Cohomology of Dehn filling quotients

Therefore,

E_1,2^p,0 ∼= ker(CQ1,p+2)M

ker(M SS₂^p,0)∼=E_2,2^p,0M

E_1,p+2^p,0 ∼=E_2,2^p,0M H₁^p.

Lemma 5.1.8. Iff^p−1is surjective andf^p is an isomorphism, thenM SS₂^p,0is an isomorphism.

Proof. Ifp60, then Remark 2.15.9,E^{k,`</sup><sub>1,2</sub> ⇒H<sub>1</sub><sup>k+`}, andE_2,2^{k,`</sup>⇒H<sub>2</sub><sup>k+`}imply

E_1,2^p,0 ∼=H₁^p, E_2,2^p,0 ∼=H₂^p.

As f^p is an isomorphism and M SS is compatible withf^p, Remark 2.15.11 implies that M SS₂^p,0 is an isomorphism.

Let us supposep>1. AsM SSis compatible withf andf^p−1is surjective,M SS_p+1^p−r−1,r is surjective forr = 0, ..., p−1, by Remark 2.15.11. It follows from Lemma 5.1.6 thatM SS_r+2^p−r−1,r is surjective for r= 0, ..., p−1. By Lemma 5.1.5,CQ_1,p+1mapsker(M SS₂^p,0)injectively intoker(M SS_p+2^p,0 ).

By Remark 2.15.11 and the assumption that f^p is an isomorphism, M SS_p+2^p,0 is injective. Thus, ker(M SS_p+2^p,0 ) = {0}. As CQ1,p+1 maps ker(M SS₂^p,0) injectively into ker(M SS_p+2^p,0 ), we have ker(M SS₂^p,0) ={0}, i.e.,M SS₂^p,0 is injective.

By Remark 2.15.11 and the assumption thatf^p is an isomorphism,M SS_p+2^p,0 is surjective. By succes- sively applying Lemma 5.1.6 (withpin place ofp−1in part (a)), we see thatM SS₂^p,0is surjective and thus is an isomorphism.

Proof. LetM SS be as in Theorem 4.0.1. Note thatM SSandN T R_Gsatisfy the assumptions of Lemma 5.1.7, which yields (5.8) and shows thatM SS₂^p,0is surjective. By Theorem 4.0.1,M SS₂^p,0can be identified withN T^p

Gand thusN T^p

Gis surjective.

Recall that for a groupG, thecohomological dimension of Gis

cd(G) = sup{`∈N|H<sup>`</sup>(G, A)6={0}for someZG-moduleA}.

If(G,{H_λ}_λ∈Λ,{N_λ}_λ∈Λ)is a group triple, let

cd(H) = sup

λ∈Λ

{cd(H_λ)}, cd(H) = sup

λ∈Λ

{cd(H_λ)}.

Also recall the following result of [7] concerning relative cohomology groups.

Proposition 5.2.2 ([7, Proposition 1.1]). LetGbe a group with a family of subgroups{H_λ}λ∈Λ. Then for everyZG-moduleA, there is a long exact sequence

· · · →H^{`</sup>(G,{H<sub>λ</sub>}<sub>λ∈Λ</sub>;A)→H<sup>`}(G;A)−−−−→^{N T RG} Y

λ∈Λ

H^{`</sup>(H<sub>λ</sub>;A)→H<sup>`+1}(G,{H_λ}_λ∈Λ;A)→ · · ·

whereN T RGis the natural map defined in Notation 2.14.2.

Corollary 5.2.3. Let (G,{H_λ}_λ∈Λ,{N_λ}_λ∈Λ) be a group triple satisfying the Cohen-Lyndon property.

Then for all`>cd(H) + 3and everyZG-moduleA, there is an isomorphism

H^{`</sup>(G,{H<sub>λ</sub>}<sub>λ∈Λ</sub>;A)∼=H<sup>`}(G;A).

For`=cd(H) + 2, there is a surjectionH<sup>`</sup>(G,{H_λ}_λ∈Λ;A)H^`(G;A).

