The Borel construction, stacks, and rebuilding

The Borel Construction and Bass-Serre Theory

6.1 The Borel construction, stacks, and rebuilding

If one understands an n-connected rigid G-CW complex Y, in particular if one understands the quotient G\Y and the stabilizers of the cells ofY, one may be able to deduce information about the groupG. The particular case in whichY is a tree, called Bass-Serre Theory, is discussed in the next section: in that case one can sometimes deduce thatGis decomposable as a free product with amalgamation or as an HNN extension. But the method is much more general, as we now explain.

Starting with a simply connected rigidG-CW complexY which might not be free, the method provides a path connected CW complex Z with funda- mental groupGand a mapq :Z →G\Y called a “stack.” A stack is like a fiber bundle¹ but the fibers over different cells may have different homotopy types. The method of improving spaces within their homotopy types intro- duced in Sect. 4.1 can be applied to the “fibers” over cells, by induction on the dimensions of the cells, to rebuildZ, thus producing a “better”q :Z →G\Y which is equivalent toq. A first application of this rebuilding process to group theory is given in the proof of Theorem 6.1.5.

Recall that the left G-CW complex Y is rigid iff for each cell ˜eofY, the stabilizerG_e˜acts trivially on ˜e. We saw in 3.2.2 that the quotientG\Y =:V

1 See 4.4.11.

of a rigidG-action receives a CW complex structure²from Y whose cells are the images of the cells ofY under the quotient mapp:Y →V.

Starting with a pointed CW complex (X, x0) such that π1(X, x0) ∼= G, we permanently identify G with π1(X, x0) by some chosen isomorphism, so that the universal cover ˜X, with some base vertex ˜x0 overx0, is a freeG-CW complex. The diagonal action ofG on the product CW complex ˜X×Y is defined by g.(x, y) = (g.x, g.y). It is a free action so, by 3.2.1, the quotient map r : ˜X ×Y →G\( ˜X ×Y) =: Z is a covering projection which imposes a natural CW complex structure onZ. The quotient maps pand rinduce a commutative diagram:

X˜ ×Y −−−−−−→^projection Y

⏐⏐

^r ⏐⏐^p Z −−−−→^q V

Clearly,q is a quotient map. This procedure is (a special case of) the Borel construction. One easily checks:

Proposition 6.1.1.The mapqis cellular and takes each cell ofZ onto a cell ofV. For each subcomplexU ofV,q⁻¹(U)is a subcomplex of Z.

LetV have base vertexv. Choose a vertex ˜vofY as base point, such that p(˜v) = v. The base point of ˜X ×Y is (˜x0,˜v), and its r-image is the base pointzofZ. Since ˜X×Y is a simply connected covering space, 3.2.3 implies π1(Z, z)∼=G.

Let En be the set of n-cells of V. For eache ∈ En make the following choices: (i) ann-cell ˜eofY such thatp(˜e) =e, (ii) a point ˜ue∈^◦˜e, (iii) a charac- teristic maph˜e: (Bⁿ, Sⁿ⁻¹)→(˜e,e). (Of course, when^•˜ n= 0, ˜uw= ˜wfor each w∈E0, and there is only one possiblehw˜.) Letue=p(˜ue), lethebe the char- acteristic mapp◦h˜efore, and letXe=q⁻¹(ue). Thenq⁻¹(V⁰) =

w∈E₀

Xw. We are going to show thatq⁻¹(Vⁿ⁺¹) is homeomorphic to the adjunction complex q⁻¹(Vⁿ)∪^f

⎛

⎝

e∈E_n+1

Xe×Bⁿ⁺¹

⎞

⎠, wheref :

e∈E_n+1

Xe×Sⁿ→q⁻¹(Vⁿ) is de- fined on eachXe×Sⁿ to agree with the mapfein the following commutative diagram (details to be explained):

2 Throughout this section,Y denotes a simply connected rigid G-CW complex.

6.1 The Borel construction, stacks, and rebuilding 145 X˜× {u˜_e} ×S^{n re×id}//

projection

Xe×Sⁿ

f_e

projection //Sⁿ

h_e|

X˜×Sⁿ

id×(h_e˜|)

˜

X×Yⁿ _r

| //q⁻¹(Vⁿ)

q| //Vⁿ

The mapr_e : ˜X × {u˜_e} →X_e :=q⁻¹(u_e) is the quotient map of the action of G_u˜e. Since the given G-action on Y is rigid, G_u˜e = G_˜e. Letting x_e = r(˜x₀,u˜_e)∈X_e, we have:

Proposition 6.1.2.For each cell e of V, Xe is homeomorphic to G˜e\X,˜

henceπ1(Xe, xe)∼=G˜e.

