Homology of 2-Types and Crossed Modules

Journal of Algebra 244, 352᎐365Ž. 2001 doi:10.1006rjabr.2001.8919, available online at http:rrwww.idealibrary.com on

OnŽ. Co Homology of 2-Types and Crossed Modules

T. Datuashvili and T. Pirashvili

A. Razmadze Mathematical Institute, Aleksidze str. 1, Tbilisi, 380093, View metadata, citation and similar papers at core.ac.ukRepublic of Georgia brought to you by CORE Communicated by Michel Broue´ provided by Elsevier - Publisher Connector

Received March 2, 2001

1. INTRODUCTION

␲ s / A connected CW-complex X is called the 2-type if i X 0 for i 1, 2. Following Elliswx 8 we investigate the homology of 2-types by means of crossed modules, which are algebraic models for 2-typesŽ seewx 1, 11 for this and more general results. . More precisely, a crossed module is a group homomorphism Ѩ: T ª G together with an action of G on T satisfying

Ѩ Ѩ Ž.g t s gѨ Ž.tgy1 and ts s tsty1 , g g G, t, s g T.

Any crossed module Ž.T, G, Ѩ has a classifying space BTŽ., G, Ѩ , which is a 2-type, with ␲ Ѩ ( Ѩ ␲ Ѩ ( Ѩ 12BTŽ., G, Coker Ž.and BT Ž.Ž., G, Ker .

Moreover any 2-type up to homotopy equivalence is of such typeŽ see, for example,wx 11.Ž . Among other things, we will prove see Theorem 4.1. that if Ž.T, G, Ѩ is a projective crossed modulewx 15 with free G, then Ѩ s HBT3 Ž.Ž., G, 0 Ѩ ( ␲ ⌫ 2␲ s HBTiiŽ.Ž., G, H y41Ž., 2, i 4,5. Ѩ Moreover for HBT6ŽŽ, G, ..one has an exact sequence

␲ ⌫ 2␲ ª ␲ ⌫ 3␲ ª Ѩ ª ␲ ⌫ 2␲ ª H31Ž.Ž., 2H 01, 2HBT 6Ž.Ž., G, H21Ž., 2 0. ␲ s ␲ Ѩ s ⌫ Here iiBTŽ., G, for i 1, 2 and *A denotes the divided power algebra on an abelian group A Žseewx 7. . This result in the dimensions 3

352 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. Ž.CO HOMOLOGY OF 2-TYPES AND CROSSED MODULES 353 and 4 and for a free crossed module Ž.T, G, Ѩ was proved by Ellis Ž see Theorem 8 ofwx 8. using the theory of quadratic complex of Baueswx 1 . Our proof is based on a theorem of Ratcliffewx 15 and uses a standard argument with Serre spectral sequence. Our next main result establishes a vanishing line for a spectral sequence related withŽ. co homology of classifying space of crossed modules. Accord- ing to the results of Lodaywx 11 , the category of crossed modules is equivalent to the category of simplicial groups, whose Moore complex has length one. This gives rise to the spectral sequence with abutment the homology of the classifying space of a crossed module Ž.ŽT, G, Ѩ see the first part of Theorem 5.3. . The E1 term of the spectral sequence is given via homology of groups, which are components of the simplicial group corresponding to a crossed module Ž.T, G, Ѩ under Loday’s equivalence of categories. The second part of Theorem 5.3 gives a vanishing line for this 2 s ) particular spectral sequence. It says that Epq 0 provided p q. The same spectral sequence was also considered by EllisŽ see Corollary 3 ofwx 8 . but without a vanishing line. The vanishing result of Theorem 5.3 is related to our previous workŽ seewx 4, 13. . Namely the main result ofwx 13 can be interpreted as a particular case of Theorem 5.3, corresponding to the case ␲ Ѩ s when 2 BTŽ., G, 0. However the proof given here is new and simpler than the one given inwx 13 . A result ofwx 4 proved in cohomological framework in much more general context of internal categories in cate- p1 s ) gories of groups with operations shows that E2 0 provided p 2. Inwx 4 the first author defined and completely described cohomologies of internal categories in categories of groups with operations. Since internal categories in the category of groups are equivalent to crossed modulesŽ see wx11. the approach ofwx 4 gives also a cohomology theory for crossed modules. This theory was further developed inwx 5 , where cohomologically trivial crossed modules are characterized. We clarify the relation between cohomologies fromwx 4 and cohomologies of the classifying spaces of crossed modules. The exact statements are stated in partŽ. i of Theorem 5.3 and in Corollary 5.5. The authors were partially supported by Grants INTAS-97-31961 and INTAS-99-00817 and by the TMR network K-theory and algebraic groups, Grant ERB FMRX CT-97-0107. The first author is grateful to DAAD for the support in preparing this paperŽ two-month fellowship in Frankfurt University. . The second author is grateful to MPI at Bonn for hospitality.

