Combinatorial description of covering spaces

Appendix: Presentations

3.2 Combinatorial description of covering spaces

LetY be a CW complex. Anautomorphism ofY is a homeomorphismh:Y → Y such that whenevereis a cell ofY so ish(e). The group of all automorphisms ofY will be denoted by AutY: the multiplication is composition, (h◦h)(y) = h(h(y)).

LetGbe a group. AG-CWcomplex is a CW complexY together with a homomorphismα:G→AutY. We writeg.yforα(g)(y). The homomorphism αis also called acell-permuting left action ofGonY. ThestabilizerofA⊂Y is GA = {g ∈ G | g(A) = A}. The action is free, and Y is a free G-CW complex, if the stabilizer of each cell is trivial. The action isrigid, andY is a rigid G-CW complex, if the stabilizer of each cell,e, acts trivially one. There

10The interest here is that there are uncountably many isomorphism classes of finitely generated groups – see [106, p. 188].

3.2 Combinatorial description of covering spaces 85 is a sense (explained at the end of Sect. 4.1) in which “rigidG-CW complex”

rather than “arbitraryG-CW complex” is the natural equivariant analog of

“CW complex”.¹¹

An action of G defines an equivalence relation on Y: y1 =y2 modG iff there is someg ∈Gsuch that g.y1=y2. The set of equivalence classes (also calledorbits) is denoted byG\Y: it is given the quotient topology.¹²

Proposition 3.2.1.Let Y be a free G-CW complex. The quotient map q : Y → G\Y =: X is a covering projection, and X admits the structure of a CW complex whose cells are{q(e)|eis a cell ofY}.

Proof. Let Xⁿ = ∪{q(e) | e is a cell of Y having dimension ≤ n}. X⁰ is discrete.X =

n

Xⁿ. IfAindexes then-cells ofY, theG-action onY induces aG-action onA. WriteB=G\A. Consider the following diagram:

Bⁿ(A)

Yⁿ⁻¹ −−−−→^p Yⁿ

⏐⏐

^r ⏐⏐^q| Bⁿ(B)

X^{n−1 p}

−−−−→ Xⁿ

Here, Bⁿ(A) is regarded as a G-space, where g ∈ G maps Bⁿ_α to B_g.αⁿ by the “identity”. ThusBⁿ(A)

Yⁿ⁻¹ is a G-CW complex. The map pis the quotient map described in 1.2.1; it can be chosen to satisfy g.p(x) = p(g.x) for all x∈ Bⁿ(A)

Yⁿ⁻¹ by defining p to agree with a characteristic map hα onBⁿ_α for one α in eachG-orbit ofA, and then defining p on the other B_αⁿ’s using the group action. The mapragrees withqonYⁿ⁻¹and mapsB_αⁿ homeomorphically ontoBⁿ_β, whereβ ∈ Bis the orbit ofα. Sinceris a quotient map (see Sect. 1.1), there is a mappmaking the diagram commute. Moreover, p |Xⁿ⁻¹ = inclusion,p(Sⁿ⁻¹(B))⊂Xⁿ⁻¹, and p maps Bⁿ(B)−Sⁿ⁻¹(B) homeomorphically ontoXⁿ−Xⁿ⁻¹. The mapp is a quotient map, since the other three maps in the diagram are quotient maps. SoXⁿ is obtained from Xⁿ⁻¹by attachingn-cells. Sinceqis a quotient map andY is a CW complex, X has the weak topology with respect to {Xⁿ}. ThusX is a CW complex.

Next we describe the inductive step in the proof thatq is a covering pro- jection. Suppose Un−1 is an open subset of Xⁿ⁻¹ having the property that q⁻¹(Un−1) =q⁻¹(Un−1)∩Yⁿ⁻¹=∪{g.Vn−1|g∈G}, whereVn−1 is open in Yⁿ⁻¹, and g1.Vn−1∩g2.Vn−1 = ∅ whenever g1 = g2. Then certainly Un−1

is evenly covered by q | Yⁿ⁻¹. Let Wn−1 = p⁻¹q⁻¹(Un−1)∩Bⁿ(A), let Sn−1 = p⁻¹(Vn−1)∩Bⁿ(A), and let Tn−1 = (p)⁻¹(Un−1)∩Bⁿ(B). Then

11Looking ahead, and using terminology from Sect. 5.3, we find a useful way of rigidifying: when Y is a regular G-CW complex, its “first derived” |sd Y| is a rigidG-CW complex.

