Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types Ronald Brown *and Christopher D. Wensley School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT, U.K. (email: r.brown, c.d.wensley @ bangor.ac.uk) January 12, 1996 Abstract We obtain some explicit calculations of crossed Q-modules induced from a* * crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and H* *iggins, this enables the computation of the homotopy 2-types and second homotopy m* *odules of certain homotopy pushouts of maps of classifying spaces of discrete groups. Introduction A crossed module M = ( : M ! P ) has a classifying space BM (see, for example, * *[4]) which is of the homotopy type of B(P=M) if is the inclusion of the normal subg* *roup M of P . Consider a homotopy pushout X of the form BP __B__//BQ B || || fflffl| fflffl| BM _____//X where : P ! Q is a morphism of groups and is the natural inclusion from the g* *roup P; regarded as a crossed module 1 ! P; to the crossed module M: It is shown in our* * previous paper [7], using results of Brown and Higgins in [3, 4], that the homotopy 2-ty* *pe of X is determined by the induced crossed Q-module *M. This explains the homotopical in* *terest in calculating induced crossed modules. Such calculations include of course the* * calculation of a weaker invariant, namely the second homotopy module of X, which is in this ca* *se just the kernel of the boundary of *M. Note also that results previous to [7] gave infor* *mation on the homotopy type of X only if : P ! Q is also surjective (see [1] for 2-type in t* *his case, and [6] for 3-type). ________________________________* The first author was supported by EPSRC grant GR/J94532 for a visit to Zarag* *oza in November, 1993, and is, with Prof. T. Porter, supported for equipment and software by EPSRC Gr* *ant GR/J63552, "Non abelian homological algebra". 1 In all these cases, the key link between the topology and the algebra is pro* *vided by a higher dimensional Van Kampen Theorem. Proofs of these theorems require non tra* *ditional concepts, for example double groupoids, as in [3], or Loday's catn-groups, as i* *n [6]. The results even on the second homotopy modules seem unobtainable by more traditional metho* *ds, for example transversality and pictures, as described by Hog-Angeloni, Metzler, and* * Sieradski in [10]. Another interest of induced crossed modules is algebraic. Consider for examp* *le the inclu- sion crossed module ( : M ! P ) of a normal subgroup M of P , and suppose : P * *! Q is an inclusion of a subgroup. Then the image of the boundary @ of the induced cro* *ssed module (@ : *M ! Q) is the normal closure NQ (M) of M in Q. Thus the induced crossed m* *odule construction replaces this normal closure by a bigger group on which Q acts, an* *d which has a universal property not usually enjoyed by NQ (M). The algebraic significance * *of the kernel of @ has yet to be exposed. The purpose of this paper is to give some new results on crossed modules ind* *uced by a morphism of groups : P ! Q in the case when is the inclusion of a normal subg* *roup. One of our main results (in section 1) determines *M, and so the kernel of *M ! Q, * *in the case P and M are normal in Q. In section 2 we use the presentation of induced crossed modules given in [3,* * 7] to describe the crossed module induced by the normal inclusion in terms of the coproduct o* *f crossed P -modules discussed in [12, 1]. This allows us to apply methods of Gilbert