2. PRELIMINARIES ON CROSSED MODULES

Let G be a group. A G-group is a group T together with a homomorphism ␰ : G ª AutŽ.T . For g g G and t g T we put g t [ ␰ Ž.Ž.gt. 354 DATUASHVILI AND PIRASHVILI

DEFINITION 2.1. A G-crossed module is a G-group T together with a group homomorphism Ѩ: T ª G satisfying

Ѩ Ѩ Ž.g t s gѨ Ž.tgy1 and ts s tsty1 , g g G, t, s g T.

By crossed module we mean a triple Ž.T, G, Ѩ , where G is a group and Ѩ: T ª G is a G-crossed module. It follows from the definition that the image ImŽ.Ѩ is a normal subgroup of G, and the kernel KerŽ.Ѩ is in the center of T. Moreover the action of G on T induces an action of Q on KerŽ.Ѩ , where Q s Coker Ѩ. A homomorphism of crossed modules is a pair of homomorphisms Ž.Ž.Ž␾, f : T, G, Ѩ ª TЈ, Ѩ Ј, GЈ .such that

f (Ѩ s Ѩ Ј(␾, ␾ Ž.g t sfŽ g .␾ Ž.t , g g G, t g T.

We let XMod be the category of crossed modules. For a group Q and for a Q-module A one denotes by X extŽ.Q, A the category of short exact sequences

Ѩ 0 ª A ª T ª G ª Q ª 0, where Ž.T, G, Ѩ is a crossed module and the action of Q on A induced from the crossed module structure coincides with the prescribed one. The morphisms in X extŽ.Q, A are commutative diagrams

6 6 Ѩ 6 6 6 0 ATGQ0

id␾ f id 6 6 6 6

6 6 Ѩ Ј 6 6 6 0 ATЈ GЈ Q 0, where Ž.␾, f is a morphism of crossed modules ŽT, G, Ѩ .Žª TЈ, GЈ, Ѩ .. We let X extŽ.Q, A be the class of the connected components of the category X extŽ.Q, A . For a simplicial group G# we let N#G# be the Moore normalization of G# Žseewx 12. , which is a chain complex, whose n-dimensional component is given by

# s Ѩ n ª ) NGniF KerŽ.: GnGny1 , n 0 i)0 # ª # Ѩ while the boundary map d: NGnnNGy10is the restriction of on # wx NGn . By 11 we know that the Moore normalization yields an equiva- # ª lence of categories N : SGF1 XMod. Here SGF1 is the category of simplicial groups whose Moore complex is of length one. Ž.CO HOMOLOGY OF 2-TYPES AND CROSSED MODULES 355

Let Ž.T, G, Ѩ be a crossed module. Since G acts on T, it acts also on T n g s gg by Ž.Žt1,...,tn t1,..., tn.. Hence we can take the semi-direct product s n i s #y1 Ѩ Gn T G. Here G0 G. Let N Ž.T, G, be a simplicial group, which is given as follows

y1 Ѩ s G NnnŽ.T , G, G , n 0, Ѩ s F F i Ž.t1 ,...,tn , g Ž.t1 ,...,ˆtin,...,t , g ,1i n, Ѩ s y1 y1 Ѩ 01Ž.t ,...,tn , g Ž.tt21 ,...,ttn 11, tg, s sti Ž.Ž1 ,...,tn , g t1 ,...,tin,0,...,t , g .. One easily checks that N#Ž N#y1 Ž..Ž.T, G, Ѩ ( T, G, Ѩ . Thus we have the following #y1 ª ¨ LEMMA 2.2. The functor N : XMod SGF1 is also an equi alence of # g #y1 # # categories and for any G SGF1 one has an isomorphism N Ž.N G ( G#. Now for a crossed module Ž.T, G, Ѩ we can define the classifying space BTŽ., G, Ѩ to be the classifying space of the simplicial group N#y1 Ž.T, G, Ѩ . ␲ Ѩ s ) Hence i BTŽ., G, 0, i 1 and ␲ Ѩ ( Ѩ ␲ Ѩ ( Ѩ 12BTŽ., G, Coker , BTŽ., G, Ker . By abuse of notation we write ␲ Ѩ s Ѩ ␲ Ѩ s Ѩ ␲ Ѩ s ) 12Ž.T , G, Coker , Ž.T , G, Ker , i Ž.T , G, 0, i 2. Thus BTŽ., G, Ѩ is a 2-type, meaning a path-connected CW-complex X ␲ s G Ѩ with i X 0 for i 3. Therefore the homotopy type of BTŽ., G, is ␲ ␲ completely determined by 12, and the corresponding k-invariant of Ѩ 3 ␲ ␲ Ѩ BTŽ., G, which is an element in H Ž.12, . We let kT Ž, G, .be this element. The following result was first proved by Lodaywx 10 .

