aa r X i v : . [ m a t h . G R ] D ec A LIMIT APPROACH TO GROUP HOMOLOGY
IOANNIS EMMANOUIL AND ROMAN MIKHAILOV
Abstract.
In this paper, we consider for any free presentation G = F/R of a group G thecoinvariance H ( G, R ⊗ nab ) of the n -th tensor power of the relation module R ab and show thatthe homology group H n ( G, Z ) may be identified with the limit of the groups H ( G, R ⊗ nab ),where the limit is taken over the category of these presentations of G . We also consider thefree Lie ring generated by the relation module R ab , in order to relate the limit of the groups γ n R/ [ γ n R, F ] to the n -torsion subgroup of H n ( G, Z ). Introduction
It is well-known that one may use a presentation of a group G as the quotient F/R , where F is a free group, in order to calculate its (co-)homology. Besides Hopf’s formula for the secondhomology group H ( G, Z ) (cf. [1, Chapter II, Theorem 5.3]), another example supporting thatclaim is the existence of the Gruenberg resolution [3]. Using Quillen’s description of the cyclichomology of an algebra over a field of characteristic 0 as the limit of a suitable functor overthe category of extensions of the algebra (cf. [6]), the homology groups H n ( G, Q ) are describedin [2] as the limits of certain functors over the category of group extensions G = K/H (here,the group K is not necessarily free).Working in the same direction, we obtain in this paper a description of the even homologyof G with coefficients in an arbitrary Z G -module M as the limit of a functor over the category P of all free presentations G = F/R . More precisely, we use the associated relation module R ab = R/ [ R, R ] and prove that there is an isomorphism H n ( G, M ) ≃ lim ←− H ( G, M ⊗ R ⊗ nab ) , where the limit is taken over P . We note that the technique used in the present paper allowsus to interpret only the even homology of G as a limit. Together with the free associativering T R ab on R ab (which is built up by the tensor powers R ⊗ nab , n ≥ L R ab on R ab . The Lie ring L R ab is graded and its homogeneous componentin degree n ≥ γ n R/γ n +1 R , where ( γ i R ) i ≥ is the lower centralseries of R . Then, the inclusion L R ab ⊆ T R ab induces a natural map l n : γ n R/ [ γ n R, F ] −→ H ( G, R ⊗ nab )for all n ≥
1. The group γ n R/ [ γ n R, F ] is the kernel of the free central extension1 −→ γ n R/ [ γ n R, F ] −→ F/ [ γ n R, F ] −→ F/γ n R −→ H ( F/γ n R, Z ). Ithas been studied by many authors; a survey of the corresponding results may be found in [9].As an example, we note that the torsion subgroup of γ n R/ [ γ n R, F ], which is shown in [loc.cit.]
Research co-funded by European Social Fund and National Resources (EPEAEK II) PYTHAGORAS. to be an n -torsion group if n ≥
3, may be identified with the kernel of the so-called Guptarepresentation of F/ [ γ n R, F ] (cf. [4,8,10]). Confirming the existence of a close relationshipbetween the groups γ n R/ [ γ n R, F ] and the torsion in the homology of G , we show that the l n ’sinduce an additive map ℓ n : lim ←− γ n R/ [ γ n R, F ] −→ lim ←− H ( G, R ⊗ nab ) ≃ H n ( G, Z ) , whose image is contained in the n -torsion subgroup of H n ( G, Z ).The contents of the paper are as follows: In Section 1, we explain how one can use dimensionshifting by the powers of the relation module R ab , which is associated with a presentation G = F/R , in order to embed the homology groups H n ( G, ) into H ( G, ⊗ R ⊗ nab ) for all n ≥
1. In the following Section, we record some generalities about limits and prove a simplecriterion for them to vanish. In Section 3, we define the presentation category P of G andprove the existence of an isomorphism between H n ( G, ) and the limit of the H ( G, ⊗ R ⊗ nab )’s.Finally, in the last Section, we consider the free Lie ring on the relation module R ab and relatethe limit of the quotients γ n R/ [ γ n R, F ] to the n -torsion subgroup of H n ( G, Z ).It is a pleasure for both authors to thank I.B.S. Passi and R. St¨ohr for helpful commentsand suggestions.1. Relation modules and dimension shifting in homology