12This actionαis a left action ofGonY, hence the notationG\Y; we reserveY /G for right actions.

Tn−1 is evenly covered by Wn−1 (via r) and r |: Sn−1 → Tn−1 is a homeo- morphism. The setsSn−1andWn−1are open inSⁿ⁻¹(A), whileTn−1is open inSⁿ⁻¹(B). ChooseSn open inBⁿ(A) so thatSn∩Sⁿ⁻¹(A) =Sn−1. Write Wn=

{gSn|g∈G}. ThenWn is open inBⁿ(A) andTn =r(Wn) =r(Sn) is open in Bⁿ(B), while Tn∩Sⁿ⁻¹(B) =Tn−1. We leave it as an exercise to show that Sn can be so chosen that gSn∩Sn =∅ wheneverg ∈ G. Define Vn =p(Sn∪Vn−1) and Un =p(Tn∪Un−1). ThenUn is open inXⁿ and is evenly covered byp(Wn∪Vn−1) =

{gVn|g∈G}.

To see thatqis a covering projection, letx∈X. There is a uniquemand a unique celle^m_δ such that x∈ e^◦m_δ . We have q⁻¹(^◦e^m_δ ) = ∪{g.^◦e^m_γ

δ | g ∈ G} wheree^m_γ

δ is an arbitrarily chosenm-cell overe^m_δ . Clearly,g1.^◦e^m_γ ∩g2.^◦e^m_γ =∅ wheneverg1 =g2. Write Um=^◦e^m_δ . By induction, chooseUm⊂Um+1 ⊂. . . as above. LetU =

k≥m

U_k. ThenU is an open subset ofX evenly covered by

q.

Neither part of 3.2.1 need hold if the action is not free: for example, take G=Z2,Y = [−1,1] with 0-cells at±1 and one 1-cell, and let the non-trivial element ofGact onY byt→ −t. However, we have:

Proposition 3.2.2.Let Y be a rigidG-CW complex and letq:Y →G\Y be the quotient map. ThenG\Y admits a CW complex structure whose cells are {q(e)|eis a cell ofY}.

Proof. Similar to the first part of the proof¹³ of 3.2.1.

The map q in 3.2.2 is not, in general, a covering projection (unlike the situation in 3.2.1): for example, take G = Z2, Y = [−1,1] with the CW structure consisting of 0-cells at−1, 0 and 1, and 1-cells [−1,0] and [0,1]; let the non-trivial element ofGact onY byt→ −t.

We will sometimes refer to then-cells ofX (in 3.2.1 and 3.2.2) as then-cells of Y mod G. We sayY isfinite mod GifG\Y is a finite CW complex.¹⁴

Our next task is to show that when Y is a simply connected free G-CW complex, andvis a vertex ofX :=G\Y, thenπ1(X, v)∼=G. For this we must define lifts of edge paths inX to edge paths inY. Ifτ1 is an edge ofX with initial pointv=q(˜v), there is a unique edge ˜τ1ofY with initial point ˜vwhich maps to τ1, by 2.4.6 and 2.4.7. Call ˜τi the lift of τi at ˜v. By induction we define the lift of an edge path τ := (τ1, . . . , τk) to be the unique edge path

˜

τ:= (˜τ1, . . . ,τ˜k) with initial point ˜v such thatq(˜τ) =τ.

Pick a vertex ˜v∈Y with q(˜v) =v. AssumingY simply connected, define χ : G → π1(X, v) as follows: choose an edge path ˜τ inY from ˜v tog.˜v, and letχ(g) be the element of π1(X, v) represented by the edge loop q(˜τ). Path connectedness ensures that ˜τ exists.

13The reader should consider why the argument would fail without rigidity.

14Alternatively, one says thatGactscocompactly onY.