THEOREM 2.3. There is a well-defined map k: X extŽ.Q, A ª H 3 Ž.Q, A which assigns the element kŽ. T, G, Ѩ to a crossed moduleŽ. T, G, Ѩ . More- o¨er this map is a bijection. A morphism of crossed modules Ž.Ž.Ž␾, f : T, G, Ѩ ª TЈ, Ѩ Ј, GЈ .is said ¨ ␲ Ѩ ( to be a weak equi alence if it induces isomorphisms 1Ž.T, G, ␲ Ј Ѩ Ј Ј ␲ Ѩ ( ␲ Ј Ѩ Ј Ј 122Ž.Ž.Ž.T , , G and T, G, T , , G . It is well known that there is an equivalence of categories HoŽ.Ž 2-types ª Ho crossed modules . 356 DATUASHVILI AND PIRASHVILI between corresponding homotopy categoriesŽ see, for example,wx 1, 11. . Here HoŽ.ŽŽ 2-types resp. Ho crossed modules .. is the localization of the category of 2-typesŽ. resp. crossed modules with respect to weak equiva- lences.

3. ON HOMOLOGY OF CROSSED MODULES

All results of this section are well known to experts. We use these results in Section 4 and Section 5.

DEFINITION 3.1. Let Ž.T, G, Ѩ be a crossed module and let M be a ␲ Ѩ wx 1Ž.T, G, -module. According to 8 we define the homology H#ŽŽT, G, Ѩ ., M . and cohomology H*ŽŽT, G, Ѩ ., M .of ŽT, G, Ѩ . with coefficients in M to be the homology and cohomology of BTŽ., G, Ѩ with coefficients in the local system corresponding to M. The following result is well knownŽ see, for example,wx 8. .

LEMMA 3.2. If Ž.Ž.Ž␾, f : T, G, Ѩ ª TЈ, Ѩ Ј, GЈ .is a weak equi¨alence then it induces isomorphisms H#Ž.Ž.Ž.T , G, Ѩ , M ª H# ŽTЈ, Ѩ Ј, GЈ ., M , H*Ž.Ž.Ž.TЈ, Ѩ Ј, GЈ , M ª H* Ž.T , G, Ѩ , M . s ␲ s Ѩ s Let us note that if T 0, then 1 G and BTŽ.Ž., G, KG,1 . Hence in this case one obtains the classical homology and cohomology of groups. The second extreme case is when G s 1. In this case one has Ѩ ( ␲ BTŽ.Ž.,G, K 2 , 2 . In order to state the next result we need to fix some notation. For an abelian group A we let ⌫*A be the divided power algebra generated by A wx7 . For ) s 2 we obtain Whitehead’s universal quadratic functor ⌫ 2. Moreover, let R: Ab ª Ab be the first simplicial derived functor of the functor ⌫ 2. This functor was introduced by Eilenberg and Mac Lanewx 7 . According to these authors one has a functorial decomposi- tion RAŽ.Ž.Ž.Ž.[ B ( RA[ RB[ Tor A, B . Moreover R preserves all filtered colimits and vanishes on torsion-free groups and RŽ.Zrn s Zr Ž.2, n . HereŽ. 2, n is the ideal in Z generated by 2 and n. The homology of Eilenberg᎐Mac Lane spaces KAŽ., 2 is well understood thanks to the work of Eilenberg and Mac Lane, and Serre and Cartan. We will summa- rize the information of these groups in the following theorem. Ž.CO HOMOLOGY OF 2-TYPES AND CROSSED MODULES 357

THEOREM 3.3.Ž. i For any abelian group A one has isomorphisms s s HK13Ž. A,2 0 HK Ž. A,2 , ( ( ⌫ 2 HK24Ž. A,2 A, HK Ž. A,2 A. s Ž.ii If A is a torsion-free abelian group, then Hi KŽ. A,2 0 ifiisodd ( ⌫ n G and H2 n KŽ. A,2 A, n 0. Ž.iii For any abelian group one has