Proof. By Proposition 5.2.2, there is a long exact sequence

· · · →H^{`</sup>(G,{H<sub>λ</sub>}<sub>λ∈Λ</sub>;A)→H<sup>`}(G;A) ^{N T}

`

−−−→G Y

λ∈Λ

H^{`</sup>(Hλ;A)→H<sup>`+1}(G,{H_λ}_λ∈Λ;A)→ · · ·

By Theorem 5.2.4, if` >cd(H) + 2, thenN T<sup>`</sup>

Gis surjective andker(N T^`

G) ∼=H^`(G;A), which implies the desired result.

Corollary 5.2.4. Let (G,{H_λ}_λ∈Λ,{N_λ}_λ∈Λ) be a group triple satisfying the Cohen-Lyndon property.

Then

cd(G)6max{cd(G), cd(H) + 1, cd(H)}.

Proof. Ifcd(G)6cd(H) + 1, then the desired conclusion already holds. Thus, let us assume thatcd(G)>

cd(H) + 2. Let ` > cd(H) + 2, and let N T R_G be the natural map defined by Notation 2.14.2, then Q

λ∈ΛH^{`</sup>(H<sub>λ</sub>;A) = {0}andN T R<sub>G</sub> mapsH<sup>`−1}(G;A)surjectively ontoQ

λ∈ΛH^`−1(H_λ;A) = {0}. It follows from Theorem 5.2.1 that

H^`(G;A)∼= Y

λ∈Λ

H^`(Hλ;A)

!

MH^`(G;A),

which impliescd(G)6max{cd(G), cd(H)}.

Proof of Theorem 1.2.15. By Theorem 3.0.1, for sufficiently deepN H, the group triple(G, H, N)has the Cohen-Lyndon property. Thus, Theorem 1.2.15 follows from the case |Λ| = 1of Theorem 5.2.1 and Corollary 5.2.4.

Our next result concerns another finiteness property. Recall that a groupGis of typeF P∞if there is a projective resolutionP →ZoverZGsuch thatP<sub>`</sub>is finitely generated for`∈N. Also recall the following characterization ofF P∞.

Theorem 5.2.5 ([10, Chapter VIII Theorem 4.8] (see [9, Theorem 3] for a proof)). A groupGis of type F P∞if and only ifH^∗(G;·)preserves direct limits.

Lemma 5.2.6. LetF be a free group of finite rank, letN be a normal subgroup ofF, letF = F/N, let {A_i}_i∈Ibe a directed system ofZF-modules, and letA= lim−→A_i. IfF is of typeF P∞, then forp, q ∈Z, the natural mapsA_i →A, i∈I,induce an isomorphism

lim−→H^p(F;H^q(N;Ai))∼=H^p(F;H^q(N;A)). (5.9)

Proof. Let

E₂^p,q=H^p(F;H^q(N;A))⇒H^p+q(F;A)

be the LHS spectral sequence for the triple(F, N, A). Fori∈I, let

E_i,2^p,q=H^p(F;H^q(N;Ai))⇒H^p+q(F;Ai)

be the LHS spectral sequence for the triple(F, N, A_i).

Being a subgroup of the free groupF,N is also free. By the Stallings-Swan theorem [32, Corollary to Theorem 1],cd(N)61. It follows that

(CD1) E_i,2^p,q=E₂^p,q={0}wheneverq6∈ {0,1}.

Thus, ifq6∈ {0,1}, then both sides of (5.9) are{0}. Therefore, it suffices to prove (5.9) forq ∈ {0,1}.

Note that if p 6 −1, then both sides of (5.9) are{0}, and ifq = 0, then (5.9) follows from Theorem 5.2.5 as there are natural isomorphisms

H⁰(N;A)∼=A, H⁰(N;Ai)∼=Ai, fori∈I.

Thus, it suffices to prove (5.9) forp>0andq = 1.

