Show that the existence of maximal trees is equivalent to the Axiom of Choice

In this section we introduce a class of spaces whose homotopy type depends only on their fundamental group. These spaces arise many places in topology, especially in its interactions with group theory.

A path-connected space whose fundamental group is isomorphic to a given group G and which has a contractible universal covering space is called a K ( G , 1) space. The

‘1’ here refers toπ₁. More generalK(G, n) spaces are studied in§4.2. All these spaces are called Eilenberg–MacLane spaces, though in the casen= 1 they were studied by

Hurewicz before Eilenberg and MacLane took up the general case. Here are some examples:

E

xample 1B.1. S¹ is a K(Z, 1). More generally, a connected graph is a K(G, 1) with G a free group, since by the results of§1.A its universal cover is a tree, hence con-tractible.

E

xample 1B.2. Closed surfaces with infinite π₁, in other words, closed surfaces other than S² and RP², are K(G, 1) ’s. This will be shown in Example 1B.14 below. It also follows from the theorem in surface theory that the only simply-connected surfaces without boundary are S² and R², so the universal cover of a closed surface with infinite fundamental group must be R² since it is noncompact. Nonclosed surfaces deformation retract onto graphs, so such surfaces areK(G, 1) ’s with G free.

E

xample 1B.3. The infinite-dimensional projective space RP^∞ is a K(Z₂, 1) since its universal cover isS^∞, which is contractible. To show the latter fact, a homotopy from the identity map ofS^∞ to a constant map can be constructed in two stages as follows.

First, define f_t:R^∞

→

^{R∞ by ft(x1, x2,}···) = (1 − t)(x1, x₂,···) + t(0, x1, x₂,···).

This takes nonzero vectors to nonzero vectors for all t∈ [0, 1], so f_t/|f_t| gives a ho-motopy from the identity map ofS^∞ to the map(x₁, x₂,···)

,

^{(0, x1, x2,}···). Then a homotopy from this map to a constant map is given byg_t/|gt| where gt(x₁, x₂,···) = (1− t)(0, x1, x₂,···) + t(1, 0, 0, ···).

E

xample 1B.4. Generalizing the preceding example, we can construct a K(Z_m, 1) as an infinite-dimensional lens space S^∞/Zm, where Zm acts on S^∞, regarded as the unit sphere inC^∞, by scalar multiplication bym^throots of unity, a generator of this action being the map (z₁, z₂,···)

,

^{e2π i/m(z1, z2,}···). It is not hard to check that this is a covering space action.

E

xample 1B.5. A product K(G, 1)×K(H, 1) is a K(G×H, 1) since its universal cover is the product of the universal covers ofK(G, 1) and K(H, 1) . By taking products of circles and infinite-dimensional lens spaces we therefore get K(G, 1) ’s for arbitrary finitely generated abelian groups G . For example the n dimensional torus Tⁿ, the product of n circles, is a K(Zⁿ, 1) .

E

xample 1B.6. For a closed connected subspace K of S³ that is nonempty, the com-plementS³−K is a K(G, 1). This is a theorem in 3 manifold theory, but in the special case that K is a torus knot the result follows from our study of torus knot comple-ments in Examples 1.24 and 1.35. Namely, we showed that forK the torus knot K_m,n there is a deformation retraction of S³− K onto a certain 2 dimensional complex X_m,n having contractible universal cover. The homotopy lifting property then implies that the universal cover of S³− K is homotopy equivalent to the universal cover of X_m,n, hence is also contractible.

E

xample 1B.7. It is not hard to construct a K(G, 1) for an arbitrary group G , us-ing the notion of a ∆ complex defined in §2.1. Let EG be the ∆ complex whose n simplices are the ordered (n+ 1) tuples [g₀,··· , g_n] of elements of G . Such an n simplex attaches to the (n− 1) simplices [g0,··· , bg_i,··· , gn] in the obvious way, just as a standard simplex attaches to its faces. (The notation gb_i means that this vertex is deleted.) The complex EG is contractible by the homotopy h_t that slides each point x∈ [g₀,··· , g_n] along the line segment in [e, g₀,··· , g_n] from x to the vertex [e] , where e is the identity element of G . This is well-defined in EG since when we restrict to a face[g₀,··· , bg_i,··· , gn] we have the linear deformation to [e]

in[e, g₀,··· , bg_i,··· , gn] . Note that h_t carries[e] around the loop [e, e] , so h_t is not actually a deformation retraction of EG onto [e] .