and* * Higgins in [9] to generalise the result of section 1, and to deduce a result on the index * *2 case from the results on coproducts in [1]. For crossed modules, and modules, the action is a crucial part of the struct* *ure, and this is reflected in our Theorems and Examples. The initial motivation of this set of papers was a conversation with Rafael * *Sivera in Zaragoza, in November 1993, which suggested the lack of explicit calculations o* *f induced crossed modules. This led to discussions at Bangor on the use of computational * *group theory packages which culminated in a GAP [11] program for computing finite induced cr* *ossed modules. In some cases the form of the resulting calculations suggested some ge* *neral results, such as those in [7]. A separate paper [8] in preparation discusses the algorithmic aspects of the* * GAP program and includes a table of explicit calculations. The work with GAP is being exten* *ded by the second author to a general package of calculations with crossed modules. Reports on some of the results of this and the other papers were given by th* *e authors at Groups in Galway in May, 1994, and by the first author at the European Category* * Theory Meeting at Tours, July, 1994 (see [2]). Induced constructions may be thought of in terms of `change of base'. For m* *ore back- ground on related contexts, see [2]. 2 1 Inducing from a normal subgroup P of Q This section contains the following main result, which is proved by a direct ve* *rification of the universal property for an induced crossed module. We assume as known the de* *finition of induced crossed modules given in [3, 7]. If n 2 M, then the class of n in Ma* *b is written [n]. If R is a group, then I(R) denotes the augmentation ideal of R: The augmen* *tation ideal I(Q=P ) of a quotient group Q=P has basis {t- 1 | t 2 T 0} where T is a transve* *rsal of P in Q; T 0= T \ {1} and qdenotes the image of q in Q=P: Theorem 1.1 Let M P be normal subgroups of Q, so that Q acts on P and M by co* *n- jugation. Let : M ! P; : P ! Q be the inclusions and let M denote the cros* *sed module ( : M ! P ) with the conjugation action. Then the induced crossed Q-modu* *le *M is isomorphic as a crossed Q-module to (i : M x (Mab I(Q=P )) ! Q) where for m; n 2 M; x 2 I(Q=P ) : (i) i(m; [n] x) = m 2 Q; (ii) the action of Q is given by (m; [n] x)q = (mq; [mq] (q - 1) + [nq] xq): The universal map i : M ! M x (Mab I(Q=P )) is given by m 7! (m; 0); and if* * (fi; ) is a morphism from M to the crossed module C = (O : C ! Q), then the morphism nn7C7??_____ finnnnnfflffl_______ nnnn _OEfflffl____ nnn __fflffl M _i__//_Z Offlffl (1) | | fflffl | i| fflffl fflffl||fflfflfflffl P ____//_Q OE : M x (Mab I(Q=P )) ! C induced by fi is, for m; n 2 M; q 2 Q, given by i i -1jjq OE(m; [n] (q - 1)) = (fim) (fin)-1 fi nq : (2) The following corollary is immediate. Corollary 1.2 The homotopy 2-type of X = BQ [BP B(P=M) is determined by the crossed Q-module (i : M x (Mab I(Q=P )) ! Q) above. In pa* *rticular, the second homotopy module of X is isomorphic to the Q=M-module Mab I(Q=P ): Proof of Theorem 1.1 Let Z = M x (Mab I(Q=P )). The proof that Z = (i : Z ! Q) with the given action is indeed a crossed module is straightforward and is omit* *ted. Clearly we have a morphism of crossed modules (i; ) : M ! Z. We verify that* * this morphism satisfies the universal property of the induced crossed module. Consider diagram (1). We prove below: 3 1.3 If OE : Z ! C is a morphism Z ! C of crossed Q-modules such that OEi = fi; * *then OE is given the formula (2). We next prove that this formula does define a morphism of crossed Q-modules. Let q 2 Q. We define a function i i -1jjq flq : M ! C; m 7! (fim)-1 fi mq : We prove in turn: 1.4 flq(M) is contained in the centre of C. 1.5 flq is a morphism, which factors through Mab: 1.6 The morphisms flq depend only on the classes q of q in Q=P , and so define * *a morphism of groups fl : Mab I(Q=P ) ! C; [m] (q - 1) 7! flq(m). 