In this Section, we consider a group G and fix a presentation of it as the quotient of a freegroup F = F ( S ) on a set S by a normal subgroup R . We note that the conjugation actionof F on R induces an action of F on the abelianization R ab = R/ [ R, R ], which is obviouslytrivial when restricted to R . Therefore, the latter action induces an action of G on R ab . Theabelian group R ab , endowed with the G -action defined above, is referred to as the relationmodule of the given presentation.The augmentation ideal f ⊆ Z F of F is well-known to be free as a Z F -module; in fact, itis free on the set { s − s ∈ S } . In particular, the Z G -module Z G ⊗ Z F f is free on the set { ⊗ ( s −
1) : s ∈ S } . Moreover, it follows from [1, Chapter II, Proposition 5.4] that there isan exact sequence of Z G -modules(1) 0 −→ R ab µ −→ Z G ⊗ Z F f σ −→ Z G ε −→ Z −→ , where µ maps r [ R, R ] onto 1 ⊗ ( r −
1) for all r ∈ R , σ maps 1 ⊗ ( s −
1) onto sR − s ∈ S and ε is the augmentation homomorphism. We note that R , being a subgroup of thefree group F , is itself free; therefore, the relation module R ab is Z -free. Since this is also thecase for the other three terms of the exact sequence (1), we conclude that the latter is Z -split.We shall refer to the exact sequence (1) as the relation sequence associated with the givenpresentation of G . The map µ therein was defined by Magnus in [5]; it will be referred to asthe Magnus embedding. Lemma 1.1.
Let M be a Z G -module. Then, there are natural isomorphisms H i ( G, M ) ≃ H i − ( G, M ⊗ R ab ) for all i ≥ , where G acts on M ⊗ R ab diagonally.Proof. Since the relation sequence (1) is Z -split, we may tensor it with M and obtain theexact sequence of Z G -modules (with diagonal action)(2) 0 −→ M ⊗ R ab −→ M ⊗ ( Z G ⊗ Z F f ) −→ M ⊗ Z G −→ M −→ . LIMIT APPROACH TO GROUP HOMOLOGY 3 If N is a free Z G -module, then the Z G -module M ⊗ N (with diagonal action) is known tobe isomorphic with an induced module (cf. [1, Chapter III, Corollary 5.7]); in particular, thehomology of G with coefficients in M ⊗ N vanishes in positive degrees. Since the Z G -modules Z G ⊗ Z F f and Z G are free, we may use the exact sequence (2) and dimension shifting, in orderto obtain the existence of natural isomorphisms, as claimed. (cid:3) Corollary 1.2.
Let M be a Z G -module. Then, there are natural isomorphisms H n ( G, M ) ≃ H ( G, M ⊗ R ⊗ n − ab ) and H n +1 ( G, M ) ≃ H ( G, M ⊗ R ⊗ nab ) for all n ≥ .Proof. The result follows by induction on n , using Lemma 1.1. (cid:3) Corollary 1.3.
There are isomorphisms H n ( G, Z ) ≃ H ( G, R ⊗ n − ab ) and H n +1 ( G, Z ) ≃ H ( G, R ⊗ nab ) for all n ≥ . (cid:3) Remark 1.4.
The dimension shifting in the homology of a group G , which is associatedwith the relation module R ab as above, may be alternatively described by using cap products;see, for example, [11, § . χ ∈ H ( G, R ab ) be the cohomology class thatclassifies the group extension1 −→ R/ [ R, R ] −→ F/ [ R, R ] −→ G −→ , as in [1, Chapter IV, Theorem 3.12]. Then, the dimension shifting isomorphisms above areinduced by the cap product maps with χ or with suitable powers of it.We consider a Z G -module M and note that the Lyndon-Hochschild-Serre spectral sequenceassociated with the extension 1 −→ R −→ F −→ G −→ −→ H ( G, M ) −→ H ( G, H ( R, M )) −→ H ( F, M ) −→ H ( G, M ) −→ . Since M is trivial as a Z R -module, we have H ( R, M ) = M ⊗ R ab and hence the latter exactsequence reduces to0 −→ H ( G, M ) −→ H ( G, M ⊗ R ab ) −→ H ( F, M ) −→ H ( G, M ) −→ . We note that the above embedding of H ( G, M ) into H ( G, M ⊗ R ab ), which is provided bythe d -differential of the spectral sequence, is known to coincide (up to a sign) with the capproduct map with the cohomology class χ ∈ H ( G, R ab ) defined in Remark 1.4. In particular,replacing M by M ⊗ R ⊗ n − ab , we conclude that there is an exact sequence0 → H ( G, M ⊗ R ⊗ n − ab ) χ ∩ → H ( G, M ⊗ R ⊗ nab ) → H ( F, M ⊗ R ⊗ n − ab ) → H ( G, M ⊗ R ⊗ n − ab ) → n ≥ Proposition 1.5.
