taking its objects to be internal discrete fibrations, defined as arrows FC 0F \to C_0 equipped with the data of an action by the internal category CC, considered as a monoid in spans from C 0C_0 to C 0C_0. (This is a standard usage of the term “discrete fibration”; see Johnstone’s Topos Theory for instance.) Looking over this again, I guess I really should have had F=Mor(Π 1(B))F = Mor(\Pi_1(B)), and B|B| here means the underlying set of BB. But hopefully my meaning is now clear.

David Roberts: Yes, I see now.

(as a span from 11 to B|B|) equipped with the obvious (covariant) action of the monad Π 1(B)\Pi_1(B) (as a span from B|B| to B|B|). This is essentially the “regular representation” of the fundamental groupoid. The tensor product of the previous section,

Path¯(B) Π 1(B)cod,\overline{Path}(B) \otimes_{\Pi_1(B)} cod,

is a way of realizing the universal covering space over BB.

Here is a way of thinking of this construction which links it to the description of universal bundles by Roberts and Schreiber, which is based on considering tangent spaces of the fundamental groupoid. If the fundamental groupoid G=Π 1(B)G = \Pi_1(B) is connected, its universal bundle (as a fibration of groupoids) may be realized as the “tangent groupoid at bb” or slice

T b(G):=(b/G)GT_b(G) := (b/G) \to G

for a chosen basepoint bBb \in B. Note that this slice groupoid is the pullback

(b/G) G I ev 0 {b} G \array{ (b/G) & \to & G^I & \\ \downarrow & & \downarrow & ev_0\\ \{b\} & \to & G & }

with II the groupoid (01)(0 \overset{\sim}{\to} 1). This is then a groupoid over GG by the restriction of ev 1ev_1.

Since the set of arrows of GG is obtained as a quotient of the set of paths in BB, it inherits naturally a topology (a quotient of the compact-open topology on B IB^I) which, together with the given topology on G 0=BG_0 = B, makes GG a topological groupoid. Then we recover the universal covering space B b (1)B^{(1)}_b (I prefer this notation for the 1-connected cover, rather than the usual B˜\tilde{B}, because it generalises to B (n)B^{(n)} for nn-connected covers - DR) over BB by pulling back along the functor BGB \to G, where we consider BB as a topological groupoid with only identity arrows. The assumptions on the topology of BB mean that GG is a locally trivial groupoid? with discrete hom-spaces, which implies that B b (1)B^{(1)}_b is a locally trivial bundle with discrete fibres. Local path-connectedness implies that it is locally trivial, and the local condition on π 1\pi_1 holds if and only if the fibres are discrete - this last result is due to Daniel Bliss.

Remark

Another way to consider the topological conditions on BB is to realise that Π 1(B)\Pi_1(B), with its inherited topology, is equivalent to a topologically discrete groupoid (in some appropriate localisation of the 2-category of topological groupoids) if and only if BB is locally path-connected and semi-locally simply-connected. Otherwise one has to consider the pro-homotopy 1-type of BB, as in the theory of algebraic fundamental groups (recall that varieties with appropriate topologies - e.g Zariski - are topologically badly behaved).

David Roberts: Is there a prodiscrete completion of a topological groupoid? Maybe we need to assume it is locally trivial, so it is weakly equivalent (in the said localised 2-category) to a groupoid enriched in TopTop, considered as being internal to TopTop. We could then talk about quotients by wide subgroupoids being topologically discrete. Or even quotients being discrete and having finite Leinster cardinality?? Hmm

In this analysis, the universal covering space E bE_b of (path-connected) BB is retrieved as the quotient of the space of paths which start at the basepoint bb, modulo homotopy-rel-boundary; the projection to BB takes a class of a path ϕ\phi to its terminal point ϕ(1)\phi(1). This last description is what one would find in any textbook on algebraic topology dealing with covering spaces. This covering space is, strictly speaking, universal among connected covering spaces

More generally, if SBS \subset |B| is a set of basepoints (Thanks, Ronnie Brown! - DR), we can form the pullback