1.7 The function OE defined in the theorem satisfies OEi = fi and is a well def* *ined morphism of crossed modules. Proof of 1.3 Let OE : Z ! C be a morphism of crossed Q-modules such that OEi = * *fi: Let m; n 2 M; q 2 Q: Then OE(1; [n] (q - 1))=OE((n-1;i0)(n;j[n]ii(q - 1))) -1 qjj = fi n-1 OE nq ; 0 i i -1 jjq = fi(n)-1 OE nq ; 0 i i -1jjq = fi(n)-1 fi nq = flq(n): The result follows since (m; [n] (q - 1)) = (m; 0)(1; [n] (q - 1)). * * 2 Proof of 1.4 This follows from the facts that if m 2 M; then Oflq(m) = 1; and t* *hat C is a crossed module. 2 Proof of 1.5 Let m; n 2 M: Then i i -1jjq flq(mn) = (fi(mn))-1 fi (mn)q i i -1jjqi i -1jjq = (fin)-1(fim)-1 fi mq fi nq i i -1jjq = (fin)-1(flqm) fi nq i i -1jjq = (flqm)(fin)-1 fi nq = flq(m)flq(n): This proves that flq is a morphism of groups. By 1.4, flq(m)flq(n) = flq(n)flq(* *m); and so flq factors through Mab: 2 Proof of 1.6 The first part follows from the fact that fi is a P -morphism. The* * second follows from the fact that the elements q- 1; q2 Q=P; form a basis of I(Q=P ): * * 2 4 Proof of 1.7 The function OE is clearly a well-defined morphism of groups since* * it is of the form OE(m; u) = (fim)(flu); where fi; fl are morphisms of groups and flu belong* *s to the centre of C: Further, OEi = fi; and OOE = i since Ofl is trivial. Next we prove that OE preserves the action.iThisiisjthejcrucialqpart of the * *argument. Recall that fl([n] (q - 1)) = flq(n) = (fin)-1 fi nq-1 : Let m; n 2 M; r; q 2 Q: Then OE ((m; [n] (r - 1))q) = OE(mq; [mq] (q - 1) + [nq] (r - 1)q) = (fi(mq))fl([mq] (q - 1) + [nq] ((rqi-i1) - (q - 1))) -1r-1)jjrq q -1 = (fi(mq)) (fi(mq))-1(fim)q(fi(nq))-1fi nq(q (flq(n )) i i -1jjrq = (fim)q(flq(nq))-1 (fi(nq))-1fi nr i i -1jjrq = (fim)q((fin)q)-1fi nr = (OE(m; [n] (r - 1)))q: 2 This completes the proof of the theorem. i i jj* *q 2 An intuitive explanation of this result is that the part (fin)-1 fi nq-1 * *measures the deviation of fi from being a Q-morphism. Corollary 1.8 In particular, if the index [Q : P ] is finite, and P is the cros* *sed module 1 : P ! P; then *P is isomorphic to the crossed module (pr1 : P x (P ab)[Q:P]-1* *! Q) with action as above. Remark 1.9 It might be imagined from this that the Postnikov invariant of this* * crossed module is trivial, since one could argue that the projection pr2 : P x P ab I(Q=P ) ! P ab I(Q=P ) should give a morphism from *P to the crossed module 0 : P ab I(Q=P ) ! Q=P; wh* *ich represents 0 in the cohomology group H3(Q=P; P ab I(Q=P )) (see [7]). However, * *the pro- jection pr2is a P -morphism, but is not in general a Q-morphism, as the above r* *esults show. In fact, in the next Theorem we give a precise description of the Postnikov inv* *ariant of *P when Q=P is cyclic of order n. This generalises the result for the case P = Cn;* * Q = Cn2 in Theorem 5.4 of [7]. Theorem 1.10 Let P be a normal subgroup of Q such that P=Q is isomorphic to Cn* *; the cyclic group of order n. Let t be an element of Q which maps to the