3.2 Combinatorial description of covering spaces 87 Proposition 3.2.3.χ is well defined and is an isomorphism.

Proof. We first show thatχ is well defined. Let ˜σand ˜τ be edge paths from

˜

v to g.˜v. Since Y is simply connected, ˜σ and ˜τ are equivalent. (To see this [usingfor equivalence] note thatσσ.(g.˜v)σ.(σ⁻¹.τ) = (σ.σ⁻¹).τ τ;

the first and last equivalences are elementary, the other comes from 3.1.3.) Each elementary equivalence on the way from ˜σ to ˜τ induces an elementary equivalence on the way fromq(˜σ) toq(˜τ). Hence [q(˜σ)] = [q(˜τ)].

χ is obviously a homomorphism. Letχ(g) = 1. If ˜τ is an edge path in Y from ˜v tog.˜v, the corresponding edge loopτ atv is equivalent to the degen- erate edge loopv. If two loops atvdiffer by an elementary equivalence, their lifts at ˜v have the same final point. Hence this also holds if they are merely equivalent. It follows that g.˜v = ˜v, hence g = 1. So χ is a monomorphism.

Butχis clearly onto, for given an edge loopτ atv in X, letg.˜v be the final point of ˜τ. Thenχ(g) is the element ofπ₁(X, v) represented byτ.

We immediately conclude, using 3.1.17:

Theorem 3.2.4.Let Y be a simply connected free G-CW complex such that there are only finitely many 1-cells [resp. 1-cells and 2-cells] mod G. Then G

is finitely generated [resp. finitely presented].

We have seen in 3.2.3 that the quotient of a simply connected freeG-CW complex has fundamental groupG. Conversely, given a path connected CW complexX, we now show how to construct a simply connected freeπ1(X, v)- CW complex ˜X having quotientX. This is the “universal cover” construction.

It is common to define ˜X as a quotient space of a function space – an efficient but non-constructive procedure. We prefer to construct ˜X as a CW complex, skeleton by skeleton. We shall see in Chaps. 13, 14 and 16 that even the 1- skeleton and 2-skeleton of ˜X, as constructed here, exhibit interesting “end”

invariants of the groupπ1(X, v), so the work involved in the construction will be worthwhile.

LetX be a path connected CW complex. Choose an orientation forX, a maximal treeT ⊂X, and a vertexv∈X as base point. Writeπ=π₁(X, v).

Giveπthe discrete topology.

Let ˜X⁰=π×X⁰, letp₀: ˜X⁰→X⁰be projection on theX⁰-factor, and let πact on ˜X⁰by ¯g.(g, vα) = (¯gg, vα). Thisπ-action is free andp0is its quotient map;p0is a covering projection. Pick as base vertex ˜v:= (1, v)∈π×X⁰= ˜X⁰. Next, we define the 1-skeleton ˜X¹. Part of the 1-skeleton isπ×T, but we will attach more 1-cells. Let the (already oriented) 1-cells ofX which are not inT be{e¹_β|β ∈ B}. For eachβ∈ B, letgβ ∈πbe the element represented by the edge loopλ.e¹_β.µ⁻¹, whereλandµare the unique reduced edge paths in T fromvto the initial and final points ofe¹_β; see the proof of 3.1.12. By 3.1.16, the elementsg_βgenerateπ. Pick a characteristic maph_β: (B¹, S⁰)→(e¹_β,^•e¹_β) representing the chosen orientation; let f_β :S⁰ →X⁰ be the corresponding attaching map. For eachβ ∈ B and eachg ∈πattach a 1-cell e¹_β,g toπ×T

by the attaching mapfβ,g :S⁰ →X˜⁰⊂π×T,−1→ (g, hβ(−1)) and 1→ (ggβ, hβ(1)). The resulting CW complex is ˜X¹ =

⎛

⎝(π×T)

β,g

B_β,g¹

⎞

⎠/∼, where∼is defined in the obvious way by the mapsfβ,g. Note thatπ×T is a subcomplex of ˜X¹. Now,X¹ =

⎛

⎝T

β

B_β¹

⎞

⎠/∼where ∼is defined by the mapsf_β. The map (π×T)

β,g

B_β,g¹ →T

β

B¹_βwhich is “projection ontoT” onπ×T and is “identity”:B_β,g¹ →B_β¹ onB_β,g¹ induces a mapp₁: ˜X¹→X¹ extendingp0. For each ¯g∈π, the self-homeomorphism ˜d¯gof (π×T)

