Theory and Applications of Categories Vol No pp ON FINITE INDUCED CROSSED MODULES AND THE HOMOTOPY TYPE OF MAPPING CONES Dedicated to the memory of J H C Whitehead RONALD BROWN AND CHRISTOPHER D WENSLEY Transmitted byLawrence Breen ABSTRACT Results on the niteness of induced crossed mo dules are proved b oth algebraically and top ologically Using the Van Kamp en typ e theorem for the fundamen tal crossed mo dule applications are given to the typ es of mapping cones of classifying spaces of groups Calculations of the cohomology classes of some nite crossed mo dules are given using crossed complex metho ds Intro duction Crossed mo dules were intro duced by JHC Whitehead in They form a part of what can b e seen as his programme of testing the idea of extending to higher dimensions the metho ds of combinatorial group theory of the s and of determining some of the extra structure that was necessary to mo del the geometry Other pap ers of Whitehead of this era show this extension of combinatorial group theory tested in dierent directions In this case he was concerned with the algebraic prop erties satised by the b oundary map X A A of the second relative homotopy group together with the standard action on it of the fundamental group A This is the fundamental crossedmodule X Aofthepair X A In order to determine the second homotopy group of a CW complex he formu lated and proved the following theorem for this structure g be obtainedfrom the connectedspace A by attaching Theorem W Let X A fe cel ls Then the second relative homotopy group X A may bedescribed as the free crossed Amodule on the cel ls The pro of in uses transversality and knot theory ideas from the previous pap ers See for an exp osition of this pro of Several other pro ofs are available The The rst author was supp orted by EPSRC grant GRJ for a visit to Zaragoza in November and is with Dr T Porter supp orted for equipment and software by EPSRCGrant GRJ Non ab elian homological algebra Received by the editors March and in revised form April Published on May Mathematics Sub ject Classication G F P Q Key words and phrases crossed mo dules homotopytyp es generalised van Kamp en Theo remcrossed resolution Postnikovinvariant c Ronald Brown and Christopher D Wensley Permission to copy for private use granted Theory and Applications of Categories Vol No survey byBrown and Huebschmann and the b o ok edited by HogAngeloni Metzler and Sieradski give wider applications The pap er of Mac Lane and Whitehead uses Theorem W to show that the dimensional homotopy theory of p ointed connected CW complexes is completely mo d elled by the theory of crossed mo dules This is an extra argument for regarding crossed mo dules as dimensional versions of groups One of our aims is the explicit calculation of examples of the crossed mo dule A V A A of a mapping cone when V A are nite The key to this is the dimensional Van Kamp en Theorem VKT proved by Brown and Higgins in This implies a generalisation of Theorem W namely that the crossed mo dule is induced from the identity crossed mo dule V V by the morphism V A Presentations of induced crossed mo dules are given in and from these we prove a principal theorem Theorem that crossed mo dules induced from nite crossed mo dules by morphisms of nite groups are nite We also use top ological metho ds to prove a similar result for nite pgroups Corollary These results give a new range of nite crossed mo dules Sequels to this pap er discuss crossed mo dules induced by a normal inclusion and calculations obtained using a group theory package The origin of the VKT was the idea of extending to higher dimensions the notion of the fundamental group oid as suggested in in This led to the discovery of the relationship of dimensional group oids to crossed mo dules in work with Sp encer This relationship reinforces the idea of higher dimensional group theory and was essential for the pro of of the VKT for the fundamental crossed mo dule In view of the results of Mac Lane and Whitehead and of metho ds of classifying spaces of crossed mo dules byLoday and Brown and Higgins see section the VKT allows for the explicit computation of some homotopy typ es in the form of the crossed mo dules which mo del them In some cases the Postnikovinvariant of these typ es can b e calculated as the following example shows CorollaryLet C denote the cyclic group of order n and let BC denote its n n classifying space The second homotopy mo dule of the mapping cone X BC BC n n is a particular cyclic C mo dule A say The cohomology group H C A is a cyclic n n n n group of order n and the rst Postnikovinvariantof