generator t* *of Cn under the quotient map. Then the first Postnikov invariant k3 of the mapping cone X =* * BQ[ BP of the inclusion BP ! BQ lies in a third cohomology group H3(Cn; P ab I(Cn)) 5 This group is isomorphic to P ab Cn; and under this isomorphism the element k3 is taken to the element [tn] t: Proof We have to determine the cohomology class represented by the crossed mod* *ule : P x P ab I(Cn) ! Q: Let A = P ab I(Cn). As in [7] for the case Q = Cn2; P = Cn, we consider the dia* *gram ZZ[Cn]_ffi//4_ZZ[Cn]//ffi3_ZZ[Cn]//ffi2_C1//_Cn 0|| f3|| |f2| |f1| 1|| fflffl| fflffl| fflffl| |fflffl fflffl| 0________//_A__i__//_P x_An__//Q_____//Cn: Here the top row is the begining of a free crossed resolution of Cn. The free C* *n-modules ZZ[Cn] have generators y4; y3; y2 respectively, C1 has generator y1 and ffi2(y2) = yn1* *; ffi3(y3) = y2:(t-1) (here C1 operates on each ZZ[Cn] via the morphism to Cn); ffi4(y4) = y3:(1+t+t2* *+. .+.tn-1). Further, we define f1(y1) = t; f2(y2) = (tn; 0); f3(y3) = [tn](t-1), and i(a) =* * (1; a); a 2 A. Thus the diagram gives a morphism of crossed complexes, and the cohomology clas* *s of the cocycle f3 is the Postnikov invariant of the crossed module. As in [7], Theorem 5.4, since ZZ[Cn] is a free Cn-module on one generator, t* *he cohomology group H3(Cn; A) is isomorphic to the homology group of the sequence ffi*4 ffi*3 A oo__A_oo___A where ffi*4is multiplication by 1 + t+ t2+ . .+.tn-1and ffi*3is multiplication * *by t- 1: It follows that ffi*4= 0; and it is easy to check that I(Cn)=I(Cn)(t - 1) is a cyclic grou* *p of order n generated by t- 1: The cocycle f3 determines the element f3(y3) = [tn] (t- 1) * *of A, and the result follows. * * 2 Remark 1.11 The reason for the success of this last determination is that we h* *ave a conve- nient small free crossed resolution of the cyclic group Cn. 2 Coproducts of crossed P -modules We refer to [7] for further background information that we require on crossed m* *odules. Let X M =P be the category of crossed modules over the group P . It is well * *known that arbitrary coproducts exist in this category. They may be constructed in the fol* *lowing way, which is given in essence, but not with this terminology, in [12]. Let T be an indexing set and let {Mt = (t : Mt ! P ) | t 2 T } be a family o* *f crossed P -modules. Let Y be the free product of the groups Mt; t 2 T . Let @0: Y ! P b* *e defined 6 by the morphisms t: The operation of P on the Mt extends to an operation of P o* *n Y , so that (@0 : Y ! P ) becomes a precrossed P -module. The standard functor from pr* *ecrossed modules to crossed modules, obtained by factoring out the Peiffer subgroup [5, * *10], is left adjoint to the inclusion of crossed modules into precrossed modules, and so tak* *es coproducts into coproducts. Applying this to (@0 : Y ! P ) gives the coproduct (@ : Ot2TMt* * ! P ) in the category X M =P; determined by the canonical morphisms of crossed P -modules iu : Mu ! Y ! Ot2TMt; where the first morphism is the inclusion to the coproduct of groups, and the s* *econd is the quotient morphism. As is standard for coproducts in any category, the coproduct* * in X M =P is associative and commutative up to natural isomorphisms. We now assume that P is a normal subgroup of Q, and show in Theorem 2.2 that* * the coproduct of crossed P -modules may be used to give a presentation of induced c* *rossed P - modules analogous to known presentations of induced modules. Suppose first