The group G acts on EG by left multiplication, an element g ∈ G taking the simplex [g₀,··· , gn] linearly onto the simplex [gg₀,··· , ggn] . Only the identity e takes any simplex to itself, so by an exercise at the end of this section, the action of G on EG is a covering space action. Hence the quotient map EG

→

EG/G is the universal cover of the orbit space BG= EG/G, and BG is a K(G, 1).

Since G acts on EG by freely permuting simplices, BG inherits a ∆ complex structure from EG . The action of G on EG identifies all the vertices of EG , so BG has just one vertex. To describe the ∆ complex structure on BG explicitly, note first that every n simplex of EG can be written uniquely in the form

[g₀, g₀g₁, g₀g₁g₂,··· , g₀g₁··· g_n]= g₀[e, g₁, g₁g₂,··· , g₁··· g_n]

The image of this simplex in BG may be denoted unambiguously by the symbol [g₁|g₂| ··· |g_n] . In this ‘bar’ notation the g_i’s and their ordered products can be used to label edges, viewing an

edge label as the ratio between the two labels on the vertices at the endpoints of the edge, as indicated in the figure. With

g g

g

0 g₀

g₃ g₃

g₀

1 g₁ g₁ g₀g₁

g₀g_{1 2} g₀g_{1 2}g g₀g_{1 2}g g₃

g₂ g₃

g g_{1 2}

g g_{1 2}

g₂ g₂

g g_{1 2}

this notation, the boundary of a simplex [g₁| ··· |gn] of BG

consists of the simplices [g₂| ··· |gn] , [g₁| ··· |g_n−1] , and [g₁| ··· |gig_i+1| ··· |gn] for i= 1, ··· , n − 1.

This construction of a K(G, 1) produces a rather large space, since BG is al-ways infinite-dimensional, and if G is infinite, BG has an infinite number of cells in each positive dimension. For example, BZ is much bigger than S¹, the most efficient K(Z, 1). On the other hand, BG has the virtue of being functorial: A homomorphism f : G

→

H induces a map Bf : BG

→

BH sending a simplex [g₁| ··· |g_n] to the simplex [f (g₁)| ··· |f (gn)] . A different construction of a K(G, 1) is given in§4.2. Here one starts with any 2 dimensional complex having fundamental group G , for example

the complex X_G associated to a presentation of G , and then one attaches cells of di-mension 3 and higher to make the universal cover contractible without affecting π₁. In general, it is hard to get any control on the number of higher-dimensional cells needed in this construction, so it too can be rather inefficient. Indeed, finding an efficientK(G, 1) for a given group G is often a difficult problem.

It is a curious and almost paradoxical fact that ifG contains any elements of finite order, then every K(G, 1) CW complex must be infinite-dimensional. This is shown in Proposition 2.45. In particular the infinite-dimensional lens space K(Zm, 1) ’s in Example 1B.4 cannot be replaced by any finite-dimensional complex.

In spite of the great latitude possible in the construction of K(G, 1) ’s, there is a very nice homotopical uniqueness property that accounts for much of the interest in K(G, 1) ’s:

T

heorem 1B.8. The homotopy type of a CW complex K(G, 1) is uniquely determined by G .

Having a unique homotopy type of K(G, 1) ’s associated to each group G means that algebraic invariants of spaces that depend only on homotopy type, such as ho-mology and cohoho-mology groups, become invariants of groups. This has proved to be a quite fruitful idea, and has been much studied both from the algebraic and topological viewpoints. The discussion following Proposition 2.45 gives a few references.

The preceding theorem will follow easily from:

P

roposition 1B.9. Let X be a connected CW complex and let Y be a K(G, 1) . Then every homomorphism π₁(X, x₀)

→

^{π1(Y , y0}) is induced by a map (X, x₀)

→

^{(Y , y0)}

that is unique up to homotopy fixing x₀.

To deduce the theorem from this, letX and Y be CW complex K(G, 1) ’s with iso-morphic fundamental groups. The proposition gives maps f : (X, x₀)

→

^{(Y , y0) and}

g : (Y , y₀)

→

^{(X, x0}) inducing inverse isomorphisms π₁(X, x₀)≈ π1(Y , y₀) . Then f g and gf induce the identity on π₁ and hence are homotopic to the identity maps.

