Generators for Group Homology and a Vanishing Conjecture
GGenerators for Group Homology and a VanishingConjecture
Joshua Roberts
Abstract
Letting G = F/R be a finitely-presented group, Hopf’s formulaexpresses the second integral homology of G in terms of F and R .Expanding on previous work, we explain how to find generators of H ( G ; F p ). The context of the problem, which is related to a conjectureof Quillen, is presented, as well as example calculations. Exploiting a classical theorem due to Hopf, we presented a series of algo-rithms in [14] that give upper bounds on group homology in homologicaldimensions one and two, provided coefficients are taken in a finite field. Inparticular, examples confirmed the results in [3], as well a new result, con-cerning the rank two special linear group over rings of number theoreticinterest. This paper can be viewed as both a sequel and expansion of theresults in [14].The initial motivation for constructing the algorithms was to gain in-sight into special cases of a conjecture originally given by Quillen in 1971,which we briefly discuss in Section 2. However, since the algorithms in [14]depend only upon Hopf’s formula for H , the usefulness of these algorithmsextends to groups beyond the scope of Quillen’s Conjecture. Moreover, thealgorithms are distinct from existing methods of calculating low dimensionalgroup homology in that they give an upper bound on the homology of anyfinitely-presented group, though the upper bound is, at times, very large.The main contribution of this paper is in Section 3 wherein we present atechnique that expounds on the algorithms in [14] to find explicit generatorsof these homology groups. The technique relies heavily upon the above men-tioned Hopf’s formula for the second homology group of a finitely-presentedgroup; the calculations are carried out with the computational algebra pro-gram GAP [6]. 1 a r X i v : . [ m a t h . K T ] S e p s a byproduct of the calculations related to Quillen’s Conjecture we areinvolved in a long term project of preparing a database for low dimensionalgroup homology of linear groups over number fields and their rings of inte-gers. This work will be extended to other classes of finitely-presented groupsof interest to computational group theory and algebraic topology. The firstset of these calculations is found in Section 4.We note that when it is clear from the context, we occasionally omitexplicitly writing the ground ring of linear groups as well as homology coef-ficients. One motivational problem for low dimensional group homology, which isrelated to algebraic K-theory, is the study of homology for groups GL j ( R ),where GL j is a finite rank j general linear group and R is the ring of in-tegers in a number field. An approach to this problem is to consider thediagonal matrices inside GL j . Let D j denote the subgroup formed by thesematrices. Then the canonical inclusions D j ⊂ GL j for j = 0 , , ... inducehomomorphisms on group homology with k -coefficients ρ : H i ( D j ( R ); k ) → H i ( GL j ( R ); k ) . (2.1)In [13] Quillen conjectured: Conjecture 2.1.
The homomorphism ρ , as given above, is an epimorphismfor R = Z [ ζ p , /p ] , p a regular odd prime, ζ p a primitive p th root of unity, k = F p and any values of i and j . Conjecture 2.1 has been proved in a few cases and disproved in infinitelymany other cases. For R = Z [1 /
2] it was proved by Mitchel in [12] for j = 2and by Henn in [8] for j = 3. Anton gave a proof for R = Z [1 / , ζ ] and j = 2 in [1].Dwyer gave a disproof for the conjecture for R = Z [1 /
2] and j = 32 in [5]which Henn and Lannes improved to j = 14 in [9]; this is an improvementin light of Henn’s result in [7] that states that if Conjecture 2.1 is falsefor j then it is false for all j ≥ j . Anton disproved the conjecture for R = Z [1 / , ζ ] and j ≥
27 also in [1]. The interested reader should consult[10] for more details.This conjecture was reformulated and, in a sense, corrected by Anton:
Conjecture 2.2. [2]
Given p, k and R as above, H ( GL ( R ); k ) ∼ = H ( D ( R ); k ) . (2.2)2nton’s conjecture led to a proof of Conjecture 2.1 for Z [1 / , ζ ] and i = j = 2. For a survey on the current status of conjectures 2.1 and 2.2 wecite [3]. Given a group extension 1 → N → G → Q → E p,q ∼ = H p ( Q ; H q ( N ; k )) = ⇒ H p + q ( G ; k ) , (2.3)where we take coefficients in a field k regarded as a trivial G -module. Weuse this spectral sequence to reduce a special case of Quillen’s conjecture toan exercise in linear algebra. Lemma 2.3.