given a crossed P -module M = ( : M ! P ). Let ff be an automo* *rphism of P . The proof that the following definition does give a morphism of crossed * *modules is left to the reader. Definition 2.1The crossed module Mff= (ff: Mff! P ) associated to an automorphi* *sm ff and an isomorphism (kff; ff) : M ! Mff, kff M ____//_Mff || |ff| fflffl|fflffl| P __ff_//_P: are defined as follows. The group Mffis just M x {ff} and kffm = (m; ff); m 2 * *M. The morphism ffis given by (m; ff) 7! ffm. The action of P is given by (m; ff)p = (* *mff-1p; ff): We shall apply this construction to the case ff = fft: p 7! t-1pt, for some * *t 2 Q, and we write Mfftas Mt= (t: Mt! P ) where t(m; t) = t-1(m)t. Given a set T of elements of Q, we write M"OT = (@ : M "OT ! P ) for the coproduct crossed P -module Ot2TMt, and it: Mt! M"OT; t 2 T for the canonical morphisms of crossed P -modules defining the coproduct. Theorem 2.2 Let M = ( : M ! P ) be a crossed P -module, and let : P ! Q be an inclusion of a normal subgroup. Let T be a right transversal of P in Q. For t* * 2 T , let Mt= (t: Mt! P ) be the crossed P -module in which the elements of Mtare (m; t);* * m 2 M with -1 t(m; t) = t-1(m)t; (m; t)p = (mtpt ; t): 7 Then there is a unique action of Q on M "OT which satisfies (it(m; t))q = iu(mp; u); (3) for q 2 Q; p 2 P; t; u 2 T , such that tq = pu. This action makes M"OT = (@ : M* * "OT ! Q) a crossed Q-module and the morphism (i1; ) : M ! M"OT has the universal propert* *y of the induced crossed Q-module *M = (@ : *M ! Q) as shown in the diagram kkC55;;____ fikkkkk_ff______ kkkk ___ffff__ kkkkk __OEffff_____ ____//_kk ffff M i1 M "OT fOfff | | ffff | @| ffff fflffl| |fffffflffl P ______//_Q Further, given a morphism (fi; ) : M ! C = (O : C ! Q), the induced morphism OE : M"OT ! C is given by OE(it(m; t)) = (fim)t: (4) Proof The construction of the induced crossed module given in [3] and used in [* *7] is to form the precrossed module @0: Y ! Q where Y is the free product *t2TMt, where the Mtare copies of M, with elements * *(m; t); m 2 M and action as above. The new aspect of the current situation is that the part* * t: Mt! P of @0 is also a crossed P -module. Now we see that both the induced crossed Q-module and the coproduct crossed * *P -module are obtained by factoring Y by the Peiffer subgroup, which is the same whether * *Y is considered as a precrossed P -module Y ! P or as a precrossed Q-module Y ! Q. This prov* *es the theorem. 2 We remark that the result of Theorem 2.2 is analogous to descriptions of ind* *uced modules, except that here we have replaced the direct sum which is used in the module ca* *se by the coproduct of crossed modules. Corresponding descriptions in the non-normal case* * look to be considerably harder. As a consequence of the theorem we obtain: Proposition 2.3 If M is a finite p-group and P is normal and of finite index in* * Q, then the induced crossed module *M is a finite p-group. Proof The coproduct of two crossed P -modules is shown in [1] to be obtained as* * a quotient of their semidirect product, so that the coproduct of two, and hence of a finit* *e number, of finite crossed P -modules is finite. * * 2 Note that a similar result is proved in [7] by topological methods, without * *the normality condition, but assuming that Q also is a finite p-group. We can now apply a result of Gilbert and Higgins [9] to obtain a description* * of an induced crossed module in more general circumstances than in section 