Let M be a Z G -module and consider the cohomology class χ ∈ H ( G, R ab ) defined in Remark 1.4. Then, there is an exact sequence → H n ( G, M ) χ n ∩ → H ( G, M ⊗ R ⊗ nab ) → H ( F, M ⊗ R ⊗ n − ab ) → H ( G, M ⊗ R ⊗ n − ab ) → IOANNIS EMMANOUIL AND ROMAN MIKHAILOV for all n ≥ . In particular, there is an exact sequence −→ H n ( G, Z ) χ n ∩ −→ H ( G, R ⊗ nab ) −→ H ( F, R ⊗ n − ab ) −→ H ( G, R ⊗ n − ab ) −→ for all n ≥ . (cid:3) Some generalities on limits
Let C be a small category, Ab the category of abelian groups and F : C −→ Ab a functor.Then, the limit lim ←− F of F is the subgroup of the direct product Q c ∈ C F ( c ), consisting of thosefamilies ( x c ) c which are compatible in the following sense: For any two objects c, c ′ ∈ C andany morphism a ∈ Hom C ( c, c ′ ), we have F ( a )( x c ) = x c ′ ∈ F ( c ′ ). We often denote the abeliangroup lim ←− F by lim ←− F ( c ).Let F , G be two functors from C to Ab . Then, a natural transformation η : F −→ G inducesan additive map lim ←− η : lim ←− F −→ lim ←− G , by mapping any element ( x c ) c ∈ lim ←− F onto ( η c ( x c )) c ∈ lim ←− G . In this way, lim ←− itself becomesa functor from the functor category Ab C to Ab .The proof of the following result is straightforward. Lemma 2.1.
The limit functor lim ←− : Ab C −→ Ab is left exact. (cid:3) We recall that the coproduct of two objects a, b of C is an object a ⋆ b , which is endowed withtwo morphisms ι a : a −→ a ⋆ b and ι b : b −→ a ⋆ b , having the following universal property:For any object c of C and any pair of morphisms f : a −→ c and g : b −→ c , there is a uniquemorphism h : a ⋆ b −→ c , such that h ◦ ι a = f and h ◦ ι b = g . The morphism h is usuallydenoted by ( f, g ).As an example, we note that the coproduct of two abelian groups M and N in the category Ab is the direct sum M ⊕ N , endowed with the obvious inclusion maps. For any abeliangroup T and any pair of additive maps f : M −→ T and g : N −→ T , the additive map( f, g ) : M ⊕ N −→ T is given by ( m, n ) f ( m ) + g ( n ), ( m, n ) ∈ M ⊕ N .The following elementary vanishing criterion will be used twice in the sequel. Lemma 2.2.
Let C be a small category and F : C −→ Ab a functor. We assume that:(i) Any two objects a, b of C have a coproduct ( a ⋆ b, ι a , ι b ) as above.(ii) For any two objects a, b of C the morphisms ι a : a −→ a ⋆ b and ι b : b −→ a ⋆ b inducea monomorphism ( F ( ι a ) , F ( ι b )) : F ( a ) ⊕ F ( b ) −→ F ( a ⋆ b ) of abelian groups.Then, the limit lim ←− F is the zero group.Proof. Let ( x c ) c ∈ lim ←− F be a compatible family and fix an object a of C . We consider thecoproduct a ⋆ a of two copies of a and the morphisms ι : a −→ a ⋆ a and ι : a −→ a ⋆ a .Then, we have F ( ι )( x a ) = x a⋆a = F ( ι )( x a ) LIMIT APPROACH TO GROUP HOMOLOGY 5 and hence the element ( x a , − x a ) is contained in the kernel of the additive map( F ( ι ) , F ( ι )) : F ( a ) ⊕ F ( a ) −→ F ( a ⋆ a ) . In view of our assumption, this latter map is injective and hence x a = 0. Since this is the casefor any object a of C , we conclude that the family ( x c ) c is the zero family, as needed. (cid:3) A limit formula for H n ( G, )We fix a group G and define the category of presentations P = P ( G ), as follows: Theobjects of P are pairs of the form ( F, π ), where F is a free group and π : F −→ G a surjectivegroup homomorphism. Given two objects ( F, π ) and ( F ′ , π ′ ) of P , a morphism from ( F, π ) to( F ′ , π ′ ) is a group homomorphism ϕ : F −→ F ′ such that π ′ ◦ ϕ = π . Since the groups thatare involved are free, we note that for any two objects ( F, π ) and ( F ′ , π ′ ) of P there is at leastone morphism from ( F, π ) to ( F ′ , π ′ ).Given an object ( F, π ) of P , we may consider the group ring Z F , the augmentation ideal f ,the kernel R = ker π , the relation module R ab and the associated Magnus embedding µ : R ab −→ Z G ⊗ Z F f . It is clear that all these depend naturally on the object (
F, π ) of P . Moreover, this is alsotrue for the cohomology class χ ∈ H ( G, R ab ) defined in Remark 1.4. Therefore, invoking thenaturality of the low degrees exact sequence which is induced by the Lyndon-Hochschild-Serrespectral sequence with respect to the group extension and the coefficient module, we concludethat the dimension shifting isomorphisms as well as the exact sequences of Proposition 1.5 arenatural with respect to the morphisms of P . In view of the left exactness of the limit functor(cf. Lemma 2.1), we thus obtain an exact sequence(3) 0 −→ H n ( G, M ) −→ lim ←− H ( G, M ⊗ R ⊗ nab ) −→ lim ←− H ( F, M ⊗ R ⊗ n − ab )for all n ≥
1, where the limits are taken over the category P . Lemma 3.1.
Let ( F, π ) and ( F ′ , π ′ ) be two objects of the presentation category P of G .(i) The coproduct ( F, π ) ⋆ ( F ′ , π ′ ) is provided by the object ( F ′′ , π ′′ ) of P , where F ′′ is thefree product of F and F ′ and π ′′ : F ′′ −→ G the homomorphism which extends both π and π ′ .(ii) Let ι : ( F, π ) −→ ( F ′′ , π ′′ ) and ι ′ : ( F ′ , π ′ ) −→ ( F ′′ , π ′′ ) be the structural morphisms ofthe coproduct ( F ′′ , π ′′ ) . Then, the induced maps ι ∗ : R ab −→ R ′′ ab and ι ′∗ : R ′ ab −→ R ′′ ab betweenthe corresponding relation modules are both split monomorphisms of Z G -modules.Proof. Assertion (i) is clear and, because of symmetry, we only have to prove assertion (ii)for the structural morphism ι . We note that the additive map ι ∗ : R ab −→ R ′′ ab is obtained byrestricting ι and then passing to the quotients. We choose a morphism ϕ : ( F ′ , π ′ ) −→ ( F, π )in P and consider the morphism λ = ( id F , ϕ ) : ( F ′′ , π ′′ ) −→ ( F, π ), which extends both theidentity of (
F, π ) and ϕ . Then, λ restricts to a group homomorphism λ : R ′′ −→ R , which isa left inverse of the restriction ι : R −→ R ′′ of ι and satisfies the equality λ ( ι ( x ) r ′′ ι ( x ) − ) = xλ ( r ′′ ) x − for all x ∈ F and r ′′ ∈ R ′′ . It follows that the additive map λ ∗ : R ′′ ab −→ R ab , which is inducedfrom λ by passage to the quotients, is a Z G -linear left inverse of ι ∗ . (cid:3) We can now state and prove our first main result.
IOANNIS EMMANOUIL AND ROMAN MIKHAILOV
Theorem 3.2.