By Proposition 4.3.4, the mapsA_i →A, i∈I,induce morphisms

M SSi :Ei →E

between spectral sequences. Fori∈I andp∈Z, Proposition 4.3.4 implies that the map

M SS_i,2^p,1:E_i,2^p,1 →E₂^p,1

can be identified with the natural map

H^p(F;H¹(N;A_i))→H^p(F;H¹(N;A))

induced byAi→A. It suffices to show that forp>0,

lim−→M SS_i,2^p,1: lim−→H^p(F;H¹(N;A_i))→H^p(F;H¹(N;A)). (5.10)

is an isomorphism.

Fixp>0. We have the following commutative diagram.

E_i,2^p,1 E₂^p,1

E_i,2^p+2,0 E₂^p+2,0

M SS^p,1_i

d^p,1_i,2 d^p,1₂

M SS_i^p+2,0

(5.11)

Note thatH^p+2(F;A) = {0}. AsE₂^{k,`</sup> ⇒ H<sup>k+`}(F;A), we haveEr^p+2,0 = {0}for sufficiently large r. By (CD1) and the definition of spectral sequences,Er^p+2,0 =E₃^p+2,0for allr>3. Thus,E^p+2,0₃ ={0}

and, as a consequence,d^p,1₂ is surjective. Similarly,d^p,1_i,2 is surjective.

Ifp > 1, then asH^p+1(F;A) = {0}andE^{k,`</sup><sub>2</sub> ⇒ H<sup>k+`}(F;A), we haveEr^p,1 = {0}for sufficiently larger. By (CD1),Er^p,1 =E₃^p,1for allr >3. Thus,E₃^p,1={0}. Using (CD1) once again, we see thatd^p,1₂ is injective and thus is an isomorphism. Similarly,d^p,1_i,2 is an isomorphism.

Taking direct limit of (5.11), we obtain

lim−→E_i,2^p,1 E₂^p,1

lim−→E_i,2^p+2,0 E₂^p+2,0

(5.12)

By Theorem 5.2.5, the lower horizontal map of (5.12) is an isomorphism. Being direct limits of isomor- phisms, the vertical maps of (5.12) are isomorphisms. Thus, the upper horizontal map of (5.12) is also an isomorphism, which proves thatlim−→M SS_i,2^p,1is an isomorphism forp>1.

Supposep = 0. Thend^p,1_i,2 =d^0,1_i,2 andd^p,1₂ =d^0,1₂ are not necessarily injective. Letkeri(resp. ker) be the kernel ofd^0,1_i,2 (resp.d^0,1₂ ). By Remark 2.15.9 andE₂^{k,`</sup>⇒H<sup>k+`}(F;A), there is an exact sequence

1→E₃^1,0→H¹(F;A)→ker→1. (5.13)

By the same argument, we see that there is an exact sequence similar to (5.13) holds for everyi∈I. As

E_i,2^−1,1=E₂^−1,1={0}, we have

E_i,3^1,0 =E_i,2^1,0, E₃^1,0 =E₂^1,0. Combining these observations, we obtain a commutative diagram

1 E_i,2^1,0 H¹(F;A_i) ker_i 1

1 E₂^1,0 H¹(F;A) ker 1

M SS^1,0_i,2 M SS_i,2^0,1 (5.14)

By taking direct limit of (5.14) and using the fact thatlim−→is an exact functor, we obtain the following commutative diagram with exact rows.

1 lim−→E_i,2^1,0 lim−→H¹(F;A_i) lim−→ker_i 1

1 E₂^1,0 H¹(F;A) ker 1

(5.15)

As F has finite rank, F is of typeF P∞. By Theorem 5.2.5, the first and the second vertical maps of (5.15) are isomorphisms. Thus, the five lemma implies that the last vertical map of (5.15) is also an isomorphism.

Consider the commutative diagram

1 ker_i E^0,1_i,2 E_i,2^2,0 1

1 ker E^0,1₂ E₂^2,0 1

M SS_i,2^0,1

d^0,1_i,2

M SS_i,2^0,1 M SS_i,2^2,0

d^0,1₂

(5.16)

By taking direct limit of (5.16) and using the fact thatlim−→is an exact functor, we obtain the following

commutative diagram with exact rows.