( ⌫ 3 ( HK65Ž. A,2 A and HKŽ. A,2 RA Ž.. Before we return to the general crossed modules, let us consider the ␲ ␲ Ѩ s case when the action of 12on is trivial and kTŽ., G, 0. In this Ѩ ␲ = case BTŽ., G, has the same homotopy type as the product KŽ.1,1 ␲ KŽ.2 , 2 and hence the Kunneth¨ theorem describes theŽ. co homology of Ѩ ␲ ␲ Ž.T, G, in terms of Ž. co homology of 12and KŽ., 2 . Such a situation arises when Ž.T, G, Ѩ is abelian. We recall that Ž.T, G, Ѩ is abelian if both G and T are abelian groups and the action of G on T is trivial. In this case the k-invariant is trivial because it lies in the image of

2 ␲ ␲ ª ␲ ␲ ( 3 ␲ ␲ Ext Z Ž.12, X ext Ž.Ž. 12, H 12, 2 s and Ext Z 0. This simple observation shows that the homology of abelian # Ѩ ␲ ␲ crossed modules H Ž.T, G, can be described in terms of 12and Žcompare with Theorem 24 ofwx 8. .

PROPOSITION 3.4.Ž. i For any crossed module Ž T, G, Ѩ ., there exist spectral sequences

2 s ␲ ␲ « Ѩ EpqH qŽ.1 , HKq Ž.Ž.2 ,2 ,M HTpqq Ž. Ž, G, ., M pq s q ␲ q ␲ « pqq Ѩ E212H Ž., H Ž.K Ž.,2 ,M H Ž. ŽT , G, ., M .

Ž.ii One has the isomorphisms

#2 s # ␲ #2 s #2 s # ␲ m ␲ E 011212H Ž., M , E 0, E H Ž., M , #2 s # ␲ Z ␲ #2 s # ␲ m ⌫ 2␲ E 3112412H ŽŽ, Tor Ž.M, , E H , M . and exact sequences

иии ª # ␲ m ␲ ª #2 ª # ␲ Z ⌫ 2␲ H Ž.125112, M RŽ. E H Ž., Tor Ž.M, ª # ␲ m ␲ ª #2 H y11Ž., M RŽ.2 E y1,5 ª # ␲ Z ⌫ 2␲ ª иии H y11Ž., Tor 1Ž.M, 2 358 DATUASHVILI AND PIRASHVILI

and

иии ª # ␲ m ⌫ 3 ␲ ª #2 ª # ␲ Z ␲ H Ž.Ž126112, M Ž. E H , Tor Ž.M, RŽ. . ª # ␲ m ⌫ 3 ␲ ª #2 H y11Ž., M Ž.2 E y1,6 ª # ␲ Z ␲ ª иии H y11Ž., Tor 1Ž.M, RŽ. 2 .

Ž.iii The first nontri¨ial differential

2 s ␲ ª ␲ m ␲ s 2 d3 : Ep0 Hp Ž.1 , M Hpy31 Ž, M 2 .Epy3,2

yl g 3 ␲ ␲ ¨ Ѩ is equal to Ž. k, where k H Ž.12, is the k-in ariant ofŽ. T, G, and l denotes the cap product. Proof. It is well known that Ž.T, G, Ѩ fits in a fibration sequence ␲ ª <<Ѩ ª ␲ K Ž.21,2 BT Ž, G, .K Ž.,1 . Thus the associated Serre spectral sequence givesŽ. i . Since the space ␲ KŽ.2 , 2 is 1-connected one can use the universal coefficient theorem and Theorem 3.3 to get isomorphisms inŽ. ii . The last part is the consequence of Theorem 10.4 ofwx 9 . Of course similar results hold for cohomology as well. For instance, the 3 py3 ␲ ␲ ª p ␲ differential d : H ŽŽ..12,Hom , M H Ž. 1, M fits in the commutative diagram

j 6 py3 ␲ ␲ k p ␲ ␲ m ␲ H Ž12, Hom Ž, MH .. Ž 122, HomŽ, M ..

¨# d3 6 e

p ␲ 6 H Ž.1, M ¨ ␲ m ␲ ª m ¬ where e : 22HomŽ., M M is given by x f fxŽ..