Since r_e is a quotient map, so is the map r_e×id in the diagram; to see this, apply 1.3.11 (n+ 1) times to conclude thatre×id : ˜X× {u˜e} ×Bⁿ⁺¹→ Xe×Bⁿ⁺¹ is a quotient map, and then restrict. The mapre×id is clearly surjective. Rigidity implies that there is a unique function, hence a map,fe, making the left half of the diagram commute. It is then obvious that the whole diagram commutes and thatfeis cellular.

For the same reasons, there is a mapHe:Xe×Bⁿ⁺¹→q⁻¹(Vⁿ⁺¹) making the following diagram commute:

X˜ × {u˜_e} ×B^{n+1 re×id}//

projection

X_e×Bⁿ⁺¹

H_e

projection //Bⁿ⁺¹

h_e

X˜ ×Bⁿ⁺¹

id×h_˜e

˜

X×Yⁿ⁺¹ _r

| //q⁻¹(Vⁿ⁺¹)

q| //Vⁿ⁺¹

Assembling the maps He,e∈En+1, we obtain the desired structure the- orem:

Theorem 6.1.3.The map q⁻¹(Vⁿ)⎛

⎝

e∈E_n+1

Xe×Bⁿ⁺¹

⎞

⎠ → q⁻¹(Vⁿ⁺¹) which agrees with inclusion on q⁻¹(Vⁿ) and withH_e onX_e×Bⁿ⁺¹ induces a homeomorphism

s_n+1:q⁻¹(Vⁿ)∪f

⎛

⎝

e∈E_n+1

X_e×Bⁿ⁺¹

⎞

⎠→q⁻¹(Vⁿ⁺¹)

for each n ≥ 0, such that sn+1 maps each cell of the adjunction complex homeomorphically onto a cell ofq⁻¹(Vⁿ⁺¹). Moreover, the following diagram commutes:

q⁻¹(Vⁿ)⎛

⎝

e∈E_n+1

Xe×Bⁿ⁺¹

⎞

⎠

quotient

^t &&MMMMMMMMMMMMMMMM

q⁻¹(Vⁿ)∪f

⎛

⎝

e∈E_n+1

X_e×Bⁿ⁺¹

⎞

⎠

s_n+1

Vⁿ⁺¹

q⁻¹(Vⁿ⁺¹)

q|

66n

nn nn nn nn nn nn nn nn nn

wheret agrees with qonq⁻¹(Vⁿ)and withh_e◦projection on X_e×Bⁿ⁺¹. Proof. The functionsn+1is clearly a continuous bijection. Moreover, it maps cells bijectively onto cells. Thus s⁻_n+1¹ | is continuous on each cell, which, by 1.2.12, is enough to imply continuity of s⁻_n+1¹ (exercise). The second part is

clear.

At the risk of repetition, the point of this theorem is that it gives a useful decomposition of the spaceZ in the Borel construction.

The Borel Construction is important and will be used in several ways in this book, so we need vocabulary to describe the result. Let π : A → C be a cellular map between CW complexes. Let he : (Bⁿ, Sⁿ⁻¹) → (e,e) be a^• characteristic map for the celleofC, and for each such celleletFebe a CW complex. We call π : A →C a stack of CW complexes with base space C, total spaceAand fiber F_eovere, if for eachn≥1 (denoting the set ofn-cells ofC byE_n) there is a cellular mapf_n :

e∈E_n

F_e×Sⁿ⁻¹ →π⁻¹(Cⁿ⁻¹) and a homeomorphismkn:π⁻¹(Cⁿ⁻¹)∪f_n

e∈E_n

Fe×Bⁿ

→π⁻¹(Cⁿ) satisfying:

(i)k_n agrees with inclusion onπ⁻¹(Cⁿ⁻¹), (ii)k_n maps each cell onto a cell, and (iii) the following diagram commutes:

6.1 The Borel construction, stacks, and rebuilding 147

π⁻¹(Cⁿ⁻¹)

e∈E_n

Fe×Bⁿ

quotient

^u &&MMMMMMMMMMMMMMM

π⁻¹(Cⁿ⁻¹)∪f_n

e∈E_n

Fe×Bⁿ

k_n

Cⁿ

π⁻¹(Cⁿ)