1. We are careful * *about giving 8 when possible the Q-action for this crossed module, since this is of course a k* *ey element of the structure. If a group M acts on a group N, then the quotient of N by the action of M is* * written NM ; it is the quotient of N by the `displacement subgroup' generated by the el* *ements n-1nm for all n 2 N; m 2 M: Theorem 2.4 Suppose that M = ( : M ! P ) is a crossed module and that the rest* *riction 0: M ! M of has a section oe : M ! M. Let : P ! Q be the inclusion of a normal subgroup. Suppose that for all q 2 Q; q-1(M)q M. Let T be a transversal of P i* *n Q and let T 0= T \ {1}. Then the group *M of the induced crossed module *M is iso* *morphic L to the group M x t2T0(Mt)M and this yields by transference of actions an iso* *morphism of *M to a crossed module of the form M X = ( = pr1: M x (Mt)M ! Q): t2T0 If, further, the section oe is P -equivariant, then the action of Q in X is * *given as follows, where m 2 M; r 2 P; t; v 2 T; q = rv 2 Q, and [m; v] denotes the class of (m; v* *) in (Mv)M : (i) ( (mq; 0) if v = 1; (m; 0)q = q r (oe((m) ); [m ; v])ifv 6= 1; (ii) if tq = pu; t 2 T 0; p 2 P; u 2 T , then 8 ><(1; [mp; t]) if v = 1; (1; [m; t])q = > (oe(mp)-1mp; -[oe((mp)v-1); v])ifv 6= 1; u = 1; : (1; -[oe((mp)uv-1); v] + [mp; u])ifv 6= 1; u 6= 1: Further, given a morphism (fi; ) : M ! C = (O : C ! Q), the induced morphism L OE : M x t2T0(Mt)M ! C is given by OE(m; 0) = fim; OE(m; [n; v]) = (fim) fi(oe((n)v))-1 (fin)v: Proof We identify M and M1, so that i = i1 : (m; 1) 7! (m; 0). Let W = Ot2T0Mt, so that by Theorem 2.2 there is an isomorphism of crossed Q* *-modules *M ~=M O W: By Proposition 2.1 and Corollary 2.3 of [9], there is an isomorphism of groups M O W ~=M x WM ; where WM is the quotient of W by the action of M via P . We next observe that since tMt M for all t 2 T 0, we have M WM ~= (Mt)M : t2T0 9 The reason is that under these circumstances the Peiffer commutators 0(m;t) (m; t)-1(m1; t1)-1(m; t)(m1; t1)@ which generate the Peiffer subgroup of *t2T0Mt reduce to ordinary commutators. The detailed description of the action in the case oe is P -equivariant is a* *rrived at by examining carefully the isomorphisms of groups given in [9]. * * 2 We now include an example for Theorem 2.4 showing the action in the case v 6* *= 1; u = 1. Example 2.5 Let n be an odd integer and let Q = D8n be the dihedral group of o* *rder 8n generated by elements {t; y} with relators {t4n; y2; (ty)2}. Let P = D4n be ge* *nerated by {x; y}, and let : P ! Q be the monomorphism given by x 7! t2; y 7! y. Then let* * M = C2n be generated by {m}. Define X = ( : M ! P ) where m = x2; mx = m and my = m-1. This crossed module is isomorphic to a sub-crossed module of (D4n ! Aut(D4n)) a* *nd has kernel {1; mn}. The image M is the cyclic group of order n generated by x2, and there is an * *equivariant section oe : M ! M; x2 7! mn+1 since (x2)(n+1)= x2 and gcd(n + 1; 2n) = 2. Th* *en Q = P [ P t, T = {1; t} is a transversal, Mt is generated by (m; t) and t(m; t)* * = x2. The action of P on Mt is given by (m; t)x = (m; t); (m; t)y = (m-1; t): Since M acts trivially on Mt, *M ~=M x Mt ~= C2nx C2n: Using the section oe given above, Q acts on *M by (m; 0)t= (mn+1; [m; t]); (m; 0)y= (m-1; 0); (1; [m; t])t= (mn; (n - 1)[m; t]); (1; [m; t])y= (1; -[m; t]): We can obtain some information on the kernel of induced crossed modules in t* *he case P is of index 2 in Q by using results of [1]. Proposition 2.6 Let ( : M ! P ) and ( : P ! Q) be inclusions of normal subgroup* *s. Suppose that P is of index 2 in Q, and t 2 Q \ P . Then the kernel of the induc* *ed crossed module (@ : *M ! Q) is isomorphic to (M \ t-1Mt) = [M; t-1Mt]: Proof By previous results, *M is isomorphic to the coproduct crossed P -module* * M O Mt with a further action of Q. The result now follows from Proposition 2.8 of [1].