Computation and Homotopical Applications
of Induced Crossed Modules
Ronald Brown
Christopher D Wensley
Mathematics Division
School of Informatics
University of Wales, Bangor
Gwynedd, LL57 1UT
U.K.
email: {r.brown, c.d.wensley}@bangor.ac.uk
September 6, 2002
Bangor Mathematics Preprint 02.04
Abstract
We explain how the computation of induced crossed modules allows the co*
*mpu-
tation of certain homotopy 2-types and, in particular, second homotopy gro*
*ups. We
discuss various issues involved in computing induced crossed modules and g*
*ive some
examples and applications.
Introduction
The interactions between topology and combinatorial and computational group the*
*ory are
largely based on the fundamental group functor
ß1 : (based spaces)! (groups).
At the beginning of the 20th century there was an aim to to generalise the non *
*commutative
fundamental group to higher dimensions, hopes which seemed to be dashed in 1932*
* by
the proof that the definition of higher homotopy groups ßn then proposed by ~Ce*
*ch led to
commutative groups for n > 2.
1
Nonetheless, in the late 1930s and 1940s J.H.C. Whitehead developed properti*
*es of the
second relative homotopy group functor
2 : (based pairs of spaces)! (crossed modules),
(X, A, a)7! (@ : ß2(X, A, a) ! ß1(A, a)) ,
where a 2 A X (see Section 4). Mac Lane and Whitehead showed in 1950 [24] th*
*at
crossed modules modelled homotopy 2-types (3-types in their notation) and evide*
*nce has
grown that crossed modules can be regarded as `2-dimensional groups'. Part of t*
*his evidence
is the 2-dimensional version of the Van Kampen Theorem proved by Brown and Higg*
*ins
in 1978 [9], which allows new computations of homotopy 2-types and so second ho*
*motopy
groups. This result should be seen as a higher dimensional, non commutative, l*
*ocal-to-
global theorem, illustrating themes in Atiyah's article [4]. It is interesting*
* to note that
the computation of these second homotopy groups is obtained through the computa*
*tion
of a larger non commutative structure. This work also throws emphasis on the p*
*roblem
of explicit computation with crossed modules, the discussion of which is the th*
*eme of this
paper.
Our main emphasis in this paper is on induced crossed modules, which were de*
*fined
in [9] and studied further in papers by the authors [14, 15]. Given the crosse*
*d module
M = (~ : M ! P ) and a morphism of groups ' : P ! Q, the induced crossed module*
* '*M
has the form (@ : '*M ! Q), a crossed module over Q, and comes with a morphism *
*of
crossed modules ('*, ') : M ! '*M :
'* //
M _______'*M
~|| |@|
fflffl| fflffl|
P ____'__//Q .
Their study requires a solution to many of the general computational problems o*
*f crossed
modules.
In the case ~ = 0, when M is simply a P -module, '*M is the usual induced Q-*
*module
M ZP ZQ.
Even in the case M = P, ~ = idP, we know of no relation between the induced *
*crossed
module (@ : '*P ! Q) and other standard algebraic constructions, although, inte*
*restingly,
im @ = NQ ('P ) the normal closure of 'P in Q. Thus the induced crossed module*
* con-
struction replaces this normal closure by a bigger group on which Q acts, and w*
*hich has a
universal property not usually enjoyed by NQ ('P ).
A long-term project at Bangor is the development of a share library for the *
*computa-
tional group theory program GAP [18], providing functions to compute with these*
* higher-
dimensional structures. The first stage of this project saw the production of *
*the library
XMod1, containing functions for crossed modules and their derivations and for c*
*at1-groups
and their sections. The manual for XMod1 was included in [28] as Chapter 73. In*
* particular,
Alp [1] enumerated all isomorphism classes of cat1-structures on groups of orde*
*r at most
47. This library has recently been rewritten for GAP4, with XMod2 included wit*
*h the 4.3
2
release. Related libraries include Heyworth's IdRel [19] for computing identiti*
*es among the
relators of a finitely presented group, and Moore's GpdGraph and XRes [25] for *
*computing
with finite groupoids; group and groupoid graphs; and crossed resolutions. Thes*
*e libraries
are available at the HDDA website [20].