4. ON HOMOLOGY OF PROJECTIVE CROSSED MODULES

Let G be a fixed group and let ␦: Z ª G be a function from a set Z to G. Consider the category of crossed modules Ž.T, G, Ѩ together with maps ␣: Z ª T satisfying Ѩ␣ s ␦. One can prove that this category has an initial object, which is called a free crossed module generated by ␦ Žseew 15 and references given therex.Ž . A G-crossed module T, G, Ѩ .is called projective if it is a projective object in the category of G-crossed modules Žseewx 15. . Clearly any free crossed module is projective. A theorem of Ž.CO HOMOLOGY OF 2-TYPES AND CROSSED MODULES 359

wx Ѩ Ratcliffe 15 claims that Ž.T, G, is projective iff Tab is projective as ␲ s Ѩ Ѩ# ª s Ѩ 12Coker -module and : HT HN2is trivial. Here N Im . The aim of this section is to prove the following theorem.

THEOREM 4.1. Let G be a free group and letŽ. T, G, Ѩ be a projecti¨e ␲ s Ѩ ␲ s Ѩ ␲ G-crossed module. Let 12Coker and Ker . Let M be a 1-module. Then there are isomorphisms

Ѩ s Ѩ ( ␲ ⌫ 2␲ m HT340Ž.Ž.Ž., G, , M 0, HT Ž., G, , M H Ž.1, 2M , Ѩ ( ␲ ⌫ 2␲ m HT51Ž.Ž., G, , M H Ž.1, 2M and one has an exact sequence

␲ ⌫ 2␲ m ª ␲ ⌫ 3␲ m H31Ž.Ž., 2 M H01, 2 M ª Ѩ ª ␲ ⌫ 2␲ m ª HT62Ž.Ž., G, , M H Ž.1, 2M 0.

Remark. The results on H34and H were obtained by EllisŽ see Theorem 8 ofwx 8. . His proof uses results ofwx 1 on quadratic complexes. We leave the task to the interested reader to make the trivial reformulation for the corresponding result for cohomology groups. Proof. Let N s Im Ѩ. Then one has short exact sequences ª ␲ ª ª ª 0 2 T N 0 and ª ª ª ␲ ª 0 N G 1 0. Since N is free and the first short exact sequence is central, it splits and hence induces short split exact sequences of abelian groups ª ␲ ª ª ª 0 2ababT N 0, ª ␲ m ª m ª m ª 0 2aM T bM N abN 0. However, these are in general nontrivial extensions in the category of 1 ␲ -modules. The class of the first short exact sequence in Ext w␲ xŽ.N , ␲ 1 Z 1 ab 2 is denoted by k1. Since G is free, the second short exact sequence yields the following short exact sequence ª ª rwxª ␲ ª 0 Nab G N, G 1 0. 2 ␲ s j Denote its corresponding class in H Ž.1ab, N by k 2. Clearly k k1k 2. Since G is free it follows from the cup product reduction theoremwx 16 that yl ␲ ª ␲ m k2 : HnŽ.1 , M Hny21 Ž, M N ab . 360 DATUASHVILI AND PIRASHVILI is an isomorphism if n G 3Ž this fact is also a consequence of the Hochschild᎐Serre spectral sequence for the following short exact sequence ª ª ª ␲ ª of groups 1 N G 1 1 and the fact that homology of free G groups vanishes in dimension 2. . By the theorem of Ratcliffe Tab is a ␲ m ␲ projective 1a-module. Hence T bM is also a projective 1-module and

yl ␲ m ª ␲ m ␲ k1 : HnŽ.Ž.1ab, M N Hny11, M 2 is an isomorphism if n G 2 and a monomorphism for h s 1. Then it follows that the differential

syl ␲ ª ␲ m ␲ d3 k : HnŽ.1 , M Hny31 Ž, M 2 . of the Serre spectral sequence

2 s ␲ ␲ « Ѩ EpqH pŽ.1 , HKq Ž.Ž.2 ,2 ,M HTpqq Ž. Ž, G, ., M

G s ␲ is an isomorphism for n 4 and a monomorphism for n 3. Since 2 is wx␲ a subgroup of a projective Z 1 -module, it is free as an abelian group and 2 s by the second part of Theorem 3.3 we have Epq 0 for odd q and 2 s ␲ m ⌫ n␲ s G EpqH pŽ , M 2 . for q 2n, n 0 and the result follows.