Fix R = SL ( Z [ ζ p , /p ]) and field of coefficients k = F p , H ( GL ( R ); k ) ∼ = ( H ( SL ( R ); k ) GL ( R ) /Im ( τ ) ⊕ H ( GL ( R ); k ) , (2.4) where, for a group G and a G -module M , M G is the group of co-invariantsand τ is the transgression map E , → E , .Proof. We first note that R is a Euclidean ring [4], which, by Lemma 7.2[2] implies that SL ( R ) is a perfect group. Thus, applying the spectralsequence 2.3 to the extension1 → SL ( R ) → GL ( R ) → GL ( R ) → , (2.5)we see that the entries E p, are all 0. Thus for q < E page is equalto the E page.We also note that GL ( R ) ∼ = D ( R ) ∼ = R × , (2.6)where R × is the group of units of R .Figure 1 displays the E page of this spectral sequence, and we haveincluded the transgression τ : E , → E , for reference. Note that since E p,q ∼ = E p,q for all p and for all q < E p, ∼ = E ∞ p, . Moreover, E p,q ∼ = E ∞ p,q for p, q + 1 < E , ∼ = H ( SL ( R )) GL ( R ) /Im ( τ ). Since we have chosen3igure 1: E page with τ : E , → E , displayedfield coefficients, any extension problems are trivial. Thus we have thefollowing decomposition. H ( GL ( R )) ∼ = E , ⊕ E , ⊕ E , (2.7) ∼ = H ( SL ( R )) GL ( R ) /Im ( τ ) ⊕ H ( GL ( R )) . (2.8)This immediately implies the following corollary. Corollary 2.4.
As vector spaces over k ,dim H ( GL ( R ); k ) ≥ dim H ( GL ( R ); k ) (2.9)Recall from Section 2 that the Quillen Conjecture implies that the mapinduced by inclusion H ( D ( R )) (cid:16) H ( GL ( R )) (2.10)4s surjective. Anton’s reformulation of Quillen’s conjecture in [3] and resultsin [2] imply that map 2.10 factorizes thusly: H ( D ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:37) (cid:37) (cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75)(cid:75) H ( GL ) H ( D ) (cid:56) (cid:56) (cid:56) (cid:56) (cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114) . (2.11)Then H ( D ( R )) (cid:16) H ( GL ( R )) is surjective anddim H ( D ) ≥ dim H ( GL ) . (2.12)Then equations 2.9, 2.12, and 2.6 imply Conjecture 2.2 H ( GL ( R )) ∼ = H ( GL ( R )) , (2.13)which, by Equation 2.8, is equivalent to H ( SL ( R )) GL ( R ) ∼ = Im ( τ ) , (2.14)which is true if and only if τ is surjective. Moreover, for Conjecture 2.2 to betrue, it is sufficient for the finitely-presented group SL ( Z [1 /p, ζ p ]) to havetrivial second dimensional F p -homology.In this context, the purpose of [14] was to give a series of algorithmsthat estimated the second homology group of any finitely-presented group.More precisely, given a finitely-presented group G and a finite field k , thesecond homology group H ( G ; k ) with coefficients in k is a finite dimensionalvector space over k . Our algorithms give an upper bound for the dimensionof H ( G ; k ) and, in particular cases, the algorithms calculate precisely thisdimension.The algorithms confirmed results by Anton that Conjecture 2.1 holds for R = SL ( Z [1 /p, ζ p ]) and k = F p for p = 3 and p = 5 ([2] and [3]). Let 1 → K i → F q → G → F is afinitely generated free group and K is finitely generated as an F -module withthe F -action given by conjugation, i and q denote inclusion and quotienthomomorphism, respectively. That is, G has finite presentation given bythe generators of F modulo then normal closure of K in F .5 heorem 3.1 (Hopf) . Given