1 Crossed modules
A crossed module M (over P ) consists of a morphism of groups ~ : M ! P , calle*
*d the
boundary of M, together with an action of P on M, written (m, p) 7! mp, satisfy*
*ing for all
m, n 2 M, p 2 P the axioms:
CM1) ~(mp) = p-1(~m)p , CM2) n~m = m-1nm .
When CM1) is satisfied, but not CM2), the structure is a pre-crossed module [11*
*, 21], having
a Peiffer subgroup C generated by Peiffer commutators = m-1n-1m n~m , an*
*d an
associated crossed module (~0: M=C ! P ) with ~0induced by ~.
Some standard algebraic examples of crossed modules are:
(i)normal subgroup crossed modules (i : N ! P ) where i is an inclusion of a *
*normal
subgroup, and the action is given by conjugation;
(ii)automorphism crossed modules (Ø : M ! Aut(M)) in which (Øm)(n) = m-1nm;
(iii)abelian crossed modules (0 : M ! P ) where M is a P -module;
(iv)central extension crossed modules (~ : M ! P ) where ~ is an epimorphism w*
*ith kernel
contained in the centre of M.
For our purposes, an important standard construction is the free crossed Q-m*
*odule
F! = (@ : F (!) ! Q)
on a function ! : ! Q, where is a set and Q is a group. The group F (!) ha*
*s a
presentation with generating set x Q and relators
(m, q)-1 (n, p)-1 (m, q) (n, pq-1(!m)q) 8 m, n 2 , p, q 2 Q .
The action is given by (m, q)p = (m, qp) and the boundary morphism is defined o*
*n generators
by @(m, q) = q-1(!m)q. This construction will be seen later as a special case o*
*f an induced
crossed module. The reader should be warned that the group F (!) can be very fa*
*r from a
free group: in fact, if ! maps all of to {1Q}, then F (!) is just the free Q-*
*module on the
set , and in particular is a commutative group.
The major geometric example of a crossed module can be expressed in two ways*
*. Let
(X, A, a) be a based pair of spaces, with a 2 A X. The second relative homoto*
*py group
ß2(X, A, a) consists of homotopy classes rel J1 of continuous maps
ff : (I2, `I2, J1) ! (X, A, a)
3
where I = [0, 1] and J1 = (I x {0, 1}) [ ({1} x I) I2. Each such ff is a map *
*from the unit
square I2 to the space X mapping three sides of the square to the point a and t*
*he fourth
side to a loop at a. Whitehead showed in [30] that there is a crossed module 2*
*(X, A, a)
with boundary map
@ : ß2(X, A, a) ! ß1(A, a), ff 7! fi = ff(I x {0}) .
The image of ff1 2 ß2(X, A, a) under the action of fi2 2 ß1(A, a) is illustrate*
*d in the right-
hand square of Figure 1.
|___________________a||| |___________________a| |___________________a|
| | | | | | | |@ |
| | | | | a| | |@ fi2 fi2 |
| | | | | | | | @I a ` |
| | | | | |______|| | | @|_________| |
| | | | | | | | | | | |
|a a| |a a| |a a | |a a| | | | |
| | | | ~ | | | | ~ |a | | a |
| | | | | | | | | | | |
|ff-1 | ff | ff | |ff-1 | ff | ff | | | | |
| 2 | 1 | 2 | | 2 | 1 | 2 | | a| ff1 |a |
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|____oe___-______-_|_|| |____oe__-_______-_|_|| |_____oe__-_______-_|_*
*||
fi2 fi1 fi2 fi2 fi1 fi2 fi2 fi1 fi2
ff-12ff1ff2 ff@ff21
Figure 1: Verification of CM2) for 2(X, A, a) .
Whitehead's main result in [29, 30, 31] was:
Theorem 1.1 (Whitehead) If X is obtained from A by attaching 2-cells, then ß2*
*(X, A, x)
is isomorphic to the free crossed ß1(A, x)-module on the attaching maps of the *
*2-cells.
Later Quillen observed that if F ! E ! B is a based fibration, then the ind*
*uced
morphism of fundamental groups ß1F ! ß1E may be given the structure of a cross*
*ed
module. This fact is of importance in algebraic K-theory.