(S/G) G I ev 0 S G \array{ (S/G) & \to & G^I & \\ \downarrow & & \downarrow & ev_0\\ S & \to & G & }

which is again a groupoid over GG by restriction of ev 1ev_1. Then pullback of (S/G)G(S/G) \to G along the inclusion BGB \to G is a covering space which is the sum

B (1)S= bSB b (1) B^{(1)}\langle S\rangle = \sum_{b\in S} B^{(1)}_b

of connected, 1-connected covering spaces based at the points in SS. Thus for not-necessarily-connected BB, taking SS such that it intersects each component of BB once we can get a universal covering space of BB (universal among covering spaces EBE \to B that induce isomophisms Π 0(E)Π 0(B)\Pi_0(E) \to \Pi_0(B)).

This construction is functorlal (for general SBS\subset |B|), since a map (B,S)(B,S)(B,S) \to (B',S') of pairs (remember we are giving S,SS,S' the discrete topology, not the subspace topology) induces a functor of (topological) groupoids Π 1(B)Π 1(B)\Pi_1(B) \to \Pi_1(B'), which by universality of the pulbacks in the above construction gives a map

B (1)SB (1)S B^{(1)}\langle S\rangle \to B'^{(1)}\langle S'\rangle

covering the given map BBB \to B'.

The dependence on basepoints is of course spurious; we can make this explicit by considering the colimit obtained by pasting together the universal covering spaces B b (1)B^{(1)}_b along isomorphisms induced by paths bcb \to c. But this is in effect how our tensor product construction of the universal covering space works: Path¯(B)\overline{Path}(B) is precisely the sum

cBB c (1)\sum_{c \in |B|} B^{(1)}_c

which can be viewed as a topological span from B|B| to BB. The fundamental groupoid acts contravariantly on this sum, and the tensor product

Path¯(B) Π 1(B)(cod:Π 1(B)B)\overline{Path}(B) \otimes_{\Pi_1(B)} (cod: \Pi_1(B) \to |B|)

is the same thing as the coequalizer of the pair of arrows

[ϕ]:bc cB c (1) cB c (1)\sum_{[\phi]: b \to c} \sum_c B^{(1)}_c \overset{\to}{\to} \sum_c B^{(1)}_c

in Top/BTop/B, where one arrow is projection and the other is given by the action of pulling back along classes of paths; this coequalizer is a precise description of the pasting colimit alluded to above. It should be noted that this coequaliser is isomorphic to the covering space B (1)SB^{(1)}\langle S\rangle when SS has one point in each component of BB, but the description as the tensor product is a priori functorial without reference to a set of basepoints.

David Roberts: I think, though, due to the lifting theorems for covering spaces, that given a map f:BBf:B \to B' and basepoint sets SBS \subset |B|, SBS' \subset |B'| that are not necessarily preserved by ff, there should be a unique lift of B (1)SBB^{(1)}\langle S\rangle \to B' to B (1)SB'^{(1)}\langle S'\rangle anyway. This would also make this construction independent, up isomorphism, of the choice of basepoints and probably also functorial.

David Roberts: It won’t be functorial - the lift referred to isn’t unique. The up-to-isomorphism is a non-canonical isomorphism.

(David or Urs: please feel free to sprinkle your own sugar over this, by adapting or even copying what David wrote below based on your paper.)

(David Roberts: unless someone feels the discussion below is essential, it can be deleted.)

David Roberts: My personal favourite way of doing this is to topologise the fundamental groupoid, then form the following strict pullback of topological groupoids

B˜ T bΠ 1(B) B Π 1(B) \array{ \widetilde{B} & \to & T_b\Pi_1(B) \\ \downarrow && \downarrow\\ B & \to &\Pi_1(B) }

where bBb\in B is a chosen basepoint and T bΠ(B)T_b\Pi(B) is the tangent groupoid at the object bb. This links the ideas that the tangent groupoid is the contractible cover of a groupoid, that the fundamental groupoid is the 1-type of a space and the Whitehead construction of connected covers (pull back the path-fibration along the inclusion of a space into the appropriate Postnikov section).