P

roof of 1B.9: Let us first consider the case that X has a single 0 cell, the base-pointx₀. Given a homomorphism ϕ : π₁(X, x₀)

→

^{π1(Y , y0}) , we begin the construc-tion of a map f : (X, x₀)

→

^{(Y , y0) with f}∗ = ϕ by setting f (x0)= y0. Each 1 cell e_α¹ of X has closure a circle determining an element

[e¹_α] ∈ π₁(X, x₀) , and we let f on the closure of e¹_α π1 X

1

( x0)

ϕ

, ∗

π1(X,x0)

π1(Y,y0)

f

i∗

−−−−→

−−−−−−−−−→ −−−−−−−−−→

be a map representingϕ([e¹_α]) . If i : X¹

>

^{X denotes}

the inclusion, then ϕi_∗= f_∗ since π₁(X¹, x₀) is gen-erated by the elements [e¹_α] .

To extendf over a cell e²_β with attaching mapψ_β:S¹

→

^X1, all we need is for the compositionf ψ_βto be nullhomotopic. Choosing a basepoints₀∈ S¹and a path inX¹ fromψ_β(s₀) to x₀,ψ_β determines an element[ψ_β]∈ π1(X¹, x₀) , and the existence

of a nullhomotopy of f ψ_β is equivalent to f_∗([ψ_β]) being zero in π₁(Y , y₀) . We have i_∗([ψ_β]) = 0 since the cell e²_β provides a nullhomotopy of ψ_β in X . Hence f_∗([ψ_β])= ϕi_∗([ψ_β])= 0, and so f can be extended over e²_β.

Extendingf inductively over cells eⁿ_γ with n > 2 is possible since the attaching maps ψ_γ:Sⁿ⁻¹

→

^Xn−1 have nullhomotopic compositions f ψ_γ:Sⁿ⁻¹

→

Y . This is because f ψ_γ lifts to the universal cover of Y if n > 2 , and this cover is contractible by hypothesis, so the lift of f ψ_γ is nullhomotopic, hence alsof ψ_γ itself.

Turning to the uniqueness statement, if two maps f₀, f₁:(X, x₀)

→

^{(Y , y0)}

in-duce the same homomorphism onπ₁, then we see immediately that their restrictions to X¹ are homotopic, fixing x₀. To extend the resulting map X¹×I ∪ X×∂I

→

^Y

over the remaining cells eⁿ×(0, 1) of X×I we can proceed just as in the preceding paragraph since these cells have dimension n+ 1 > 2. Thus we obtain a homotopy f_t:(X, x₀)

→

^{(Y , y0}) , finishing the proof in the case that X has a single 0 cell.

The case that X has more than one 0 cell can be treated by a small elaboration on this argument. Choose a maximal tree T ⊂ X . To construct a map f realizing a given ϕ , begin by setting f (T ) = y₀. Then each edge e¹_α in X − T determines an element [e¹_α]∈ π₁(X, x₀) , and we let f on the closure of e¹_α be a map representing ϕ([e¹_α]) . Extending f over higher-dimensional cells then proceeds just as before.

Constructing a homotopy f_t joining two given maps f₀ and f₁ with f_0∗= f_1∗ also has an extra step. Leth_t:X¹

→

^X1 be a homotopy starting withh₀=¹1 and restricting to a deformation retraction of T onto x₀. (It is easy to extend such a deformation retraction to a homotopy defined on all of X¹.) We can construct a homotopy from f₀|X¹tof₁|X¹by first deformingf₀|X¹andf₁|X¹to takeT to y₀by composing with h_t, then applying the earlier argument to obtain a homotopy between the modified f₀|X¹ and f₁|X¹. Having a homotopyf₀|X¹' f1|X¹ we extend this over all ofX in

the same way as before. tu

The first part of the preceding proof also works for the 2 dimensional complexes X_G associated to presentations of groups. Thus every homomorphism G

→

^{H is}

re-alized as the induced homomorphism of some map X_G

→

^XH. However, there is no uniqueness statement for this map, and it can easily happen that different presenta-tions of a group G give X_G’s that are not homotopy equivalent.

Graphs of Groups

As an illustration of how K(G, 1) spaces can be useful in group theory, we shall describe a procedure for assembling a collection of K(G, 1) ’s together into a K(G, 1) for a larger groupG . Group-theoretically, this gives a method for assembling smaller groups together to form a larger group, generalizing the notion of free products.