We also note the following fact, shown in various texts on homological algeb*
*ra or the
cohomology of groups, e.g. [6], and which we relate to topology in section 4:
1.2 A crossed module M = (~ : M ! P ) determines algebraically a cohomology cla*
*ss
kM 2 H3(coker~, ker~),
called the k-invariant of M, and all elements of this cohomology group have suc*
*h a repre-
sentation by a crossed module.
2 Other structures equivalent to crossed modules
One aspect of the problem of higher dimensional group theory is that, whereas t*
*here is
essentially only one category of groups, there are at least five categories of *
*equationally
4
defined algebraic structures which are equivalent to crossed modules, namely:
o cat1-groups [23];
o group-groupoids [12];
o simplicial groups with Moore complex of length 1, [23];
o reduced simplicial T -complexes of rank 2, [16, 3, 26];
o reduced double groupoids with connection [13].
These categories have various geometric models. The 2-cells of some of these*
* are illus-
trated in the following pictures:
e0 [ e1 [ e2 e0 [ e1 [ e2 2-simplex square
' $ ' $ - s |s______s_|-
AA | |
s s s A | |
6 ~ AK |6 |6
&% &%- s_______sA- |s______s_|||-
crossed 2-groupoid, simplicial double
module cat1-group T -complex groupoid
There is also a polyhedral model, which allows rather general kinds of geometri*
*c objects
[22].
Thus, for computation in "2-dimensional group theory", decisions must be mad*
*e as to
which category to use to represent a given object, and to compute constructions*
*. One
reason for computing with the crossed module format is that this is closer to t*
*he familiar
realm of groups, for which many computational procedures and systems have been *
*found
and constructed. Part of the interest in computations with crossed modules is *
*that such
computations will also yield computations of these other structures, and this m*
*akes them
more familiar and understandable.
2.1 Cat1-groups
In a cat1-group C = (e; t, h : G ! R) the embedding e : R ! G is a monomorphism*
* while
the tail and head homomorphisms t, h : G ! R are surjective and satisfy:
CAT 1) te = he = idR, CAT 2) [kert, kerh] = {1G} .
When CAT1) is satisfied, but nor CAT2), the structure is a pre-cat1-group with *
*Peiffer
subgroup [kert, kerh]. A cat1-group C determines a crossed module (@ : S ! R) *
*where
S = kert and @ = h|S. Conversely, a crossed module (~ : M ! P ) determines a ca*
*t1-group
(e; t, h : P n M ! P ) where t(p, m) = p and h(p, m) = p(~m). The axiom he = i*
*dR is
5
equivalent to CM1) for a crossed module, while CAT2) is equivalent to CM2). Whe*
*n ~ is
the inclusion of the trivial subgroup in P , the associated cat1-group CP has e*
* = t = h = idP.
Note also that the semidirect product P n M admits a groupoid structure with*
* t, h as
source and target, and composition O where (p, m) O (p(~m), n) = (p, mn), makin*
*g P n M
a group-groupoid, i.e. a group internal to the category of groupoids. This no*
*tion has a
long history: the result that crossed modules are equivalent to group-groupoids*
* goes back
to Verdier, seems first to have been published in [12], and is used in [5]. Th*
*e holomorph
Aut (M) n M of a group M is the source of the cat1-group associated to the auto*
*morphism
crossed module (Ø : M ! Aut(M)).
Now a colimit of cat1-groups colimi(ei; ti, hi: Gi! Ri) is easy to describe.*
* One takes the
colimits G0, R0of the underlying groups Gi, Ri, and finds that the endomorphism*
*s ei, ti, hi
induce endomorphisms e0: R0! G0and t0, h0: G0! R0satisfying axiom CAT1). The re-
quired colimit is the cat1-group C00= (e00; t00, h00: G00! R0) which has G00= G*
*0=[kert0, kerh0]
and e00, t00, h00induced by e0, t0, h0.
When C = (e; t, h : G ! R) and ' : R ! Q is an inclusion, the induced cat1-g*
*roup '*C is
obtained as the pushout of cat1-morphisms (e, idR) : CR ! C and (', ') : CR ! C*
*Q . See Alp
[1], [2] for further details.