Let M be a Z G -module. Then, there is an isomorphism of abelian groups H n ( G, M ) ∼ −→ lim ←− H ( G, M ⊗ R ⊗ nab ) , where the limit is taken over the category P of presentations of G for all n ≥ . In particular,there is an isomorphism H n ( G, Z ) ∼ −→ lim ←− H ( G, R ⊗ nab ) for all n ≥ .Proof. Let F : P −→ Ab be the functor which maps an object ( F, π ) of P onto the abeliangroup H ( F, M ⊗ R ⊗ n − ab ). In view of the exact sequence (3), the result will follow if we showthat lim ←− F = 0. To that end, we shall apply the criterion established in Lemma 2.2. We haveto verify that conditions (i) and (ii) therein are satisfied. To that end, we fix two objects( F, π ) and ( F ′ , π ′ ) of P and denote by R ab and R ′ ab the corresponding relation modules.In view of Lemma 3.1(i), the objects ( F, π ) and ( F ′ , π ′ ) have a coproduct in P , which isprovided by ( F ′′ , π ′′ ), where F ′′ is the free product of F and F ′ . Let R ′′ ab denote the relationmodule that corresponds to the coproduct ( F ′′ , π ′′ ). We have to prove that the map H ( F, M ⊗ R ⊗ n − ab ) ⊕ H ( F ′ , M ⊗ R ′ ⊗ n − ab ) −→ H ( F ′′ , M ⊗ R ′′ ⊗ n − ab ) , which is induced by the inclusions of F and F ′ into F ′′ , is injective. To that end, we note thatthe corresponding Mayer-Vietoris exact sequence shows that the natural map H ( F, M ⊗ R ′′ ⊗ n − ab ) ⊕ H ( F ′ , M ⊗ R ′′ ⊗ n − ab ) −→ H ( F ′′ , M ⊗ R ′′ ⊗ n − ab )is injective. Therefore, it only remains to prove that the natural maps H ( F, M ⊗ R ⊗ n − ab ) −→ H ( F, M ⊗ R ′′ ⊗ n − ab )and H ( F ′ , M ⊗ R ′ ⊗ n − ab ) −→ H ( F ′ , M ⊗ R ′′ ⊗ n − ab )are injective. We may now complete the proof invoking Lemma 3.1(ii), which itself impliesthat the natural map R ⊗ n − ab −→ R ′′ ⊗ n − ab (resp. R ′ ⊗ n − ab −→ R ′′ ⊗ n − ab ) is a split monomorphismof Z G -modules and hence of Z F -modules (resp. of Z F ′ -modules). (cid:3) The limit of the γ n R/ [ γ n R, F ] ’s Let H be a group. We recall that the lower central series ( γ n H ) n ≥ of H is given by γ H = H and γ n +1 H = [ γ n H, H ] for all n ≥
1. Then, the graded Lie ring
Gr H = L ∞ n =1 Gr n H of H isdefined in degree n to be the (additively written) abelian group Gr n H = γ n H/γ n +1 H . TheLie bracket on Gr H is defined by letting( xγ n +1 H, yγ m +1 H ) = [ x, y ] γ n + m +1 H, where [ x, y ] = x − y − xy for all x ∈ γ n H and y ∈ γ m H (cf. [7, Chapter 2]).On the other hand, if A is an abelian group then we may consider the free associative ringon A , i.e. the tensor ring T A = L ∞ n =0 A ⊗ n . We recall that the multiplication in T A is definedby concatenation of tensors. The associated Lie ring
LT A is equal to
T A as an abelian group,whereas its Lie bracket is defined by letting ( x, y ) = xy − yx for all x, y ∈ T A . The free Liering on A is the Lie subring L A of LT A generated by A . In fact, L A is a graded subring LIMIT APPROACH TO GROUP HOMOLOGY 7 of LT A , whose homogeneous component L n A ⊆ A ⊗ n of degree n is generated as an abeliangroup by the left normed n -fold commutators ( x , . . . , x n ), x , . . . , x n ∈ A .We now consider a group H and its abelianization H ab = H/ [ H, H ]. Then, in view of theuniversal property of the free Lie ring L H ab , the identity map of H ab = L H ab into H ab = Gr H extends to a graded Lie ring homomorphism κ : L H ab −→ Gr H.