β,g

B_β,g¹

which is (g, u)→ (¯gg, u) on π×T and is “identity”: B_β,g¹ →B_β,¯¹_gg on B_β,g¹ induces a self-homeomorphismd_¯g of ˜X¹. Clearly,p₁◦d_¯g=p₁. Moreover, the homomorphism ¯g →dg¯ makes ˜X¹ into a free π-CW complex containing the previously defined π-CW complex ˜X⁰ as a π-subcomplex. By 3.2.1, p₁ is a covering projection, and the cells ofX¹ are thep₁-images of the cells of ˜X¹. It is easy to check that ˜X¹is path connected.

We remark that ifX has just one vertex, thenT ={v} and our construc- tion of ˜X¹ is called the Cayley graph¹⁵ of π with respect to the generators {gβ}: a vertex for each element of π, and an edge joining g to ggβ for each g∈πand each (g, β)∈G× B.

Proposition 3.2.5.Let τ be an edge loop at v ∈X, and let τ˜ be the lift of τ with initial point (g, v)∈X˜¹. The final point of τ˜ is(g¯g, v)where ¯g is the element ofπ represented by τ. In particular, either every lift of τ is an edge loop, or none is.

Proof. Let (g, u)∈π×X⁰= ( ˜X¹)⁰. A non-degenerate edgeτ_iinT fromuto wlifts to an edge in ˜X¹from (g, u) to (g, w). Ifτ_β is the edgee¹_β(β∈ B) with the preferred orientation, having initial pointuand final pointw, τ_β lifts to an edge of ˜X¹with initial point (g, u) and final point (ggβ, w);τ_β⁻¹lifts to an edge of ˜X¹with initial point (g, w) and final point (gg_β⁻¹, u). Applying these remarks inductively toτ := (τ₁ⁱ¹, . . . , τ_k^ik), we see that ˜τhas initial point (g, v)

and final point (g¯g, v) as claimed.¹⁶

15The Cayley graph of a group with respect to a finite set of generators is an important construction in group theory. We will see that the number of ends of this graph is a quasi-isometry invariant (Sect. 18.2) and gives information about the structure of the group (Sect. 13.5). It is the basis for the “word metric” on the group (Sect. 9.1) and its geometry determines whether or not the group is

“hyperbolic.” Some examples are discussed in the Appendix.

16To simplify notation, some details are omitted here: when τi is in T the π- coordinate is unchanged; whenτi=τ_β^iβ the π-coordinate is right multiplied by

3.2 Combinatorial description of covering spaces 89 Lethγ : (B², S¹)→(e²_γ,^•e²_γ) be a characteristic map representing the given orientation ofe²_γ. Letfγ :S¹→X¹be the corresponding attaching map. By 3.2.5, any representative edge loopµγ of the cyclic edge loop∆e²_γ lifts to an edge loop in ˜X¹. From this we deduce:

Proposition 3.2.6.There are maps f˜γ :S¹→X˜¹ such that p1◦f˜γ =fγ. If f˜γ is one such, then the others aredg◦f˜γ where g∈π.

Proof. Letµγ = (τ₁ⁱ¹, . . . , τ_m^im) whereτj has the chosen orientation, andij=

±1. Let K be the CW complex structure onS¹ having vertices at the m^th roots of unity (compare 1.2.17). By 1.4.2 and 3.1.1,fγ is homotopic to a map f : S¹ → X¹ such that the restriction of f maps the interior of the j^th 1-cell ofKhomeomorphically ontoτ^◦j with orientation indicated byij; or else m= 1,τ₁is degenerate andf is constant. By 3.2.5,f lifts. By 2.4.7, if ˜f is one lift, then the others ared_g◦f˜ whereg∈π. By 2.4.6, the same is true of

f˜γ.