Further investigation is needed to see whether the use of cat1-groups can be*
* shown to
be more efficient than the direct method for the computation of some colimits o*
*f crossed
modules, particularly induced crossed modules,. The procedure has three stages:*
* convert a
crossed module M to a cat1-group C; calculate '*C; then convert '*C to '*M.
3 Computing colimits of crossed modules
The homotopical reason for interest in computing colimits of crossed modules is*
* the 2-
dimensional Van Kampen Theorem (2-VKT) due to Brown and Higgins [9]. The formu-
lation and proof of this theorem was found through the notion of double groupoi*
*d with
connection, since such structures yield an appropriate algebraic context in whi*
*ch to handle
both ä lgebraic inverses to subdivision", and the öh motopy addition lemma" (wh*
*ich gives
a formula for the boundary of a 3-cube).
One form of the 2-VKT states that Whitehead's fundamental crossed module fun*
*ctor
2 : (based pairs of spaces)! (crossed modules)
preserves certain colimits. So for the calculation of certain homotopy invarian*
*ts, we need
to know how to calculate colimits of crossed modules. To this end, we start by *
*using some
elementary category theory.
The forgetful functor (crossed modules) ! (groups), (~ : M ! P ) 7! P , has *
*a right
adjoint P 7! (i : P ! P ), and so preserves colimits. This shows how to comput*
*e the
1-dimensional part of the colimit crossed module in terms of colimits of groups.
The aim now is to transfer the problem to computing colimits of crossed modu*
*les over
a fixed group P . To do this, suppose given a morphism of groups ' : P ! Q. The*
*n there is
6
a pullback functor
'* : (crossed modules over Q)! (crossed modules over P.)
This functor has a left adjoint
'* : (crossed modules over P!)(crossed modules over Q),
which gives our induced crossed module. This construction can be described as a*
* "change
of base" [8]. To compute a colimit colimi(~i: Mi! Pi), one forms the group P = *
*colimiPi,
and uses the canonical morphisms OEi : Pi ! P to form the family of induced cro*
*ssed P -
modules ((~i)* : (OEi)*Mi! P ). The colimit of these in the category of crossed*
* P -modules
is isomorphic to the original colimit. Now if M0 is the colimit in the category*
* of groups of
the (OEi)*Mi, then there is a canonical morphism M0 ! P and an action of P on M*
*0. The
resulting (M0 ! P ) is a pre-crossed module, and quotienting by its Peiffer sub*
*group gives
the required crossed module.
Presentations for induced crossed modules were given in [9], and more recent*
*ly fami-
lies of explicit examples have been computed, partly by hand and partly using G*
*AP [14].
Computation of induced crossed modules is here reduced to problems of computati*
*on in
combinatorial group theory. A key fact which makes one expect successful comput*
*ations is
that if (~ : M ! P ) is a crossed module with M finite, and if ' : P ! Q is a m*
*orphism of
finite index, then the induced crossed Q-module '*M is also finite [14, Theorem*
* 2.1].
Example 3.1 When ~ : M ! P and ' : P ! Q are subgroup inclusions, there are c*
*omplete
descriptions of '*M in the following cases:
(i)If ' is surjective then '*M ~=M=[M, ker'], ([9, Proposition 9]).
(ii)If M is abelian and '~(M) is normal in Q then '*M is abelian and is the us*
*ual induced
Q-module M ZP ZQ, ([14, Corollary 1.6]).
(iii)If M and P are normal subgroups of Q then '*M ~=M x (Mab I(Q=P )), whe*
*re I
denotes the augmentation ideal. If in addition M = P then '*P ~=P x (P ab*
*)[Q:P]-1,
([15, Theorem 1.1]).
(iv)If M = P = C2, the cyclic group of order 2, ~ = idP, and ' : C2 ! D2nis th*
*e inclusion
to a reflection in the dihedral group D2n, then '*P ~= D2n ([14, Example 1*
*.4]). The
action is not the usual conjugation: when n is odd the boundary is an isom*
*orphism,
but when n is even the kernel and cokernel are isomorphic to C2.