π|

77n

nn nn nn nn nn nn nn nn n

where u agrees withπ onπ⁻¹(Cⁿ⁻¹) and withh_e◦ projection on F_e×Bⁿ. Thus,π:A→C is “built” by induction on the skeleta ofC so that over the interior,^◦e, of ann-celle,πis “like”³the projection:F_e×e^◦→e. An immediate^◦ consequence of Theorem 4.1.7 is:

Proposition 6.1.4. (Rebuilding Lemma) If for each cell e of C we are given a CW complex F_e of the same homotopy type as Fe, then there is a stack of CW complexes π : A → C with fiber F_e over e, and a homotopy equivalence h making the following diagram commute up to homotopy over each cell:⁴

A ^h //

π?????

?? A

π

~~}}}}}}}

C

The content of Theorem 6.1.3 is thatq:Z→V is a stack of CW complexes with base spaceV, in which the fiber over the celleisG_e˜\X. We can use 6.1.4˜ to replaceG_˜e\X˜ by a more desirable CW complex of the same homotopy type and thereby produce a more desirable spaceZ of the same homotopy type as Z. The first of a number of uses of this method appears in:

Theorem 6.1.5.Let Y be a simply connected rigid G-CW complex. (i) If Y has finite 1-skeleton mod G and if the stabilizer of each vertex is finitely generated, thenGis finitely generated. (ii) If Y has finite 2-skeleton mod G, if the stabilizer of each vertex is finitely presented, and if the stabilizer of each 1-cell is finitely generated, then Gis finitely presented.

3 Readers may recognize the connection with fiber bundles and with block bundles in special cases.

4 i.e. the homotopy can be chosen so that for each cell e of C its restriction to π⁻¹(e)×I has its image ine.

Proof. By 6.1.1,q⁻¹(V¹) is a subcomplex ofZ containingZ¹. By hypothesis in (i), for eachw∈E0,π1(Xw, xw) is finitely generated. By 4.1.14, eachXwis homotopy equivalent to a CW complexX_w having finite 1-skeleton. By 3.1.13 and 4.1.9, when e ∈ E1, Xe is homotopy equivalent to a CW complex X_e having one vertex. By applying 6.1.4 to the stack q : Z → V, we obtain a stack q :Z →V in whichZ is homotopy equivalent toZ, and the fiber of q overe[resp. overw] isX_e [resp.X_w].

The CW complexesZand (q)⁻¹(V¹) have the same 1-skeleton. It is finite because: (a)Γ⁰ is finite and ((q)⁻¹(V⁰))¹is the finite graph

w∈E₀

(X_w)¹; (b) V¹ is finite and (sinceX_e×B¹is a product and X_e has one vertex) there is only one 1-cell of (q)⁻¹(V¹) mapped onto each 1-cell ofΓ. (Note that (1-cell ofX_e)×(vertex ofB¹) does not give a new 1-cell of the adjunction complex.) ThusZ has finite 1-skeleton. By 3.1.17,G∼=π1(Z, z)∼=π1(Z, z) is finitely generated. Thus (i) holds.

The proof of (ii) is similar. We replace eachXw byX_w , whose 2-skeleton is the wedge of a finite 2-complex and a bouquet of 2-spheres (see 4.1.14); for e∈E1, we replaceXe by X_e having finite 1-skeleton; fore∈E2 we replace Xe by X_e having a single vertex. The resulting CW complex Z has finite 1-skeleton (for reasons already explained) and possibly infinite 2-skeleton.

Indeed, its 2-cells are of three kinds: 2-cells ofX_w wherew∈E₀, cells coming from (1-cells ofX_e)×B¹fore∈E₁, and cells coming from (vertex ofX_e)×B² fore∈E₂. Thus all but finitely many of the 2-cells are 2-spheres (i.e., have trivial attaching maps) so they contribute nothing significant to the resulting presentation ofG∼=π1(Z, z)∼=π1(Z, z).

Exercise

Prove that ifπ : A → C is a stack of CW complexes in which every fiber Fe is contractible, thenπis a homotopy equivalence. In fact,πis ahereditary homotopy equivalence, meaning that for every subcomplex D of C, π| : π⁻¹(D) → D is a homotopy equivalence. (Hint: The proof of the Rebuilding Lemma allows us to take A=C,π= idC andh=π.)

6.2 Decomposing groups which act on trees (Bass-Serre