5. ANOTHER SPECTRAL SEQUENCE

In this section we consider the spectral sequence constructed in Corol- lary 3 ofwx 8 which converges to H#Ž.T, G, Ѩ . It is a consequence of a wx 2 s ) general result of 14 that Epq 0 for p q. We give here a more direct proof of this fact, which allows us to get more information on E 2. # ␲ # Let G be a simplicial group and let M be a 0Ž.G -module. Since ␲ # s ␲ # 10BG G one can consider M as a local system on the classifying space BG# of G#. By abuse of notation we will denote the homology and cohomology of BG# with coefficients in M by H#Ž.G#, M and H*Ž.G#, M . Clearly for G# s N#y1 Ž.T, G, Ѩ one recovers the groups #y1 ª considered in Definition 3.1. Here N : Xmod SGF1 is the same as in Lemma 2.2. We give also an algebraic definition of the same groups. For a group G and G-module A we let C#Ž.ŽŽ..G, A resp. C* G, A be the standard chain Ž.resp. cochain complex for the group homology Ž resp. cohomology . . Now # ␲ # if G is a simplicial group and if M is a 0Ž.G -module, then one can form the bicomplex C#Ž.ŽŽ..Ž.G#, M resp. C* G#, M whose n, m -compo- n nent is CGnmŽ.Ž, M resp. CGŽ..m, M . The following fact is well known. Ž.CO HOMOLOGY OF 2-TYPES AND CROSSED MODULES 361

# ␲ # LEMMA 5.1. Let G be a simplicial group and let M be a 0Ž.G -module. Then one has the isomorphisms and spectral sequences H#Ž.TotŽ.C#Ž.G#, M ( H# Ž.G#, M , H*Ž. TotŽ.C*Ž.G#, M ( H* Ž.G#, M 1 s « # EpqHG qŽ. p, M HGpqq Ž., M , pq s q « pqq # E1 H Ž.Gp , M H Ž.G , M .

Proof. If G# is a simplicial group, then BGŽ.# is a diagonal of the bisimplicial set associated to the simplicial set given by wx¬ n BGŽ.n . Now the result is a consequence of the Eilenberg᎐Zilberg᎐Cartier theoremwx 6 . #1 Let us observe that E q is a simplicial abelian group. So we can take # #1 U q the Moore normalization of it to obtain N E q. Similarly, E1 is a cosimplicial abelian group and we still can take the normalization of it to U q wx wx obtain N*E1 Žsee 2. . Then the classical normalization theorem 12 together with Theorem 5.1 ofwx 14 imply the following # ␲ COROLLARY 5.2. Let G be a simplicial group and let M be a 00Ž.G - module. Then there exists a spectral sequence

#1 « # NEpqHG pqq Ž., M , which has the same E##r terms as the pre¨ious one as soon as r G 2. ¨ # s ) #1 s s r ¨ Moreo er if Nnp G 0 for n k then N E q0 Epqpro ided that p ) kq and r G 2. Similar results hold in cohomology.

THEOREM 5.3. LetŽ. T, G, Ѩ be a crossed module and let M be a ␲ Ѩ 1Ž.T, G, -module. Ž.i Then there exist spectral sequences

pq s q « pqq Ѩ E1 H Ž.Gp , M H Ž.Ž.T , G, , M , 1 s « Ѩ EpqHG qŽ. p, M HTpqq Ž.Ž., G, , M .

# s #y1 Ѩ s p i Here G N Ž.T, G, and hence Gp T G. The cohomology of Ѩ wx p1 Ž.T, G, with coefficients in M in the sense of 4 is isomorphic to E2 pro¨ided that p ) 0. #1 s s r ) G Ž.ii One has Npq E 0 E pq as soon as p q and r 2. A similar ¨anishing result holds also for cohomology. 362 DATUASHVILI AND PIRASHVILI

Ž.iii One has isomorphisms

m #1ab( p m NEppHG0 ž/, Ž.T M m p U p ( ab p NE1 HomG ž/Ž.T , M . Here p G 1 and T ab s TrwxT, T . Remarks.Ž. 1 The spectral sequence is the same as that in Corollary 3 ofwx 8 . Ž.2 The existence of the spectral sequences as well as that of part Ž. ii is a consequence of Corollary 5.2. Indeed it suffices to take G# s NTy1 Ž., G, Ѩ in Corollary 5.2. However we give also a new proof forŽ. ii , which is far simpler then the original proof of the vanishing result in Corollary 5.2 given inwx 14 . The proof of partsŽ. ii and Ž iii . of Theorem 5.3 is based on the notion of a degree of a functorwx 7 . Since we are in a slightly different situation than that inwx 7 we give definitions of a cross-effect and of a degree of a functor in our framework. We let GG be the category of G-groups. For a functor ª s F: GG Ab with FŽ.0 0 one can define the cross-effect functors n ª G s crnGF: G Ab, n 0 as follows. We put cr1 F F. Then the bifunctor cr2 F is defined by the short exact sequence ª ª = ª = ª 0 cr212FTŽ.Ž, T FT 1T 2 .Ž.Ž.FT 1FT 2 0. In general