4 Homotopical applications
As explained in the introduction, the fundamental crossed module functor 2 ass*
*igns a
crossed module (@ : ß2(X, A, a) ! ß1(A, a)) to any based pair of spaces (X, A, *
*a). Theorem
C of [9] is a 2-dimensional Van Kampen type theorem for this functor. We will *
*use the
following consequence:
7
Theorem 4.1 ([9], Theorem D) Let (B, V, b) be a cofibred pair of spaces, let *
*f : V ! A be
a based map, and let X be the pushout A [f B in the left-hand diagram below. Su*
*ppose also
that A, B, V are path-connected, and (B, V, b) is 1-connected. Then the based p*
*air (X, A, a)
is 1-connected and the right-hand diagram
f ~*
V ____//_A ß2(B, V, b)____//_ß2(X, A, a)
|| || ffi|| ffi0||
fflffl|fflffl| fflffl| fflffl|
B ____//_X ß1(V, b)________//ß (A, a)
~ 1
presents ß2(X, A, a) as the crossed ß1(A, a)-module ~*(ß2(B, V, b)) induced fro*
*m the crossed
ß1(V, b)-module ß2(B, V, b) by the group morphism ~ : ß1(V, b) ! ß1(A, a) induc*
*ed by f.
As pointed out earlier, when P is a free group on a set and ~ is the ident*
*ity, the
induced crossed module '*P is the free crossed Q-module on the function '| : *
* ! Q.
Thus Theorem 4.1 implies Whitehead's Theorem as stated in Theorem 1.1. A consid*
*erable
amount of work has been done on this case, because of the connections with iden*
*tities among
relations, and methods such as transversality theory and "pictures" have proved*
* successful
([11, 27]), particularly in the homotopy theory of 2-dimensional complexes [21]*
*. However,
the only route so far available to the wider geometric applications of induced *
*crossed modules
is Theorem 4.1. We also note that this Theorem includes the relative Hurewicz T*
*heorem in
this dimension, on putting A = V , and f : V ! V the inclusion.
We will apply this Theorem 4.1 to the classifying space of a crossed module,*
* as defined by
Loday in [23] or Brown and Higgins in [10]. This classifying space is a functor*
* B assigning to
a crossed module M = (~ : M ! P ) a based CW -space BM with the following prope*
*rties:
4.2 The homotopy groups of the classifying space of the crossed module M = (~ :*
* M ! P )
are given by 8
< coker~ for i = 1,
ßi(BM) ~= ker~ for i = 2,
: 0 for i > 2 .
The first Postnikov invariant of BM is precisely the k-invariant of M as in 1.2.
4.3 The classifying space BP = B(i : 1 ! P ) is the usual classifying space of *
*the group
P , and BP is a subcomplex of BM. Further, there is a natural isomorphism of c*
*rossed
modules
2(BM, BP, x) ~= M .
4.4 If X is a reduced CW -complex with 1-skeleton X1, then there is a map
X ! B( 2(X, X1, x))
inducing an isomorphism of ß1 and ß2.
8
It is in these senses that it is reasonable to say, as in the Introduction, *
*that crossed
modules model all based homotopy 2-types.
We now give two direct applications of Theorem 4.1.
Corollary 4.5 Let M = (~ : M ! P ) be a crossed module, and let ' : P ! Q be a
morphism of groups. Let fi : BP ! BM be the inclusion. Consider the pushout
fi
BP _______//BM .
B'|| ||
fflffl| fflffl|
BQ ___fi0__//X
Then the fundamental crossed module of the pair (X, BQ, x) is isomorphic to the*
* induced
crossed module (@ : '*M ! Q), and this crossed module determines the 2-type of *
*X. In
particular, the second homotopy group ß2(X, x) is isomorphic to ker@.