G, F, K as above, there is an exact sequence → [ F, R ] → [ F, F ] → H ( G, Z ) → . This gives an exact sequence1 → H ( G, Z ) → R [ F, R ] → F [ F, F ] → FR [ F, F ] → . The last two terms are finitely generated abelian groups and algorithmsexist to give their structure. Also in [14], we explain how to use this exactsequence to find an upper bound on the dimension of H ( G ; k ), where k isthe finite field of prime characteristic p .The inclusion homomorphism i : K → F induces a homomorphism i ∗ : A → B where we have denoted K/K p [ F, K ] by A and F/K p [ F, F ] by B .Note that for k ∈ K and f ∈ F we have that [ k, f ] = k f k − = 1 in A . Thus k f = k in A which gives that A is a trivial F -module. Let S K be the set ofgenerators of K as an F -module.We note that the image of i ∗ is generated by the set of all i ∗ ( k ) for k ∈ S K . Then since B is a vector space over k , there is a subset S (cid:48) K ⊂ S K such that i ∗ ( k (cid:48) ) with k ∈ S (cid:48) K is a basis for the image of i ∗ .The primary interest is on the kernel of i ∗ , which is isomorphic to H ( G ; k ). As stated above, a previous paper gives an upper bound n onthe dimension of this vector space. We seek an explicit description of these n elements of S K . To this end, we restate two facts: • A is a vector space that is spanned by S K • S (cid:48) K ⊂ S K is a subset with i ∗ ( S (cid:48) K ) a basis for the image of i ∗ in B Let v ∈ A , then v = (cid:88) λ ∈ S K c λ λ , where c λ ∈ k , by (1). Note that i ∗ ( v ) = 0is equivalent to (cid:88) λ ∈ S K c λ i ∗ ( λ ) = 0 in B .Moreover, each i ∗ ( λ ) = (cid:88) µ ∈ S (cid:48) K a λ,µ i ∗ ( µ ), where a λ,µ ∈ k , by (2). There-fore, i ∗ ( v ) = 0 in B if and only if (cid:88) µ ∈ S (cid:48) K (cid:88) λ ∈ S K c λ a λ,µ i ∗ ( µ ) = 0 in B, (cid:88) λ ∈ S K c λ a λ,µ = 0for all µ ∈ S (cid:48) K . So we need to solve for the c λ ’s and find a basis for thesolutions.If a ∈ A then a = k f k f · · · = k k · · · = k + k + · · · in F p . We want touse linear algebra in B to find a basis for the image of i ∗ { i ∗ ( k ) : k ∈ K } . FK p [ F, F ] (cid:36) (cid:36) (cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73)(cid:73) KK p [ F, K ] i ∗ (cid:58) (cid:58) (cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117) j ∗ (cid:47) (cid:47) FK p [ F, K ] We consider the group G = SL ( Z [1 / , ζ ]), where ζ p is a primitive p th rootof unity. In [3] it is proven that this group is generated by S = { z, u , u , u , a, b, b , b , b , b , b , b , b , w } modulo the relations R = { b − t z t bz t a, w − z u u u , z , [ z, u i ] , [ u i , u j ] , a , [ a , z ] , [ a , u i ] ,a − zaz, a − u i au i , [ b s , b t ] , b − a , b − b b b b b b b , b − t w − b − t w, ( b b − a − u ) , ( b b − a − u ) , ( b b − a − u ) , ( b b − b − b a − u u ) , ( b b − b − b a − u u ) , ( b b − b − b a − u u ) , ( b b − b − b b b b − a − u u u ) , a − b − u i bz − i b − b − z i bz − i u i } where i, j ∈ { , , } and s, t ∈ { , , , , , } .That is, there is a short exact sequence 1 → N ( R ) → F ( S ) → G → S generating the free group F ( S ) and the set R normallygenerating the subgroup N ( R ) ⊂ F ( S ).We begin by reducing the number of generators and relators in F ( S ) /N ( R )in order to simplify the final calculations. Via GAP, it is easy to verify thefollowing. 7 roposition 3.2. There is an isomorphism of finitely-presented groups thatmaps the generators of the free group F ( S ) to the free group generated by S (cid:48) = { z, u , u , u , a, b } in the following way: z (cid:55)→ zu (cid:55)→ u u (cid:55)→ u u (cid:55)→ u a (cid:55)→ ab (cid:55)→ z − b z a − b (cid:55)→ z − b z b (cid:55)→ b b (cid:55)→ z b z − b (cid:55)→ z − b zb (cid:55)→ z b z − b (cid:55)→ z − b z b (cid:55)→ zb z − w (cid:55)→ z − u z − u u . Moreover, the isomorphic finitely-presented group has set of 32 relations R (cid:48) = { zu z − u − , u u u − u − ,u u u − u − ,u au a − ,u au a − ,zu z − u − ,a ,u u u − u − ,zaza − ,zu z − u − ,u au a − ,z ,b z − b zb − z − b − z,b z − b z b − z − b − z ,z − b z − a − b z − a − b z a,b z − b z − b − z − a − u a − z − b − u ,b z − b − z − b a − z u a − z − b − u ,b z − b z − b − z − u − a − z − b − u ,b z − b z b − z − b − z ,z − b − zu − zu − u − zb − z − u z − u u ,b − u − zu − u − z b − z − u u u , − b − z − u − z − u − u − z − b − u u u ,zb − u − u − u − z b − z − u z − u u ,z b − u − z − u − u − z − b − zu u u ,z b z − b z − b zb zb z b z − b a − ,z − b z − b − u − a − b z − b − z − u − a − z − b z − b − z − u − a − ,z − b − z − b z u − a − z − b − z − b z u − a − z − b − z − b z u − a − ,b − z − b za − z − u b − z − b z u − a − b − z − b z u − a − ,b − z − b z − b z − b − a − z u u z b − zb z b zb − u − a − u z b − zb z b zb − u − a − u ,b − z − b − z − b z − b z − u − z − a − u z − b z b − zb − z b u − z − a − u z − b − z − b z b − z b u − z − a − u ,b − z b − z − b z − b za − z − u u z − b z − b − z b zb − zu − a − u z − b z − b − z b zb − zu − a − u z ,zb z − b − z − b zu − za − u u z − b z − b − z − b z b z b − zb − z − u − a − u u z − b z − b − z − b z b z b − zb − z − u − a − u u b − z b zb − } . By [14], the dimension of H ( G ; F ) as a vector space over F is at most6. We now seek generators of of this vector space. For simplicity, we denote F ( S (cid:48) ) by F and N ( R (cid:48) ) by N . An application of the FindBasis algorithmfrom the same paper gives that NN [ F, N ] is generated by the 12 elements [ f1*f5*f1*f5^-1,f2*f3*f2^-1*f3^-1,f2*f5*f2*f5^-1,f7*f5^-1*f7*f5^-1*f7*f5,f3*f7*f1^-2*f7*f1*f7^-2*f5^-1*f3*f1^-1*f5^-1*f7^-1,f4*f7*f1^2*f7^-2*f1^2*f7*f5^-1*f1^-2*f4*f1^-1*f5^-1*f7^-1,f2*f7*f1*f5^-1*f1^-2*f5*f7^-2*f1^3*f7*f5^-1*f2*f1^-1*f5^-1*f7^-1,f1^2*f7^-7*f1^-1*f3^-1*f1^-1*f4^-1*f2^-1*f1^-2*f7^-1*f1^2*f2*f3*f4,f7*f1*f7*f1^2*f7*f1*f7*f1^2*f7*f1^3*f7*f1^3*f7*f1*f5^-1*f1^-1*f5^-1,f7*f1^2*f7^-1*f1^-1*f4^-1*f1^-1*f5^-1*f7*f1^2*f7^-1*f4^-1*f1^-2* By reducing these elements in F [ F, F ] N we obtain [
J. Pure Appl.Algebra , 144(1):1–20, 1999.[2] M. F. Anton. An elementary invariant problem and general linear groupcohomology restricted to the diagonal subgroup.
Trans. Amer. Math.Soc. , 355(6):2327–2340 (electronic), 2003.[3] M. F. Anton. Homological symbols and the Quillen conjecture.
J. PureAppl. Algebra , 213(4):440–453, 2009.[4] P. M. Cohn. On the structure of the GL of a ring. Inst. Hautes ´EtudesSci. Publ. Math. , (30):5–53, 1966.[5] W. G. Dwyer. Exotic cohomology for GL n ( Z [1 / Proc. Amer. Math.Soc. , 126(7):2159–2167, 1998.[6] The GAP Group.
GAP – Groups, Algorithms, and Programming, Ver-sion 4.4.10 , 2007. URL .[7] H.-W. Henn. Commutative algebra of unstable K -modules, Lannes’ T -functor and equivariant mod- p cohomology. J. Reine Angew. Math. ,478:189–215, 1996.[8] H.-W. Henn. The cohomology of SL ( Z [1 / K -Theory , 16(4):299–359, 1999.[9] H.-W. Henn, J. Lannes, and L. Schwartz. Localizations of unstable A -modules and equivariant mod p cohomology. Math. Ann. , 301(1):23–68,1995.[10] K. P. Knudson.
Homology of linear groups , volume 193 of
Progress inMathematics . Birkh¨auser Verlag, Basel, 2001.1211] J. McCleary.
A user’s guide to spectral sequences , volume 58 of
Cam-bridge Studies in Advanced Mathematics . Cambridge University Press,Cambridge, second edition, 2001.[12] S. A. Mitchell. On the plus construction for
BGL Z [ ] at the prime 2. Math. Z. , 209(2):205–222, 1992.[13] D. Quillen. The spectrum of an equivariant cohomology ring. I, II.
Ann.of Math. (2) , 94:549–572; ibid. (2) 94 (1971), 573–602, 1971.[14] J. Roberts. An algorithm for low dimensional group homology.
Ho-mology, Homotopy and Applications , 12:27–37, 2010. URL