We now define ˜X² and p2 : ˜X² → X². For each 2-cell e²_γ of X choose hγ, as above; choose a lift ˜fγ offγ, and, for eachg ∈π, attach a 2-celle²_γ,g to ˜X¹ by the attaching map dg◦f˜γ. The resulting CW complex is ˜X² =

X˜¹

γ,g

B_γ,g²

/∼ where ∼ is defined by the maps dg◦f˜γ. Then X² =

X¹

γ

B_γ²

/∼, where∼comes from the mapsf_γ. Just as before, the map X˜₁

γ,g

B_γ,g² →X₁

γ

B_γ² which isp₁on ˜X₁ and is “identity”:B²_γ,g →B_γ² onB_γ,g² induces a mapp₂: ˜X²→X² extending p₁. And, just as before, the freeπ-action on ˜X¹extends to make ˜X²into a freeπ-CW complex for which p2 is the quotient map. By 3.2.1, p2 is a covering projection, and the cells of X²are thep2-images of the cells of ˜X².

Theorem 3.2.7.X˜² is simply connected.

Proof. Let ˜τ be an edge loop in ˜X² at ˜v. Then τ:=p2(˜τ) is an edge loop in X at v. By 3.2.5,τ represents 1 ∈π1(X, v). Recall that equivalence of edge loops is defined in terms of elementary equivalences each of which is either a reduction or a formal move across a 2-cell (see Fig. 3.1). We say thatτ is of distance≤nfrom the trivial edge loop, (v), if it is possible to pass fromτ to (v) bynelementary equivalences. We prove, by induction onn, that ifτ is of distance≤nfrom (v) then ˜τis of distance≤nfrom (˜v). Ifn= 0,τ= (v) and

˜

τ = (˜v). The induction is completed by observing that ifτ differs fromσ by gⁱ_β^β. The resulting ¯gis indeed the element ofπrepresented byτ, as we saw in the proof of 3.1.16.

one elementary equivalence, then the same is true of the difference between

˜

τ and ˜σ. Indeed, when the difference is a reduction, this is clear. When the difference is an elementary equivalence across a 2-celle²_γ, this follows from the fact that ifµis an edge loop representing∆e²_γ then for eachg∈π, some lift

ofµrepresents∆e²_γ,g.

A covering transformation (ordeck transformation) of a covering projec- tion p: E → B is a homeomorphism d : E → E such that p◦d =p. The covering transformations form a group of homeomorphisms (with composition as the group multiplication).

In the present case, p2 : ˜X² → X² is a covering projection, and the elementsg ofπgive rise to covering transformationsdg :t→g.t. Sincep2 is the quotient map of theπ-action, these are the only covering transformations.

Note that wheng= ¯g∈π,dg=dg¯.

Here is a general property of covering projections, which we are about to use (compare 3.3.4):

Proposition 3.2.8.Let p: E →B be a covering projection, let n≥2, and letg:Sⁿ→B be a map. There is a map ˜g:Sⁿ→E such thatp◦g˜=g (call such ˜g a “lift” of g) and any other lift of g has the form d◦˜g where d is a

covering transformation ofp.

Now we are ready to definep: ˜X →X extendingp2. By induction, assume that for somen≥2, a freeπ-CW complex ˜Xⁿhas been defined whose quotient map isp_n : ˜Xⁿ →Xⁿ. As above, we denote by d_g : ˜Xⁿ →X˜ⁿ the covering transformation corresponding tog∈π. By 3.2.1, the cells of Xⁿ are thep_n- images of the cells of ˜Xⁿ. Choose a characteristic map h_δ : (Bⁿ⁺¹, Sⁿ) → (eⁿ⁺¹_δ ,^•eⁿ⁺¹_δ ) representing the given orientation of each (n+ 1)-cell,eⁿ⁺¹_δ , of X. Letf_δ :Sⁿ →Xⁿ be the corresponding attaching map. By 3.2.8, there is a lift ˜f_δ :Sⁿ →X˜ⁿ, and all lifts have the form d_g◦f˜_δ where g ∈π. Attach an (n+ 1)-celleⁿ⁺¹_δ,g to ˜Xⁿ by the attaching map dg◦f˜δ, for eachg∈π. The resulting CW complex is ˜Xⁿ⁺¹=