Let Γ be a graph that is connected and oriented, that is, its edges are viewed as arrows, each edge having a specified direction. Suppose that at each vertexv ofΓ we

place a group G_v and along each edgee of Γ we put a homomorphism ϕe from the group at the tail of the edge to the group at the head of the edge. We call this data a graph of groups. Now build a space BΓ by putting the space BG_v from Example 1B.7 at each vertex v of Γ and then filling in a mapping cylinder of the map Bϕe along each edgee ofΓ , identifying the two ends of the mapping cylinder with the two BGv’s at the ends of e . The resulting space BΓ is then a CW complex since the maps Bϕe

take n cells homeomorphically onto n cells. In fact, the cell structure on BΓ can be canonically subdivided into a∆ complex structure using the prism construction from the proof of Theorem 2.10, but we will not need to do this here.

More generally, instead of BG_v one could take any CW complex K(G_v, 1) at the vertex v , and then along edges put mapping cylinders of maps realizing the homo-morphisms ϕ_e. We leave it for the reader to check that the resulting space KΓ is homotopy equivalent to the BΓ constructed above.

E

xample 1B.10. Suppose Γ consists of one central vertex with a number of edges radiating out from it, and the group G_v at this central vertex is trivial, hence also all the edge homomorphisms. Then van Kampen’s theorem implies that π₁(KΓ ) is the free product of the groups at all the outer vertices.

In view of this example, we shall callπ₁(KΓ ) for a general graph of groups Γ the graph product of the vertex groups G_v with respect to the edge homomorphismsϕ_e. The name for π₁(KΓ ) that is generally used in the literature is the rather awkward phrase, ‘the fundamental group of the graph of groups.’

Here is the main result we shall prove about graphs of groups:

T

heorem 1B.11. If all the edge homomorphisms ϕ_e are injective, then KΓ is a K(G, 1) and the inclusions K(G_v, 1)

>

^KΓ induce injective maps on π1.

Before giving the proof, let us look at some interesting special cases:

E

xample 1B.12: Free Products with Amalgamation. Suppose the graph of groups is A

←

^C

→

B , with the two maps monomorphisms. One can regard this data as speci-fying embeddings of C as subgroups of A and B . Applying van Kampen’s theorem to the decomposition of KΓ into its two mapping cylinders, we see that π1(KΓ ) is the quotient ofA∗ B obtained by identifying the subgroup C ⊂ A with the subgroup C ⊂ B . The standard notation for this group is A ∗CB , the free product of A and B amalgamated along the subgroup C . According to the theorem, A∗CB contains bothA and B as subgroups.

For example, a free product with amalgamation Z ∗_ZZ can be realized by map-ping cylinders of the mapsS¹

←

^S1

→

^{S1 that are}m sheeted and n sheeted covering spaces, respectively. We studied this case in Examples 1.24 and 1.35 where we showed that the complex KΓ is a deformation retract of the complement of a torus knot in S³ ifm and n are relatively prime. It is a basic result in 3 manifold theory that the

complement of every smooth knot in S³ can be built up by iterated graph of groups constructions with injective edge homomorphisms, starting with free groups, so the theorem implies that these knot complements are K(G, 1) ’s. Their universal covers are all R³, in fact.

E

xample 1B.13: HNN Extensions. Consider a graph of groups C A

ψ

ϕ with ϕ and ψ both monomorphisms. This is analogous to the previous case A

←

^C

→

^{B ,}

but with the two groups A and B coalesced to a single group. The group π₁(KΓ ), which was denoted A∗_CB in the previous case, is now denoted A∗_C. To see what this group looks like, let us regard KΓ as being obtained from K(A, 1) by attaching K(C, 1)×I along the two ends K(C, 1)×∂I via maps realizing the monomorphisms ϕ and ψ . Using a K(C, 1) with a single 0 cell, we see that KΓ can be obtained from K(A, 1)∨ S¹ by attaching cells of dimension two and greater, soπ₁(KΓ ) is a quotient ofA∗Z, and it is not hard to figure out that the relations defining this quotient are of the form tϕ(c)t⁻¹= ψ(c) where t is a generator of the Z factor and c ranges over C , or a set of generators for C . We leave the verification of this for the Exercises.

As a very special case, taking ϕ= ψ =¹1 gives A∗A= A×Z since we can take KΓ = K(A, 1)×S¹ in this case. More generally, taking ϕ = ¹1 with ψ an arbitrary automorphism of A , we realize any semidirect product of A and Z as A∗_A. For example, the Klein bottle occurs this way, with ϕ realized by the identity map of S¹ and ψ by a reflection. In these cases when ϕ=¹1 we could realize the same group π₁(KΓ ) using a slightly simpler graph of groups, with a single vertex, labeled A, and a single edge, labeledψ .