Proof The first statement is immediate from Theorem 4.1. The second statement *
*follows
from results of [10], since the morphism Q ! ß1(X) is surjective. The final st*
*atement
follows from the homotopy exact sequence of the pair (X, BQ, x). *
* 2
Remark 4.6 An interesting special case of the last Corollary is when M is an *
*inclusion
of a normal subgroup, since then BM is of the homotopy type of B(P=M). So we h*
*ave
determined the 2-type of a homotopy pushout
Bp
BP _____//BR
B'|| ||
fflffl| fflffl|
BQ _p0_//_X
in which p : P ! R is surjective. *
* 2
Corollary 4.7 Let ' : P ! Q be a morphism of groups, and let BP denote the co*
*ne
on BP . Then the fundamental crossed module 2(BQ [B' BP, BQ, x) is isomorphi*
*c to
the induced crossed module (@ : '*P ! Q). In particular, the second homotopy *
*group
ß2(BQ [B' BP, x) is isomorphic to ker@.
We also note that in determining the crossed module representing a 2-type we*
* are also
determining the first Postnikov invariant of that 2-type. However it may be mor*
*e difficult to
describe this invariant as a cohomology class, though this is done in some case*
*s in [14, 15].
5 Computational issues
Recall from Proposition 9 of [9] that when ' : P ! Q is a surjection then '*M ~*
*=M=[M, K],
where K = ker' and [M, K] denotes the subgroup of M generated by the m-1mk for *
*all
9
m 2 M, k 2 K. When ' is neither surjective nor injective, we obtain a factorisa*
*tion ' = '2O'1
with '1 surjective and '2 injective, and construct the induced crossed module i*
*n two stages:
('1)* ('2)*
M ______//_('1)*M____//_('2)*('1)*M
~|| @1|| @2||
fflffl| fflffl| fflffl|
P ____'1__//im'____'2_____//_Q .
The first stage is easily constructed as a quotient group, so in the following *
*subsections
we discuss significant computational issues in the case when both ' and ~ are s*
*ubgroup
inclusions.
Note that computation of free crossed modules, as described in section 1, is*
* in general
difficult since the groups are usually infinite, and is not attempted in the cu*
*rrent version of
the package.
5.1 Isomorphic pairs of groups
The GAP function IsomorphismPermGroup enables the construction of a permutation*
* group
isomorphic to a finite group already obtained. Thus it is sufficient to impleme*
*nt the induced
construction for permutation groups. Similarly, the function IsomorphismFpGroup*
* enables
the construction of finitely presented groups F M, F P, F Q isomorphic to permu*
*tation groups
M, P, Q; monomorphisms F ~ : F M ! F P, F ' : F P ! F Q mimicing the inclusi*
*ons
M ! P ! Q; and an action of F P on F M.
5.2 Copower of groups
The construction of induced crossed modules, described in [9, 14], involves the*
* copower
F M ~*T , namely the free product of groups F Mt, t 2 T , each isomorphic to F *
*M. Here
T is a transversal for the right cosets of F P in F Q, in which the representat*
*ive of the
subgroup F P is taken to be the empty word. The group F Mt = {(m, t) | m 2 F M}
has product (m, t)(n, t) = (mn, t) and F Q acts by (m, t)q = (mp, u) where tq =*
* ('p)u in
F Q. If F M has fl generators, then a presentation F C of M ~*T with fl | T | g*
*enerators
may be constructed using functions in the GAP Tietze package ([18], Chapter 46*
*). The
relators of F C comprise | T | copies of the relators of F M, suitably renumber*
*ed. Define
ffi0: F C ! F Q, (m, t) 7! t-1('~m)t.
5.3 Quotient by the Peiffer subgroup
Let be a generating set for F M and let FP be the closure of under the act*
*ion of F P .
Then '*(M) ~=F C=F N where F N is the normal closure in F C of the Peiffer elem*
*ents
0(n,s) FP
<(n, s), (m, t)> = (n, s)-1(m, t)-1(n, s)(m, t)ffi (m, n 2 , s,*
*(t12)T ).
10
The homomorphism '* is induced by the projection pr1m = (m, 1FQ ) onto the firs*
*t factor,
and the boundary ffi of '*M is induced from ffi0as shown in the following diagr*
*am:
'* //
F M _______F C=F N .
~|| |ffi|
fflffl| fflffl|
F P___'_____//F Q
Thus a finitely presented group F I ~= '*M is obtained by adding to the relator*
*s of F C
further relators corresponding to the list of elements in equation (1), and the*
* presentation
may be simplified by applying Tietze transformations.