The topology on the fundamental groupoid can either be constructed with the assumption that BB is locally path-connected and semi-locally simply-connected, or be given the quotient topology from the free path space B IB^I. With this inherited topology, the fundamental groupoid is equivalent (in the bicategory of topological groupoids and anafunctors) to the same groupoid considered with the discrete topology if and only if BB satisfies the usual conditions for the universal covering space to exist. Thus even when Π 1(B)\Pi_1(B) is topologised, it still represents a 1-type for nice BB. One thing which interests me, even though I have no idea about how to approach it, is how for general BB the topologised fundamental groupoid can be considered as a pro-homotopy type, that is, the limit of discrete groupoids, taken in the appropriate (bi)category of topological groupoids.

I would like see several expositions of the construction of the universal covering space, since they illustrate different ideas. They seem tautologously related, but things show a bit more of the differences when one passes to bigroupoids.

The universal covering space is

  • the source-fibre (at a basepoint) of the topologised fundamental groupoid

  • the pullback of the tangent groupoid as described above

  • The pullback of the map (s,t):Mor(Π 1(B))Obj(B)×Obj(B)(s,t):Mor(\Pi_1(B)) \to Obj(B)\times Obj(B) along the inclusion {b}×BB×B\{b\}\times B \to B\times B

Todd I’ll get back to writing more of what I had planned soon. I haven’t had a chance to digest what you’re writing yet, but I prefer to proceed without having to choose basepoints. I’d like to get you and Urs to have a look though when I get back to this within a few days.

David: Of course - hence the theorem about functors from the fundamental groupoid and not the fundamental group. This is where the full tangent groupoid comes in: it is the pullback

TG G I dom Obj(G) G \array{ TG & \to & G^I & \\ \downarrow && \downarrow& dom\\ Obj(G) & \to & G & }

or equivalently the slice Obj(G)id GObj(G)\downarrow id_G for an internal groupoid GG (internal in TopTop, but extensions to other categories work too). The tangent groupoid at a point gg is just the subgroupoid of this gotten by pulling back TGObj(G)TG \to Obj(G) along the inclusion {g}Obj(G)\{g\} \to Obj(G). I hadn’t thought about applying this construction to my personal universal covering space recipe, so maybe we need to take the discrete topology on Obj(G)Obj(G). That’s what your pullback square above seems to indicate. Urs’ and my paper [arXiv:0708.1741] has stuff on tangent groupoids for anyone who interested in pitching in.

In terms of homotopy fibers

We want to describe here how the universal covering space of XX is the homotopy fiber of the canonical morphism XΠ 1(X)X \to \Pi_1(X), hence the Π 1(X)\Pi_1(X)-principal bundle classified by this cocycle.

We place ourselves in the context of topological ∞-groupoids and regard both the space XX as well as its path ∞-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 contant 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

References

An account using the concept of covering morphism of groupoids is given in Chapter 11 of the book

The advantage of this approach is that a map of spaces is modelled by a morphism of groupoids, and this has expository advantages. For example, if XX is a space and p:Gπ 1Xp: G \to \pi_1 X is a covering morphism of groupoids, we ask how to topologise Ob(G)Ob(G) so that it could become a covering space of XX. This leads to explicit conditions on XX which are dependent on the morphism pp.

Some of the problems of generalising covering spaces to deal with wild spaces are dealt with in:

  • Jeremy Brazas, Semicoverings, Homology, Homotopy and Applications, 14 (2012), No. 1, 33-63.

Thus semicoverings satisfy the “2 out of three rule”. I.e,, if f=ghf=gh and two of f,g,hf,g,h are semicoverings , then so is the third. This is not true for covering maps.

An account of the traditional way to think of the construction of the universal covering space is

Revised on August 18, 2013 17:58:05 by Ronnie Brown (81.129.231.119)