Here is another special case. Suppose we take a torus, delete a small open disk, then identify the resulting boundary circle with a longitudinal circle of the torus. This produces a space X that happens to be homeomorphic to a subspace of the stan-dard picture of a Klein bottle in R³; see Exercise 12 of§1.2. The fundamental group π₁(X) has the form (Z ∗ Z) ∗_ZZ with the defining relation tb^±1t⁻¹ = aba⁻¹b⁻¹ where a is a meridional loop and b is a longitudinal loop on the torus. The sign of the exponent in the term b^±1 is immaterial since the two ways of glueing the boundary circle to the longitude produce homeomorphic spaces. The groupπ₁(X)= a, b, t |||| tbt⁻¹aba⁻¹b⁻¹

abelianizes to Z×Z, but to show that π₁(X) is not iso-morphic to Z ∗ Z takes some work. There is a surjection π1(X)

→

Z ∗ Z obtained by setting b = 1. This has nontrivial kernel since b is nontrivial in π1(X) by the pre-ceding theorem. If π₁(X) were isomorphic toZ ∗ Z we would then have a surjective homomorphismZ ∗ Z

→

Z ∗ Z that was not an isomorphism. However, it is a theorem in group theory that a free group F is hopfian — every surjective homomorphism F

→

F must be injective. Hence π₁(X) is not free.

E

xample 1B.14: Closed Surfaces. A closed orientable surface M of genus two or greater can be cut along a circle into two compact surfacesM₁ andM₂ such that the

closed surfaces obtained from M₁ and M₂ by filling in their boundary circle with a disk have smaller genus than M . Each of M₁ and M₂ is the mapping cylinder of a map from S¹ to a finite graph. Namely, view M_i as obtained from a closed surface by deleting an open disk in the interior of the 2 cell in the standard CW structure described in Chapter 0, so that M_i becomes the mapping cylinder of the attaching map of the 2 cell. This attaching map is not nullhomotopic, so it induces an injection on π₁ since free groups are torsionfree. Thus we have realized the original surface M as KΓ for Γ a graph of groups of the form F1

←---

^Z

-→

^{F2 with F1 and F2 free and}

the two maps injective. The theorem then says that M is a K(G, 1) .

A similar argument works for closed nonorientable surfaces other thanRP². For example, the Klein bottle is obtained from two M¨obius bands by identifying their boundary circles, and a M¨obius band is the mapping cylinder of the 2 sheeted covering spaceS¹

→

^S1.

P

roof of 1B.11: We shall construct a covering space eK

→

^KΓ by gluing together copies of the universal covering spaces of the various mapping cylinders inKΓ in such a way that eK will be contractible. Hence eK will be the universal cover of KΓ , which will therefore be aK(G, 1) .

First a preliminary observation: Given a universal covering spacep : eX

→

^{X and a}

connected, locally path-connected subspaceA⊂ X such that the inclusion A

>

^X

in-duces an injection onπ₁, then each component eA of p⁻¹(A) is a universal cover of A . To see this, note thatp : eA

→

A is a covering space, so the induced map π₁( eA)

→

^π1(A)

is injective, and this map factors through π₁( eX)= 0, hence π₁( eA)= 0. For exam-ple, if X is the torus S¹×S¹ and A is the circle S¹×{x0}, then p⁻¹(A) consists of infinitely many parallel lines in R², each of which is a universal cover ofA .

For a map f : A

→

B between connected CW complexes, let p : fM_f

→

^{Mf be the}

universal cover of the mapping cylinderM_f. Then fM_f is itself the mapping cylinder of a map ef : p⁻¹(A)

→

^p−1(B) since the line segments in the mapping cylinder struc-ture on M_f lift to line segments in fM_f defining a mapping cylinder structure. Since Mf_f is a mapping cylinder, it deformation retracts onto p⁻¹(B) , so p⁻¹(B) is also simply-connected, hence is the universal cover ofB . If f induces an injection on π₁, then the remarks in the preceding paragraph apply, and the components of p⁻¹(A) are universal covers ofA . If we assume further that A and B are K(G, 1) ’s, then fM_f and the components of p⁻¹(A) are contractible, and we claim that fM_f deformation retracts onto each component eA of A . Namely, the inclusion eA

>

^{Mff is a}

homo-topy equivalence since both spaces are contractible, and then Corollary 0.20 implies that fM_f deformation retracts onto eA since the pair (fM_f, eA) satisfies the homotopy extension property, as shown in Example 0.15.