It is clear that κ depends naturally on H . In particular, for all n ≥ κ n : L n H ab −→ γ n H/γ n +1 H, which is natural in H . We note that if the group H is free then the map κ (and hence all ofthe κ n ’s) is bijective; cf. [7, Chapter 4, Theorem 6.1].We now fix a group G and consider an object ( F, π ) of the presentation category P of G with kernel R = ker π . We specialize the discussion above to R and note that the termsof its lower central series are normal subgroups of F ; in particular, F acts on each quotient Gr n R = γ n R/γ n +1 R , by letting x · yγ n +1 R = xyx − γ n +1 R for all x ∈ F and y ∈ γ n R . Thelatter action being trivial on R , it induces an action of G on the Gr n R ’s. Endowed with thataction, the abelian group Gr n R is referred to as the n -th higher relation module associatedwith the given presentation. (For n = 1, we recover the relation module Gr R = R ab .) It isclear that the induced action of G on Gr R is compatible with the Lie bracket. On the otherhand, the diagonal action of G on the tensor powers R ⊗ nab induces a G -action on T R ab , which iscompatible with multiplication. In particular, G acts on the associated Lie ring LT R ab by Liering automorphisms. It is easily seen that the action of any group element on LT R ab restrictsto a Lie ring automorphism of the free Lie ring L R ab . In particular, L R ab is a Z G -submoduleof LT R ab and the homogeneous component L n R ab is a Z G -submodule of R ⊗ nab for all n ≥ κ n : L n R ab −→ γ n R/γ n +1 R is Z G -linear for all n ≥
1. Moreover, since the group R is free (being a subgroup of the freegroup F ), the latter map is an isomorphism. For all n ≥ Z G -linear map λ n : γ n R/γ n +1 R −→ R ⊗ nab , which is defined as the composition γ n R/γ n +1 R κ − n −→ L n R ab ֒ → R ⊗ nab . Since the group H ( G, γ n R/γ n +1 R ) is identified with γ n R/ [ γ n R, F ], the Z G -linear map λ n defined above induces an additive map l n : γ n R/ [ γ n R, F ] −→ H ( G, R ⊗ nab )for all n ≥
1. The abelian groups J Gn ( R ab , Z ) = ker l n have been studied in [10] by M.W.Thomson, who proved that they are n -torsion for all n ≥
1. It is clear that l n dependsnaturally on the object ( F, π ) of the presentation category P of G . Therefore, taking limitsover P , we obtain the additive map ℓ n = lim ←− l n : lim ←− γ n R/ [ γ n R, F ] −→ lim ←− H ( G, R ⊗ nab ) . IOANNIS EMMANOUIL AND ROMAN MIKHAILOV
We can now state our second main result.
Theorem 4.1.
Let n be an integer with n ≥ . Then, under the isomorphism between thehomology group H n ( G, Z ) of G and the limit lim ←− H ( G, R ⊗ nab ) , which is established in Theorem3.2, the image of the additive map ℓ n defined above is contained in the n -torsion subgroup H n ( G, Z )[ n ] of H n ( G, Z ) . The proof of the Theorem will occupy the remaining of the Section. Let (
F, π ) be an objectof the presentation category P of G and consider the associated Magnus embedding µ : R ab −→ Z G ⊗ Z F f and the n -th tensor power map µ ⊗ n : R ⊗ nab −→ ( Z G ⊗ Z F f ) ⊗ n , which is also Z G -linear. The composition γ n R/γ n +1 R λ n −→ R ⊗ nab µ ⊗ n −→ ( Z G ⊗ Z F f ) ⊗ n is a Z G -module map, which induces, by applying the functor H ( G, ), the composition γ n R/ [ γ n R, F ] l n −→ H ( G, R ⊗ nab ) µ ⊗ n −→ H (cid:0) G, ( Z G ⊗ Z F f ) ⊗ n (cid:1) . We shall denote the latter composition by ϕ n . We deduce the existence of an exact sequence0 −→ J Gn ( R ab , Z ) −→ ker ϕ n l n | −→ H ( G, R ⊗ nab ) , where l n | denotes the restriction of l n to the subgroup ker ϕ n ⊆ γ n R/ [ γ n R, F ]. The key pointis that, as shown in [8], the map l n maps ker ϕ n into the n -torsion subgroup H ( G, R ⊗ nab )[ n ] of H ( G, R ⊗ nab ) for all n ≥
2. Therefore, we conclude that there is an exact sequence0 −→ J Gn ( R ab , Z ) −→ ker ϕ n l n | −→ H ( G, R ⊗ nab )[ n ] . We shall now consider the commutative diagram with exact rows0 −→ J Gn ( R ab , Z ) −→ ker ϕ n l n | −→ H ( G, R ⊗ nab )[ n ] k ↓ ↓ −→ J Gn ( R ab , Z ) −→ γ n R/ [ γ n R, F ] l n −→ H ( G, R ⊗ nab )where both unlabelled vertical arrows are the corresponding inclusion maps. Since all mapsinvolved are natural with respect to the given object ( F, π ) of the presentation category P of G , we may invoke Lemma 2.1 in order to obtain a commutative diagram with exact rows0 −→ lim ←− J Gn ( R ab , Z ) −→ lim ←− ker ϕ n ℓ n | −→ lim ←− H ( G, R ⊗ nab )[ n ] k ↓ ↓ −→ lim ←− J Gn ( R ab , Z ) −→ lim ←− γ n R/ [ γ n R, F ] ℓ n −→ lim ←− H ( G, R ⊗ nab ) As shown in [10, Proposition 1], if n ≥ ϕ n can be identified with the kernel of a certainmatrix representation of the group F/ [ γ n R, F ], which was defined by C.K. Gupta and N.D. Gupta in [4].
