Equivalence of the two definitions of the fundamental group of a CW complexfundamental group of a CW complex

Appendix: Cayley graphs

3.4 Equivalence of the two definitions of the fundamental group of a CW complexfundamental group of a CW complex

Now we will show that our two definitions of fundamental group agree. Let X be an oriented CW complex and let v be a vertex of X. Just until after Theorem 3.4.1, we will denote by π₁^edge(X, v) the “edge path” fundamental group defined in Sect. 3.1, and byπ^top₁ (X, v) the “topological” fundamental

3.4 Equivalence of the two definitions 97 group defined in Sect. 3.3. If ω1, ω2 : I¹ → X are maps such that ω1(1) = ω2(−1), their product ω1.ω2 :I¹ →X is defined by the same formula used for loops, namely,t→ω1(2t+ 1) whent≤0, andt→ω2(2t−1) whent≥0.

We callω1, ω2, ω1.ω2, etc., “paths” even though the domain isI¹rather than I. A characteristic maph: (I¹,I^•1)→ (e¹_α,^•e¹_α) of a 1-celle¹_α defines a path h: I¹ →X. For each non-degenerate edgeτi (= oriented 1-cell) in X, pick a characteristic maphτ_i for the underlying 1-cell representing the orientation ofτi; regardhτ_i as a path in X. For each degenerate edgeτi, let hτ_i be the constant path at the point τi. With each edge loop τ := (τ1, . . . , τk) at v, associate the product path h_τ := (. . .((h_τ1.h_τ2).h_τ3). . . .).h_τ

k. Thus h_τ is a loop atv, aparametrization of τ.

Theorem 3.4.1.This association induces an isomorphismα:π₁^edge(X, v)→ π₁^top(X, v).

Proof. We claim π^top₁ ( ˜X, v) is trivial. Thus the isomorphism χ of 3.3.3 is defined. By 3.2.3 and 3.2.9, the isomorphismχis well defined. Letα=χ◦χ⁻¹. Thenαis indeed induced by the associationτ→hτ.

G

χ

zzuuuuuuuuuu _χ

$$H

HH HH HH HH

π^edge₁ (X, v) _α //π^top₁ (X, v)

It remains to prove the claim. Let ω : (I¹,I^•1) → ( ˜X,˜v) be a loop. By 1.4.3,ω is homotopic to a loop in ˜X¹. Clearly, any loop in ˜X¹ is homotopic to a loop of the formhτ for some edge loopτ in ˜X at ˜v. By 3.2.7,τ can be transformed into the trivial edge loop by elementary equivalences. If σ is a reduction ofτ, it is clear that the loopshσandhτ are homotopic. Ifσdiffers fromτ by a formal move across a 2-cell, so that (in the notation of Sect. 3.1) σ.τ⁻¹ = λ.µ1.ν.ν⁻¹.µ2.λ⁻¹, then hσ.h_τ−1 h_σ.τ−1 hλ.hµ₁.µ₂.h_λ−1, where µ1.µ2 is an edge loop representing some ∆e²_γ. Careful consideration of the definition of ∆e²_γ will convince the reader that h_µ1.µ2 is homotopic to the constant loop at the final point ofλ. Henceh_σ h_τ. Finally, note that ifτ is the trivial edge loop at ˜v,h_τ is the constant loop at ˜v.

From now on, we will writeπ1(X, v) for both groups, understanding them to be identified by the isomorphismαof 3.4.1.

Just as with cellular homology, it follows that we may speak of π1(X, v) without reference to a particular CW complex structure on X, and that, by 3.1.16, different CW complex structures lead to different presentations of the same group. To take a simple example, choose for Sⁿ the CW complex structure consisting of one vertex,v, and onen-cell. Then π₁(S¹, v)∼=Zby 3.1.9, and forn ≥2, π₁(Sⁿ, v) = {1} by 3.1.10. For this, the combinatorial approach is simplest. On the other hand, the topological approach allows a trivial proof thatπ1 preserves products:

Proposition 3.4.2.Let {(Xα, xα)}^α∈A be a family of pointed spaces and let pβ :

α

Xα → Xβ be the projection map. Then p# : π1

α

Xα,(xα)

→

α

π₁(X_α, x_α)is an isomorphism, wherep(x) := (p_α(x)).

Hence, writingTⁿfor then-fold product of copies ofS¹(Tⁿis then-torus), we getπ₁(Tⁿ, v)∼=Zⁿ.

Of course, we have seen another proof thatπ₁(S¹, v) isZ. GiveRthe CW complex structure with vertex setZand 1-cells [n, n+ 1] for eachn∈Z. Let Zact onRbyn.x=x+n. ThenRis a simply connected freeZ-CW complex whose quotient is homeomorphic to S¹. By 3.2.3,π1(S¹, v) is isomorphic to Z. Similar remarks apply toTⁿ since its universal cover isRⁿ.

Since we are concerned with presentations of groups, it is useful to refor- mulate 3.1.16 topologically. Let (X, v) be a pointed CW complex. By 3.1.16, π1(X¹, v) is a free group. Choose an attaching map fγ : S¹ → X¹ for each 2-cell e²_γ of X. Then f_γ ◦¯k₁ : I¹ → X¹ is a loop, where I¹ −→^k¯1 S¹ is the quotient map I¹ −→ I¹/^•I¹ −→^k1 S¹ chosen once and for all in Sect. 2.5. Choose a path λγ in X¹ from v to fγ(1) := fγ ◦k¯1(−1). Let gγ = [λγ.(fγ◦k¯1).λ⁻_γ¹]∈π1(X¹, v).