crn FTŽ.1 ,...,Tn s = иии = ª = иии = ˆ = иии = Ker FTŽ.1 Tn Ł FTŽ.1 TinT . ž/i

We claim that if ª ª ª ª 0 F12F F 0 is a short exact sequence of functors, then ª ª ª ª 0 crn F1 crnnF cr F2 0 is also an exact sequence, n G 1. The claim is obvious for n s 1 and for n s 2 it is a consequence of a 3 = 3 lemma. The general case follows from this by induction, because ( crn FXŽ.Ž1 ,..., Xn cr21Ž.UX,..., XXny2 .Ž.ny1 , Xn . Ž.CO HOMOLOGY OF 2-TYPES AND CROSSED MODULES 363

Here ª UXŽ.1 ,..., Xny2 : GG Ab is given by s UXŽ.1 ,..., XYny2 Ž.Ž.crny11FX,..., Xny2 , Y . wx F s Following 7 we will say that a functor F is of degree n if crnq1 F 0. Thus if deg F s n and FЈ is a subquotient of F then deg FЈ F n. Proof of Theorem 5.3.Ž. i Here we have only to establish a relation- ship between the definition of cohomology inwx 4 and the one given in Section 3. Comparing the definitions one sees that the cochain complex K *ŽŽT, G, Ѩ ., M . constructed in Section 3 ofwx 4 is isomorphic to the cochain complex DerŽ.G#, M and the exact sequence of cosimplicial abelian groups 0 ª H 0 Ž.G#, M ª M ª Der Ž.Ž.G#, M ª H 1 G#, M ª 0 p Ѩ ( p1 shows that one has an isomorphism HK*ŽŽT, G, ., M . E2 provided that p ) 0. Ž.ii and Ž iii . We consider only homology. The dual argument works for cohomology as well. The case q s 0 is trivial, because in this case #1 s # E 00HGŽ., M is a constant simplicial abelian group with value ␲ ) H01Ž., M . So we assume that q 0. Let N be a G-module and let K be a G-group. We can consider N as a trivial K-module. It is a consequence of the Kunneth¨ theorem that the functor y ª ¬ HnGŽ., N : G Ab, K HKnŽ., N is of degree n and the last nontrivial cross-effect is

y s ab m иии m ab m crnnŽ.H Ž.Ž, NK1 ,...,K n .K1 K n N. For any p G 0 and q G 1 we let ª Fpq: G G Ab s be a functor given by FKpqŽ.HGp Ž, HK q Ž, N ... Since crn is exact and F preserves direct sums we see that degŽ.Fpq q and ( ab m иии m ab m crqpqFKŽ.1 ,...,K qpHGŽ., K1 K q N . ª We need also a functor FqG: G Ab given by [ i ª FKqqŽ.KerŽ.HK ŽG, N .HG q Ž, N .. 364 DATUASHVILI AND PIRASHVILI

s ᎐ It is clear that FqŽ.0 0 and by the Hochschild Serre spectral sequence one has the spectral sequence of functors 2 s « ) EpqF pqF pqq , q 0. ¬ Let us fix an integer n. Since the correspondence F crn F gives an exact 2 s « functor we have also the spectral sequence Epqcr nF pqcr nF pqq. Since F 22s - s deg Fpq q we see that Epq 0 for q n and E0 n HG0Ž.,crnnH . F s s Therefore deg Fnnn and cr Fnncr F0 n HG0Ž.,crnnH . s s n Now we take N M. One observes that the constructions of Gn T i Ѩ ) G and i, i 0 use only the action of G on T and not the homomor- Ѩ ª Ѩ phism : T G, which appears only in the definition of 0. Hence #1 Ѩ NEpqdepends only on T considered as a G-module and not on .By comparing the definitions it is obvious that for any crossed module Ѩ #1 Ѩ ( Ž.T, G, one has the isomorphism NEpqŽ.T, G, crpqFT Ž,...,T .. This completes the proof. Let us recall that for any crossed module Ž.T, G, Ѩ there is a weak homotopic equivalence Ž.Ž.TЈ, GЈ, Ѩ Ј ª T, G, Ѩ with free group GЈ Žsee, for example, p. 11 inwx 8. . Our next result is an immediate consequence of Theorem 5.3.