X is a generator of this group The metho d used for the calculation of the cohomology class here is also of interest It intro duces a new small free crossed resolution of the cyclic group of order n in order to construct an explicit co cycle corresp onding to the ab ove crossed mo dule This indicates a wider p ossibility of using crossed resolutions for explicit calculations It is also related to Whiteheads use of what he called in homotopy systems and which are simply free crossed complexes The initial motivation of this pap er was a conversation with Rafael Sivera in Zaragoza in Novemb er which suggested the lack of explicit calculations of induced crossed Theory and Applications of Categories Vol No mo dules This led to discussions at Bangor on the use of computational group theory packages which culminated in a GAP program and to the development of general theory Crossed mo dules and induced crossed mo dules In this section we recall the denition of induced crossed mo dules and of results of on presentations of induced crossed mo dules Wethengive some basic examples of these Recall that a crossedmodule M M P is a morphism of groups M P p together with an action m p m of P on M satisfying the two axioms p CM m p mp n CM m n mn for all m n M and p P Wesaythat M is nite when M is nite The category XM of crossed mo dules has as ob jects all crossed mo dules Morphisms in XM are pairs g f forming commutative diagrams M P g f Q N in which the horizontal maps are crossed mo dules and g f preserve the action in the sense p fp that for all m M p P wehave g m gm If P is a group then the category XMP of crossed P mo dules is the sub category of XM whose ob jects are the crossed P mo dules and in whicha morphism M P N P of crossed P mo dules p p is a morphism g M N of groups such that g preserves the action g m gm for all m M p P and g Standard algebraic examples of crossed mo dules are i an inclusion of a normal subgroup with action given by conjugation ii an inner automorphism crossed mo dule M Aut M inwhich m is the automorphism n m nm iii a zero crossed mo dule M P where M is a P mo dule iv an epimorphism M P with kernel contained in the centre of M Examples of nite crossed mo dules may b e found among those ab ove the induced crossed mo dules of this pap er and its sequels and copro ducts and tensor pro ducts of nite crossed P mo dules Further imp ortant examples of crossed mo dules are the free crossed mo dules referred to in the Intro duction which are rarely nite They arise algebraically in considering identities among relations which are nonab elian forms of syzygies We next dene pul lback crossedmodules Let P Q b e a morphism of groups and let N Q b e a crossed Qmo dule Let N P b e the pullbackof N by Theory and Applications of Categories Vol No so that N fp n P N j p ngand p n p Let P act on N by p p p n p p p n The verication of the axiom CM is immediate while CM is proved as follows Let p n p n N Then p n p n p n p p p n n n n p p p n p p p p n pn p n Proposition The functor XMQ XMP has a right adjoint Proof This follows from general considerations on Kan extensions The universal prop erty of induced crossed mo dules is the following Let M P C Q b e crossed mo dules In the diagram M P f Q M m m m m m m m g m m m m m m m m m m C the pair is a morphism of crossed mo dules suchthatforany crossed Qmo dule C Q and morphism of crossed mo dules f there is a unique morphism g M C of crossed Qmo dules suchthat g f It is a consequence of this universal prop ertythatifM P F R the free group on a set R and if w R Q is the restriction of to the set R then F Risthefree crossed mo dule on w in the sense of Whitehead see also Constructions of this free crossed mo dule are given in these pap ers A presentation for induced crossed mo dules for a general morphism is given in Prop o sition of We will need two more particular results The rst is Prop osition of that pap er and the second is a direct deduction from Prop osition Proposition If P Q is a surjection and M P is a crossedP MM K where K Ker and M K denotes the subgroup of M module then M k generatedbyallm m for al l m M k K The following term and notation will b e used frequentlyLetP b e a group and let T be a set We dene the copower P T to b e the free pro duct of groups P t T t each with elements p tp P and isomorphic to P under the map p t p If Q is a group then P Q will denote the cop ower of P with the underlying set of the group Q Theory and Applications of Categories Vol No Proposition If P Q is an injection and M P is a crossedP module let T be a right transversal of P in Q Let Q act on the