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Homology of 2-Types and Crossed Modules

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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 homomor- phism ␰ : 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.
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