⎛

⎝X˜ⁿ

δ,g

Bⁿ⁺¹_δ,g

⎞

⎠/∼where∼is defined by

the mapsd_g◦f˜_δ.Xⁿ⁺¹=

Xⁿ

δ

B_δⁿ⁺¹

/∼where∼comes from the maps f_δ. We definep_n+1 : ˜Xⁿ⁺¹ →Xⁿ⁺¹ extending p_n, and we extend the freeπ- action on ˜Xⁿto a freeπ-action on ˜Xⁿ⁺¹ just as before. This is easily seen to complete the induction. Let ˜X =

n

X˜ⁿ. Define p: ˜X →X byp|X˜ⁿ =pn. By 3.1.7 and 3.2.7, ˜X is simply connected. Summarizing:

Proposition 3.2.9.Given a path connected oriented CW complexX, a vertex v ∈ X, and a maximal tree T in X, the above construction yields a simply connected free π1(X, v)-CW complex X˜ and a covering projection p: ˜X→X

3.2 Combinatorial description of covering spaces 91 which is the quotient map of theπ1(X, v)-action. Moreover, the cells ofX are

the p-images of the cells of X.˜

We will review in Sect. 3.3 the well-known fact that this action can be defined in a purely topological manner.

Remark on Notation. For n ≥2, ( ˜X)ⁿ = (Xⁿ)˜. But for n = 0 or 1, these can be different; for example, considerX=B². In ambiguous cases ˜Xⁿ will always mean ( ˜X)ⁿ.

Propositions 3.2.9 and 3.1.8 imply:

Corollary 3.2.10.For any group G, there exists a simply connected free G-

CW complex.

We will recall in Sect. 3.3 that the simply connected covering space ˜X is, in a certain sense, unique and is a covering space of all other path connected covering spaces of X. Anticipating that, we call ˜X the universal cover ofX (remembering that our particular construction of ˜X appears to depend on many choices).

Knowing ˜X, we can easily construct a path connected covering space ofX with any subgroupH ofπ₁(X, v) as fundamental group. Let ¯X(H) =H\X˜. Consider the following diagram:

X˜

p



p_H

""

DD DD DD DD

X oo

q_H X(H¯ )

Here,p_His the quotient map. The construction of ˜X gave that space a natural base point ˜v= (1, v). We give ¯X(H) the base point ¯v=p_H(˜v).

Proposition 3.2.11.There is a mapqH making this diagram commute. Both pH and qH are covering projections. X¯(H) admits a CW complex structure whose cells are the pH-images of the cells of X. The cells of˜ X are theqH- images of the cells ofX¯(H).π1( ¯X(H),v)¯ ∼=H.

Proof. There is obviously a functionqH making the diagram commute;pH is a quotient map by definition, soqH is continuous. It is not hard to show that an open subset ofX evenly covered by pis evenly covered by qH. By 3.2.1, the CW complex structures on ¯X(H) andX are as claimed. The isomorphism ofπ1( ¯X(H),v) and¯ H comes from 3.2.3.

Here is a well-known corollary.

Theorem 3.2.12.Every subgroup of a free group is free.¹⁷

17This, together with Exercise 6, is the Nielsen-Schreier Subgroup Theorem.

Proof. LetF be a free group and letH ≤F be a subgroup. By 3.1.8, there is a 1-dimensional CW complex X (having exactly one vertex v) such that π1(X, v) is isomorphic toF. Form the covering space ¯X(H); by 3.2.11, it is a CW complex whose fundamental group is H. Being a covering space of a 1-dimensional complex, ¯X(H) is 1-dimensional. By 3.1.16,H is free.

If we letG=π₁(X, v) and consider the subgroupH ≤G, we may ask: what special property does ¯X(H) exhibit when the index [G:H] ofHinGis finite?

To answer this, we observe that the isomorphismχof 3.2.3 actually defines a bijection betweenG and the setp⁻¹(v)⊂X, under which˜ g∈Gis mapped tog.˜v∈p⁻¹(v). The action ofH onp⁻¹(v) partitionsp⁻¹(v) into equivalence classes in bijective correspondence with the cosets {Hg |g ∈ G}. These are also in bijective correspondence withq⁻_H¹(v)⊂X¯(H), whereqH: ¯X(H)→X is as in 3.2.11. This proves:

Proposition 3.2.13.If [G: H] =n≤ ∞ then the covering projection q_H : X(H¯ )→X is an n to 1 function. If X is a finite CW complex, then X¯(H)

is finite iff H has finite index inG.