LIMIT APPROACH TO GROUP HOMOLOGY 9
Since the limit lim ←− H ( G, R ⊗ nab )[ n ] of the n -torsion subgroups is identified with the n -torsionsubgroup of the limit lim ←− H ( G, R ⊗ nab ), the assertion in the statement of Theorem 4.1 followsfrom the next result. Lemma 4.2.
The additive map lim ←− ker ϕ n −→ lim ←− γ n R/ [ γ n R, F ] , which is induced by theinclusions ker ϕ n ֒ → γ n R/ [ γ n R, F ] , is an isomorphism for all n ≥ .Proof. In view of Lemma 2.1, the exact sequence0 −→ ker ϕ n −→ γ n R/ [ γ n R, F ] ϕ n −→ H (cid:0) G, ( Z G ⊗ Z F f ) ⊗ n (cid:1) , which is associated with an object ( F, π ) of P as above, induces an exact sequence0 −→ lim ←− ker ϕ n −→ lim ←− γ n R/ [ γ n R, F ] φ n −→ lim ←− H (cid:0) G, ( Z G ⊗ Z F f ) ⊗ n (cid:1) , where φ n = lim ←− ϕ n . Hence, the result will follow if we show that lim ←− H ( G, ( Z G ⊗ Z F f ) ⊗ n ) = 0.To that end, we shall apply the criterion established in Lemma 2.2. We have to verify thatconditions (i) and (ii) therein are satisfied. In view of Lemma 3.1, any two objects ( F, π ) and( F ′ , π ′ ) of P have a coproduct, which is provided by ( F ′′ , π ′′ ), where F ′′ is the free product of F and F ′ . Therefore, if F (resp. F ′ ) is free on the set S (resp. S ′ ), then F ′′ is free on the disjointunion S ′′ of S and S ′ . It follows that the Z G -modules Z G ⊗ Z F f , Z G ⊗ Z F ′ f ′ and Z G ⊗ Z F ′′ f ′′ are free on the sets { ⊗ ( s −
1) : s ∈ S } , { ⊗ ( s ′ −
1) : s ′ ∈ S ′ } and { ⊗ ( s ′′ −
1) : s ′′ ∈ S ′′ } respectively. Hence, the inclusions of F and F ′ into F ′′ induce an isomorphism of Z G -modules( Z G ⊗ Z F f ) ⊕ ( Z G ⊗ Z F ′ f ′ ) ∼ −→ Z G ⊗ Z F ′′ f ′′ . Therefore, considering n -th tensor powers, we conclude that the natural map( Z G ⊗ Z F f ) ⊗ n ⊕ ( Z G ⊗ Z F ′ f ′ ) ⊗ n −→ ( Z G ⊗ Z F ′′ f ′′ ) ⊗ n is a split monomorphism of Z G -modules. Therefore, applying the functor H ( G, ), we con-clude that the natural map H (cid:0) G, ( Z G ⊗ Z F f ) ⊗ n (cid:1) ⊕ H (cid:0) G, ( Z G ⊗ Z F ′ f ′ ) ⊗ n (cid:1) −→ H (cid:0) G, ( Z G ⊗ Z F ′′ f ′′ ) ⊗ n (cid:1) is a (split) monomorphism of abelian groups, as needed. (cid:3) Remarks 4.3 (i) Let (
F, π ) be an object of the presentation category P of G . Then, as shownin [8, Theorem 2], the kernel ker ϕ n of the additive map ϕ n constructed above coincides withthe torsion subgroup of γ n R/ [ γ n R, F ] for all n ≥
2. Therefore, it follows from [9] that ker ϕ n is an n -torsion group if n ≥ n = 2. Since this is also the case for thelimit of these groups, we may invoke Lemma 4.2 in order to conclude that lim ←− γ n R/ [ γ n R, F ]is an n -torsion group if n ≥ n = 2. The latter assertion providesanother proof of Theorem 4.1, in the case where n ≥ −→ lim ←− J Gn ( R ab , Z ) −→ lim ←− γ n R/ [ γ n R, F ] −→ H n ( G, Z )[ n ] , where the limits are taken over the presentation category P of G , for all n ≥
2. In order toobtain an embedding of the limit lim ←− γ n R/ [ γ n R, F ] into the n -torsion subgroup H n ( G, Z )[ n ] of the homology group H n ( G, Z ), at least in the case where n ≥