Proposition 3.4.3.Let i : (X¹, v) → (X, v). The homomorphism i_# : π₁(X¹, v)→ π₁(X, v)is an epimorphism whose kernel is the normal closure

of{g_γ |e²_γ is a2-cell ofX}.

By 3.3.1 and 3.3.2 we have:

Proposition 3.4.4.Let f :X →Y be a homotopy equivalence between CW complexes taking the vertexv to the vertexw. Then f#:π1(X, v)→π1(Y, w)

is an isomorphism.

We can now improve the theory of covering spaces of CW complexes begun in Sect. 3.2.

Let X be a path connected CW complex and let v be a vertex of X. Recall from Sect. 3.2 that for each subgroup H ≤ π1(X, v) there is a cov- ering projection qH : ( ¯X(H),v)¯ → (X, v) such thatπ1( ¯X(H),v)¯ ∼= H. We now have the language to strengthen that statement. The universal cover ˜X is a free left π1(X, v)-CW complex, hence also a free left H-CW complex.

Moreover, ˜X is simply connected. Therefore there is a canonical isomorphism χ : H → π1( ¯X(H),v) defined in Sect. 3.2. On the other hand,¯ qH induces a homomorphismqH# : π1( ¯X(H),¯v)→π1(X, v).

Proposition 3.4.5.qH#◦χ=inclusion:H →π1(X, v).

Corollary 3.4.6.q_H# is a monomorphism whose image is H.

3.4 Equivalence of the two definitions 99 It remains to show that these covering spaces ¯X(H) are essentially the only path connected covering spaces ofX. For this we need to know that CW complexes are locally path connected. This is easily proved in three steps (the details are an exercise):

Proposition 3.4.7.(i)Bⁿis locally path connected;(ii)ifY is obtained from the locally path connected spaceA by attachingn-cells, thenY is locally path connected;(iii)every CW complex is locally path connected.

Proposition 3.4.8.Letq: (E, e)→(X, v)be a covering projection, whereE is path connected andX is a CW complex. LetH =q#(π1(E, e))≤π1(X, v).

Then there is a homeomorphismh: (E, e)→( ¯X(H),v)¯ making the following diagram commute. In particular, q_# is a monomorphism.

(E, e) ^h //

qHHHHH##

HH

HH ( ¯X(H),v)¯

q_H

yyssssssssss

(X, v)

Proof. Apply 3.3.4 toqand toq_H. Uniqueness implies that the resulting lifts

are mutually inverse.

Pointed covering projections p1 : (E1, e1) → (X, x) and p2 : (E2, e2) → (X, x), whereE1andE2are path connected, are said to be equivalent if there is a homeomorphismh: (E1, e1)→(E2, e2) such thatp1=h◦p2.

So, up to equivalence, there is a bijection between the path connected pointed covering spaces of the pointed CW complex (X, v) and the subgroups ofπ₁(X, v), such that the fundamental group of the covering space correspond- ing to H is mapped isomorphically to H. Proposition 3.4.8 also completes the explanation given in Sect. 3.2 for the name “universal cover” – a path connected pointed covering space of (X, v) which covers all path connected pointed covering spaces of (X, v).

We close with two very useful theorems linking topology and group theory, one a special case of the other. Their proofs are left as exercises.

Theorem 3.4.9.Let (X, A) be a pair of path connected CW complexes. Let i : (A, v) → (X, v) be the inclusion of a subcomplex and let p : ( ˜X,v)˜ → (X, v)be the universal cover. (i) There is a bijection between the set of path components ofp⁻¹(A) and the set of cosets

{g.(imagei#)|g∈π1(X, v)}.

(ii)If A˜v denotes the path component of p⁻¹(A) containingv, then˜ π1(A˜v,v)˜

is isomorphic to ker i#.

Theorem 3.4.10.With notation as in Theorem 3.4.9, let ( ¯X(H),v)¯ be the pointed covering space corresponding toH.(i)There is a bijection between the set of path components ofq_H⁻¹(A)and the set of double cosets

{H.g.(imagei#)|g∈π1(X, v)}

.(ii)IfAv¯denotes the path component ofq_H⁻¹(A)containingv, then¯ π1(A¯v,v)¯

is isomorphic toi⁻_#¹(H).

In summary: (i) π₀(q⁻_H¹(A)) ∼= H\π₁(X, v)/imagei_#, and (ii) A_v¯ = A(i¯ ⁻_#¹(H)).

Exercises

1. Prove 3.4.7.

2. Theorem 3.3.3 establishes an isomorphism between the group of covering trans- formations of ˜X andπ₁(X, v). For a subgroupH ≤π₁(X, v) establish a similar isomorphism between the group of covering transformations of ¯X(H) and the groupN(H)/H whereN(H) denotes thenormalizer ofH, i.e., the largest sub- group ofπ₁(X, v) in whichH is normal. In particular, whenH is normal²⁰ in Gthe group of covering transformations is isomorphic toG/H.

3. Prove 3.4.9 and 3.4.10.

4. Prove that ifGis finitely generated there are only finitely many subgroups of a given finite index inG.

5. Prove that the intersection of finitely many subgroups of finite index inGhas finite index inG.

6. Prove that if H has finite index in Gthe set of subgroups conjugate toH is finite.

7. Describe a CW complex X having one vertex v, one 1-cell and one 2-cell so thatX− {v}is a path connected space whose fundamental group is not finitely generated.

8. Prove the following transitivity property of regular (= normal) covering projec- tions: given two cellse₁ande₂mapped to the same cell ofX, there is a covering transformation takinge₁ontoe₂.

20WhenH is normal in Gthe covering projection q_# : ¯X(H) →X is said to be regular ornormal.

4