Now we can describe the construction of the covering space eK of KΓ . It will be the union of an increasing sequence of spaces eK₁ ⊂ eK₂ ⊂ ···. For the first stage, let eK₁ be the universal cover of one of the mapping cylinders M_f of KΓ . By the

preceding remarks, this contains various disjoint copies of universal covers of the twoK(G_v, 1) ’s at the ends of M_f. We build eK₂ from eK₁ by attaching to each of these universal covers ofK(G_v, 1) ’s a copy of the universal cover of each mapping cylinder M_g of KΓ meeting Mf at the end of M_f in question. Now repeat the process to construct eK₃ by attaching universal covers of mapping cylinders at all the universal covers ofK(G_v, 1) ’s created in the previous step. In the same way, we construct eK_n+1 from eK_n for all n , and then we set eK=S

nKe_n.

Note that eK_n+1 deformation retracts onto eK_n since it is formed by attaching pieces to eK_n that deformation retract onto the subspaces along which they attach, by our earlier remarks. It follows that eK is contractible since we can deformation retract eK_n+1 onto eK_n during the time interval[1/2ⁿ⁺¹, 1/2ⁿ] , and then finish with a contraction of eK₁ to a point during the time interval[¹/₂, 1].

The natural projection eK

→

^KΓ is clearly a covering space, so this finishes the proof that KΓ is a K(G, 1).

The remaining statement that each inclusionK(G_v, 1)

>

^KΓ induces an injection on π₁ can easily be deduced from the preceding constructions. For suppose a loop γ : S¹

→

^K(Gv, 1) is nullhomotopic in KΓ . By the lifting criterion for covering spaces, there is a lift eγ : S¹

→

K . This has image contained in one of the copies of the universal^e cover of K(G_v, 1) , so eγ is nullhomotopic in this universal cover, and hence γ is

nullhomotopic inK(G_v, 1) . tu

The various mapping cylinders that make up the universal cover of KΓ are ar-ranged in a treelike pattern. The tree in question, call it TΓ , has one vertex for each copy of a universal cover of a K(G_v, 1) in eK , and two vertices are joined by an edge whenever the two universal covers of K(G_v, 1) ’s corresponding to these vertices are connected by a line segment lifting a line segment in the mapping cylinder structure of a mapping cylinder ofKΓ . The inductive construction of eK is reflected in an inductive construction of TΓ as a union of an increasing sequence of subtrees T1 ⊂ T2 ⊂ ···.

Corresponding to eK₁ is a subtreeT₁⊂ T Γ consisting of a central vertex with a number of edges radiating out from it, an ‘asterisk’ with possibly an infinite number of edges.

When we enlarge eK₁ to eK₂, T₁ is correspondingly enlarged to a tree T₂ by attaching a similar asterisk at the end of each outer vertex ofT₁, and each subsequent enlarge-ment is handled in the same way. The action ofπ₁(KΓ ) on eK as deck transformations induces an action onTΓ , permuting its vertices and edges, and the orbit space of T Γ under this action is just the original graph Γ . The action on T Γ will not generally be a free action since the elements of a subgroup G_v ⊂ π1(KΓ ) fix the vertex of T Γ corresponding to one of the universal covers ofK(G_v, 1) .

There is in fact an exact correspondence between graphs of groups and groups acting on trees. See [Scott & Wall 1979] for an exposition of this rather nice theory.

From the viewpoint of groups acting on trees, the definition of a graph of groups is

usually taken to be slightly more restrictive than the one we have given here, namely, one considers only oriented graphs obtained from an unoriented graph by subdividing each edge by adding a vertex at its midpoint, then orienting the two resulting edges outward, away from the new vertex.

Exercises

1. Suppose a group G acts simplicially on a ∆ complex X , where ‘simplicially’ means that each element of G takes each simplex of X onto another simplex by a linear homeomorphism. If the action is free, show it is a covering space action.

2. Let X be a connected CW complex and G a group such that every homomorphism π₁(X)

→

G is trivial. Show that every map X

→

K(G, 1) is nullhomotopic.

在文檔中 Allen Hatcher (頁 96-105)