3, one may ask whether theabelian group lim ←− J Gn ( R ab , Z ) is zero. Following M.W. Thomson, who studied the vanishingof the group J Gn ( R ab , Z ) in [10], we consider the following special cases:(ii1) Assume that G is a finite group of order relatively prime to n . Then, the homologygroup H n ( G, Z ) has no non-trivial n -torsion elements and the group J Gn ( R ab , Z ) vanishes forany presentation G = F/R (cf. [10, Theorem 2(ii)]). Therefore, taking into account the exactsequence above, it follows that lim ←− γ n R/ [ γ n R, F ] = 0 for all n ≥ G is ≤
2. Then, the group J Gn ( R ab , Z )vanishes for any presentation G = F/R (cf. [10, Theorem 2(iii)]), whereas the homology group H n ( G, Z ) vanishes for all n ≥
2. Therefore, taking into account the exact sequence above, itfollows that lim ←− γ n R/ [ γ n R, F ] = 0 for all n ≥ F, π ) be an object of the presentation category P of G and consider a Z G -module M . We also consider the Magnus embedding µ : R ab −→ Z G ⊗ Z F f and the Z G -linear map µ n,M = id M ⊗ µ ⊗ n : M ⊗ R ⊗ nab −→ M ⊗ ( Z G ⊗ Z F f ) ⊗ n . Then, as shown in [10, Lemma 8], the kernel of the induced additive map µ n,M : H ( G, M ⊗ R ⊗ nab ) −→ H (cid:0) G, M ⊗ ( Z G ⊗ Z F f ) ⊗ n (cid:1) is identified with the homology group H n ( G, M ) for all n ≥
1. Since the exact sequence0 −→ H n ( G, M ) −→ H ( G, M ⊗ R ⊗ nab ) µ n,M −→ H (cid:0) G, M ⊗ ( Z G ⊗ Z F f ) ⊗ n (cid:1) depends naturally on the object ( F, π ) of P , we may invoke Lemma 2.1 in order to obtain anexact sequence of abelian groups0 −→ H n ( G, M ) −→ lim ←− H ( G, M ⊗ R ⊗ nab ) −→ lim ←− H (cid:0) G, M ⊗ ( Z G ⊗ Z F f ) ⊗ n (cid:1) , where the limits are taken over the category P . Using exactly the same argument as in theproof of Lemma 4.2, we can show that lim ←− H ( G, M ⊗ ( Z G ⊗ Z F f ) ⊗ n ) = 0. We conclude thatthe group H n ( G, M ) is isomorphic with the limit lim ←− H ( G, M ⊗ R ⊗ nab ) for all n ≥
1, obtainingthereby an alternative proof of Theorem 3.2. References [1] Brown, K.S.: Cohomology of Groups. (Grad. Texts Math. ) Berlin Heidelberg New York: Springer1982[2] Emmanouil, I., Passi, I.B.S.: Group homology and extensions of groups. (preprint)[3] Gruenberg, K.W.: Cohomological Topics in Group Theory. Lecture Notes in Math. . Berlin HeidelbergNew York: Springer 1970[4] Gupta, C.K., Gupta, N.D.: Generalized Magnus embeddings and some applications. Math. Z. , 75-87(1978)[5] Magnus, W.: On a theorem of Marshall Hall. Ann. Math. , 764-768 (1939)[6] Quillen, D.: Cyclic cohomology and algebra extensions. K-Theory , 205-246 (1989) This argument was communicated to us by R. St¨ohr.
LIMIT APPROACH TO GROUP HOMOLOGY 11 [7] Serre, J.P.: Lie algebras and Lie groups. New York: Benjamin 1965[8] St¨ohr, R.: On Gupta Representations of Central Extensions. Math. Z. , 259-267 (1984)[9] St¨ohr, R.: On torsion in free central extensions of some torsion-free groups. Journal of Pure Appl. Alg. , 249-289 (1987)[10] Thomson, M.W.: Representations of Certain Verbal Wreath Products by Matrices. Math. Z. , 239-257(1979)[11] Zerck, R.: On the homology of the higher relation modules. Journal of Pure Appl. Alg. , 305-320 (1989) Department of Mathematics, University of Athens, Athens 15784, Greece
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