* * 2 We now give two homotopical applications of the last result. 10 Example 2.7 Let : P = D4n ! Q = D8n be as in Example 2.5, and let M = D2n be * *the subgroup of P generated by {x2; y}, so that M <| P <| Q and t-1Mt is isomorphic* * to a second D2n generated by {x2; yx}. Then M \ t-1Mt = [M; t-1Mt] (since [y; yx] = x2), and both are isomorphic to Cn generated by {x2}. It foll* *ows from Proposition 2.6 that if X is the homotopy pushout of the maps BC2 - BD4n -! BD8n; where the lefthand map is induced by D4n ! D4n=D2n ~=C2; then ss2(X) = 0. Example 2.8 Let M; N be normal subgroups of the group G, and let Q be the wrea* *th product Q = G o C2 = (G x G) o C2: Take P = G x G, and consider the crossed module (@ : Z ! Q) induced from M x N * *! P by the inclusion P ! Q. If t is the generator of C2 which interchanges the two fac* *tors of G x G, then Q = P [ P t and t-1(M x N)t = N x M. So (M x N) \ t-1(M x N)t = (M \ N) x (N \ M) and [M x N; N x M] = [M; N] x [N; M]: It follows that if X is the homotopy pushout of B(G=M) x B(G=N) BG x BG ! B(G o C2); then ss2(X) ~=((M \ N)=[M; N])2: If ([m]; [n]) denotes the class of (m; n) 2 (M \ N)2 in ss2(X), the action of Q* * is determined by ([m]; [n])(g;h)= ([mg]; [nh]); (g; h) 2 P; ([m]; [n])t= ([n]; [m]* *): References [1]Brown, R., Coproducts of crossed P -modules: applications to second homotop* *y groups and to the homology of groups, Topology 23 (1984) 337-345. [2]Brown, R., Homotopy theory, and change of base for groupoids and multiple g* *roupoids, Applied Categorical Structures (1995) to appear. [3]Brown, R. and Higgins, P.J., On the connection between the second relative * *homo- topy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978) 1* *93-212. [4]Brown, R. and Higgins, P.J., The classifying space of a crossed complex, Ma* *th. Proc. Camb. Phil. Soc. 110 (1991) 95-120. 11 [5]Brown, R. and Huebschmann, J., Identities among relations, in Low dimension* *al topology, Ed. R.Brown and T. L. Thickstun, London Math. Soc. Lecture Notes * *46, Cam- bridge University Press (1982) 153-202. [6]Brown, R. and Loday, J.-L., Van Kampen Theorems for diagrams of spaces, Top* *ology 26 (1987) 311-335. [7]Brown, R. and Wensley, C.D., On finite induced crossed modules, and the hom* *otopy 2-type of mapping cones, Theory and Applications of Categories, 1(3) (1995)* * 54-71. [8]Brown, R. and Wensley, C.D., On the computation of induced crossed modules, UWB Math. Preprint 95.05 (1995). [9]Gilbert, N.D. and Higgins, P.J., The non-Abelian tensor product of groups a* *nd related constructions, Glasgow Math. J. 31 (1989) 17-29. [10]Hog-Angeloni, C., Metzler, W. and Sieradski, A.J. (Editors), Two- dimensional homotopy and combinatorial group theory, London Math. Soc. Lect* *ure Note Series 197, Cambridge University Press, Cambridge (1993). [11]Sch"onert, M. et al, GAP : Groups, Algorithms, and Programming, Lehrstuhl D* * f"ur Mathematik, Rheinisch Westf"alische Technische Hochschule, Aachen, Germany,* * third edition, 1993. [12]Whitehead, J.H.C., On adding relations to homotopy groups, Ann. Math. 42 (1* *941) 409-428. 12