COROLLARY 5.4. For any crossed moduleŽ. T, G, Ѩ one has an exact sequence ª Ѩ ª ab ª ab ª Ѩ ª HG22HTŽ.Ž., G, HG0, T G HT1 Ž., G, 0. Moreo¨er if G is a free group then Ѩ ( rwxª ab HT2 Ž., G, KerŽ.T T , G G and there exists an exact sequence i ª Ѩ ª ab m ab HT34Ž.ŽG HT, G, .T G T ␣ ª i ª Ѩ ª HT23Ž.ŽG HT, G, .0. A result similar to this is true also for cohomology. Ѩ ( rwxª ab Remarks.Ž. 1 The isomorphism HT2 Ž, G, .Ker ŽT T, G G . for free G was proved by EllisŽ see Theorem 6 ofwx 8. using different methods. ␣ abm ab ª i Ž.2 The homomorphism : T G T HT2Ž.G is the restric- 2 i ª i abm ab ; tion of d02: HTŽ G.Ž.HT2 G on the subgroup T G T 2 i ᎐ wx HT2Ž G. and hence in terms of the Brown Loday tensor product 3 it is given by x m y ª Ž.Ž.xy1, Ѩ x m y, 1 . Here x, y g T and x denotes the class of x in T ab. Ž.CO HOMOLOGY OF 2-TYPES AND CROSSED MODULES 365

COROLLARY 5.5. LetŽ. T, G, Ѩ be a crossed module. Choose any crossed moduleŽ. TЈ, GЈ, Ѩ Ј which is weakly equi¨alent to Ž. T, G, Ѩ , such that GЈ is a free group. Then there is an isomorphism

H 2 Ž.Ž.Ž.T , G, Ѩ , A ( H1 ŽTЈ, GЈ, Ѩ Ј ., A , where the right-hand side is an internal category cohomology corresponding to the crossed moduleŽ. TЈ, GЈ, Ѩ Ј as defined in wx4. The proof is the direct consequence of Lemma 3.2 and the cohomological version of the previous corollary. It is also an immediate consequence of Theorem 3.1 ofwx 4 and Theorem 6 of wx 8 .

REFERENCES

1. H.-J. Baues, ‘‘Combinatorial Homotopy and 4-Dimensional Complexes,’’ De Gruyter, Berlin, 1991. 2. A. K. Bousfield and D. M. Kan, ‘‘Homotopy Limits, Completions and Localizations,’’ Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, BerlinrNew York, 1972. 3. R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 Ž.1987 , 311᎐335. 4. T. Datuashvili, Cohomology of internal categories in categories of groups with operations, in ‘‘Proceedings CAT-TOP International Conference, Prague, 1988,’’ pp. 270᎐283, World Scientific, Singapore, 1989. 5. T. Datuashvili, Cohomologically trivial internal categories in categories of groups with operations, Appl. Categ. Structures 3 Ž.1995 , 221᎐237. 6. A. Dold and D. Puppe, Homologie nicht-additiver Funktoren, Anwendungen, Ann. Inst. Fourier. Ž.Ž.Grenoble 11 1961 , 201᎐312. 7. S. Eilenberg and S. Mac Lane, On the groups HŽ.␲ , n , II, Ann. of Math. 60 Ž.1954 , 49᎐139. 8. G. J. Ellis, Homology of 2-types, J. London Math. Soc. 46 Ž.1992 , 1᎐27. 9. V. K. A. M. Gugenheim, On a theorem of E. H. Brown, Illinois J. Math. 4 Ž.1960 , 292᎐311. 10. J. L. Loday, Cohomologie et groupe de Steinberg relatifs, J. Algebra 54 Ž.1978 , 178᎐202. 11. J. L. Loday, Spaces with finitely many homotopy groups, J. Pure Appl. Algebra 24 Ž.1984 , 179᎐202. 12. P. May, ‘‘Simplicial Objects in Algebraic Topology,’’ Van Nostrand, Princeton, NJ, 1967. 13. T. Pirashvili, A spectral sequence of an epimorphism, I, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 70 Ž.1982 , 69᎐91. 14. T. Pirashvili, Simplicial degree of functors, Math. Proc. Cambridge Philos. Soc. 126 Ž.1999 , 45᎐62. 15. J. G. Ratcliffe, Free and projective crossed modules, J. London Math. Soc. 22 Ž.1980 , 66᎐74. 16. R. Swan, A simple proof of the cup product reduction theorem, Proc. Nat. Acad. Sci. U.S. A. 46 Ž.1960 , 114᎐117.