1 lim−→keri lim−→E_i,2^0,1 lim−→E_i,2^2,0 1

1 ker E^0,1₂ E₂^2,0 1

(5.17)

We have already proved that the first vertical map of (5.17) is an isomorphism. By Theorem 5.2.5 and the assumption thatF is of typeF P∞, the last vertical map of (5.17) is an isomorphism. Thus, the five lemma implies that the second vertical map of (5.17) is also an isomorphism, which proves thatlim−→M SS_i,2^0,1is an isomorphism.

Lemma 5.2.7. LetKbe a finite group, letF be free group of finite rank, letH =K×F, letN be a normal subgroup ofF, letH =H/N, let{A_i}_i∈I be a directed system ofZH-modules, and letA= lim−→A_i. IfH is of typeF P∞, then forp, q∈Z, the natural mapsA_i →Ainduce an isomorphism

lim−→H^p(H;H^q(N;Ai))∼=H^p(H;H^q(N;A)).

Proof. Note thatF = F/N has finite index inHand thusF is of typeF P∞. Fixp, q ∈Z. Lemma 5.2.6 asserts that the natural mapsA_i→Ainduce an isomorphism

lim−→H^p(F;H^q(N;A_i))∼=H^p(F;H^q(N;A)). (5.18)

Notice that F H andH/F ∼= K is a finite group. In particular,H/F is of typeF P∞. Fori ∈ I, letEi the LHS spectral sequence for the triple(H, F , H^q(N, Ai)). LetEbe the LHS spectral sequence for the triple(H, F , H^q(N, A)), letlim−→Ei be the direct limit of{E_i}_i∈I, and letM SS : lim−→Ei → E be the morphism induced byAi →A, i∈I. Then

E₂^k,`

∼=H^k(H/F;H^`(F;H^q(N;A)))

∼=H^k(H/F; lim−→H^`(F;H^q(N;Ai))) by (5.18) (5.19)

∼= lim−→H^k(H/F;H^`(F;H^q(N;A_i))) asH/F is of typeF P∞

∼= lim−→E_2,i^k,`.

The isomorphisms involved above are natural maps. Thus,M SS₂ : lim−→E_2,i →E₂ is an isomorphism of bigraded abelian groups. It follows thatM SSis an isomorphism between spectral sequences. AsE_i,2^{k,`</sup>⇒ H<sup>k+`}(H;H^q(N;Ai))andE₂^{k,`</sup> ⇒ H<sup>k+`}(H;H^q(N;A)), (5.19) and Lemmas 2.15.12, 2.15.18 imply the desired result.

Theorem 5.2.8. LetΛbe a finite index set and let(G,{H_λ}_λ∈Λ,{N_λ}_λ∈Λ)be a group triple satisfying the Cohen-Lyndon property. Employ the notations defined in Notation 2.14.2. Suppose thatG, H_λ, λ∈Λ,are of typeF P∞. If, for eachλ∈Λ, either one of the following conditions holds, thenGis of typeF P∞.

(F₁) N_λis of typeF P∞.

(F₂) Hλis of the formKλ×Fλ, whereKλis finite andFλis a finite rank free group, andNλ6Fλ. Proof. By Theorem 5.2.5, it suffices to prove that the functorH^∗(G;·)preserves direct limits. Let{A_i}_i∈I be a directed system ofZG-modules and letA = lim−→Ai. Fori∈I, letEi ={(E_i,r, di,r)}_r>2 be the LHS spectral sequence for the triple(G,hhN ii, A_i). LetE ={(E_r, dr)}_r>2 be the direct limit of{E_i}_i∈I. Also letEA={(E_A,r, dA,r)}_r>2be the LHS spectral sequence for the triple(G,hhN ii, A).

By Lemma 2.15.18,E₂^p,q⇒lim−→H^p+q(G;A_i). The mapsA_i→A, i∈I, induce (a) a morphism

M SS :E→EA

between spectral sequences, by Proposition 4.3.4, (b) a natural map

N A_G: lim−→H^∗(G;A_i)→H^∗(G;A) (c) a natural map

N A^p,q

G : lim−→H^p(G;H^q(hhN ii;Ai))→H^p(G;H^q(hhN ii;A)) forp, q∈Z.