5.4 Tracing Tietze transformations
As well as returning an induced crossed module, the construction should return *
*a morphism
of crossed modules ('*, ') : M ! '*M. When Tietze transformations are then appl*
*ied to the
initial presentation for F I, during the resulting simplification some of the f*
*irst fl generators
may be eliminated, so the projection pr()may be lost. In order to preserve this*
* projection,
and so obtain the morphism '*, it is necessary to record for each eliminated ge*
*nerator g a
relator gw-1 where w is the word in the remaining generators by which g was eli*
*minated.
A significant advantage of GAP is the free availability of the library code,*
* which enables
the user to modify a function so as to return additional information. For the X*
*Mod1 version
of the package, the Tietze transformation code was modified so that the resulti*
*ng presenta-
tion contained an additional field presI.remember, namely a list of (at least) *
*fl | T | relators
expressing the original generators in terms of the final ones. In more recent r*
*eleases of GAP
an equivalent facility has been made generally available using the TzInitGenera*
*torImages
function.
5.5 Polycyclic groups
Recall that a polycyclic group is a group G with power-conjugate presentation h*
*aving gen-
erating set {g1, . .,.gn} and relations
{goii= wii(gi+1, . .,.gn), ggji= w0ij(gj+1, . .,.gn) 8 1 6 j < i 6*
*(n}.2)
(see [18], Chapters 43,44). These are implemented in GAP as PcGroups. Since sub*
*groups
M 6 P 6 G have induced power-conjugate presentations, if T is a transversal for*
* the right
cosets of P in G, then the relators of M ~*T are all of the form in (2). Furth*
*ermore, all
the Peiffer relations in equation (1) are of the form ggji= gpk, so one might h*
*ope that a
power conjugate presentation would result. Consideration of the cyclic-by-cycli*
*c case in the
following example shows that this does not happen in general.
Example 5.1 Let Cn be cyclic of order n with generator x, and let ff : x 7! x*
*a be an
automorphism of Cn of order p. Take G = ~= Cp n Cn*
*. When
M = P = Cn . G cases (ii) and (iii) of example 3.1 apply, and '*Cn ~=Cpn.
11
It follows from the relators that hig = ghai, 0 < i < n, and that h-1(ghi*
*(1-a))h =
gh(i+1)(1-a). So if we put gi= ghi(1-a), 0 6 i < n, then ggji= g[j+a(i-j)]. Wh*
*en M = P =
Cp = and ' : Cp ! G, we may choose as transversal T = {1G, h, h2, . .,*
*.hn-1}. Then
M ~*T has generators {(g, hi) | 0 6 i < n}, all of order p, and relators {(g, h*
*i)p | 0 6 i < n}.
The additional Peiffer relators in equation (1) have the form
(g, hi)(g, hj) = (g, hj)(gk, hl) where hih-jghj = gkhl
so k = 1 and l = [j + a(i - j)]. Hence ` : '*M ! Q, (g, hi) 7! gi is an isomo*
*rphism,
and '*M is isomorphic to the identity crossed module on Q. Furthermore, if we *
*take
M to be a cyclic subgroup Cm of Cp then '*M is the normal subgroup crossed mod*
*ule
(i : Cm n Cn ! Cp n Cn). *
* 2
5.6 Identifying '*M
From some of the special cases listed in example 3.1 and from other examples, w*
*e know
that many of the induced groups '*M are direct products. However the generating*
* sets in
the presentations that arise following the Tietze transformation do not in gene*
*ral split into
generating sets for direct summands. This is clearly seen in the following simp*
*le illustration.
Example 5.2 Let Q = S4, the symmetric group of degree 4, and M = P = A4, the
alternating subgroup of Q of index 2. Since the abelianisation of A4 is cyclic *
*of order 3, case
(iii) in section 3 shows that '*M ~= A4 x C3. However a typical presentation fo*
*r A4 x C3
obtained from the program is
,
and one generator for the C3 summand is yzx2. Converting to a permutation group*
* H gives
a degree 12 representation with generating set
{(2, 9, 4)(3, 5, 6)(8, 12, 10), (1, 4, 2)(3, 5, 7)(10, 11, 12), (1, 8, 3)(*
*2, 10, 5)(7, 9, 12)}.