As there are natural isomorphisms

H⁰(hhN ii;A)∼=A, H⁰(hhN ii;Ai)∼=Ai, fori∈I,

forp∈Z,N A^p,0

G can be identified with the natural maplim−→H^p(G;A_i)→H^p(G;A)induced by the maps Ai → A, i ∈ I. Thus, it suffices to show thatN A^p,0

G is an isomorphism, which is done by using Lemma 5.1.8.

Forp∈Zandq6−1,N A^p,q

G is clearly an isomorphism as it is just a map from{0}to{0}.

Fixp∈Zandq >1. Leti, j∈Iwithi < j. Consider the following commutative diagram H^p(G;H^q(hhN ii;Ai)) H^p(G;H^q(hhN ii;Aj))

Q

λ∈ΛH^p(Hλ;H^q(Nλ;Ai)) Q

λ∈ΛH^p(Hλ;H^q(Nλ;Aj))

∼= ∼= (5.20)

where the horizontal maps are induced byAi →Aj, and the vertical isomorphisms are given by Proposition 4.2.1. Leti, jvary inI. (5.20) induces a commutative diagram corresponding to direct limits

lim−→H^p(G;H^q(hhN ii;A_i)) H^p(G;H^q(hhN ii;A))

lim−→

Q

λ∈ΛH^p(Hλ;H^q(Nλ;Ai)) Q

λ∈ΛH^p(Hλ;H^q(Nλ;A))

N A^p,q

G

∼= ∼= (5.21)

whose vertical maps, being direct limits of isomorphisms, are themselves isomorphisms.

Fixλ∈Λ. Consider the natural map

lim−→H^p(H_λ;H^q(N_λ;A_i))→H^p(H_λ;H^q(N_λ;A)) (5.22)

induced by the mapsAi →A, i∈I.

If (F1) holds forλ, then Theorem 5.2.5 implies that (5.22) is an isomorphism.

If (F2) holds forλ, then Lemma 5.2.7 implies that (5.22) is an isomorphism.

Letλvary inΛ. By taking direct product of (5.22), we obtain an isomorphism

Y

λ∈Λ

lim−→H^p(Hλ, H^q(Nλ, Ai))∼= Y

λ∈Λ

H^p(Hλ, H^q(Nλ, A)). (5.23)

As |Λ| < ∞, the operations Q

λ∈Λ andlim−→ commute with each other and thus isomorphism (5.23) implies that the lower horizontal map of (5.21) is an isomorphism. By Proposition 4.3.4,M SSandN A_G are compatible and forp, q ∈ Z,M SS₂^p,q can be identified withN A^p,q

G . AsGis of typeF P∞, Theorem 5.2.5 implies thatN A_G is an isomorphism. Thus, Lemma 5.1.8 implies thatN A^p,0

G is an isomorphism for allp∈Z.

Recall that a group Gis of type F P if (a) cd(G) < ∞ and (b)G is of type F P∞. The following corollary follows from Corollary 5.2.4 and Theorem 5.2.8.

Corollary 5.2.9. LetΛbe a finite index set and let(G,{H_λ}λ∈Λ,{N_λ}λ∈Λ)be a group triple satisfying the Cohen-Lyndon property. Suppose thatG, H_λ, λ∈Λ,are of typeF P. If, for eachλ∈Λ, either one of the following conditions holds, thenGalso is of typeF P.

(F₁) Nλis of typeF P∞.

(F₂) H_λis of the formK_λ×F_λ, whereK_λis finite andF_λis a finite rank free group, andN_λ6F_λ. Proof of Theorem 1.2.18. By Theorem 3.0.1, for sufficiently deepN H, the group triple(G, H, N)has the Cohen-Lyndon property. Thus, Theorem 1.2.18 follows from the case |Λ| = 1of Theorem 5.2.8 and Corollary 5.2.9.