Converting H to an PcGroup produces a 4-generator group with subnormal series
I . C2 . C22. A4 . A4 x C3 ,
where each extension adds a generator gi, i = 1 . .4.and g1g2g4 is a generator *
*for the normal
C3. In these representations, the cyclic summand remains hidden, and an explic*
*it search
among the normal subgroups must be undertaken to find it. *
* 2
6 Results
In this section we list the crossed modules induced from subgroups of groups of*
* order at
most 23 (excluding 16), except that the special cases mentioned earlier enable *
*us to exclude
12
abelian and dihedral groups; the case when P is normal in Q; and the case when *
*Q is a
semidirect product Cm n Cn.
In the first table, we assume given an inclusion ' : P ! Q of a subgroup P o*
*f a group Q,
and a normal subgroup M of P . We list the isomorphism type of the source of th*
*e crossed
module (@ : '*M ! Q) induced from (~ : M ! P ) by the inclusion '. Recall that *
*this kernel
is realised as a second homotopy group in corollary 4.5. Labels I, Cn, D2n, An*
*, Sn denote
the identity, cyclic, dihedral, alternating and symmetric groups of order 1, n,*
* 2n, n!=2 and
n! respectively. The group Hn is the holomorph of Cn and H+nis its positive sub*
*group in
degree n. SL(2, 3) and GL(2, 3) are the special and general linear groups of o*
*rder 24, 48
respectively. Labels of the form m.n refer to the nth group of order m accordi*
*ng to the
GAP4 numbering.
________________________________________
| | |Q|M| |P | Q | ' M | ker@ | |
|_|_____|_|___|________|___*____|______|_|_
| | 12C ||C | A | H+ | C | |
| | 2|| 2 | 4 | 8 | 4 | |
| | C || C | A | SL(2, 3) |C | |
| | 3|| 3 | 4 | | 2 | |
| | 18C ||C |C n C2 | 54.8 | C | |
| | 2|| 2 | 2 3 | | 3 | |
| | S| |S |C n C2 | 54.8 | C | |
| | |3| 3 | 2 3 | | 3 | |
| | 20C ||C | H | D | C | |
| | 2|| 2 | 5 | 10 | 2 | |
| | C || C2 | D | D | I | |
| | 2|| 2 | 20 | 10 | | |
| | C2|| C2 | D | D | I | |
| | 2|| 2 | 20 | 20 | | |
| | 21C ||C | H+ | H + | I | |
|_|_____3||_3_|___7____|___7____|______|_|
Table 1
The second table contains the results of calculations with Q = S4, where C2 *
*= <(1, 2)>,
C02= <(1, 2)(3, 4)>, and C22= <(1, 2), (3, 4)>. The final column contains the a*
*utomorphism
group Aut('*M) (where known).
___________________________________________
| | M |P | ' M |ker@ |Aut(' M) | |
|_|____|_____|____*______|______|_____*___|_|_
| | C |C | GL(2, 3) |C | S C | |
| | 2 | 2 | | 2 | 4 2 | |
| | C |C |C SL(2, 3) |C | 144.183 | |
| | 3 | 3 | 3 | 6 | | |
| | C |S | SL(2, 3) |C | S | |
| | 3 | 3 | | 2 | 4 | |
| | S |S | GL(2, 3) |C | S C | |
| | 3 | 3 | | 2 | 4 2 | |
| | C0 |C0 | 128.? |C C3 | | |
| | 2 | 2 | | 4 2 | | |
| | C0C|2, C | H+ | C | S C | |
| | 2 |2 4 | 8 | 4 | 4 2 | |
| | C0 |D | C3 | C | SL(3, 2) | |
| | 2 | 8 | 2 | 2 | | |
| | C2 |C2 | S C | C | S C | |
| | 2 | 2 | 4 2 | 2 | 4 2 | |
| | C2 |D | S | I | S | |
| | 2 | 8 | 4 | | 4 | |
| | C |C | 96.219 |C | 96.227 | |
| | 4 | 4 | | 4 | | |
| | C |D | S | I | S | |
| | 4 | 8 | 4 | | 4 | |
| | D |D | S C | C | S C | |
|_|__8_|_8___|____4_2____|__2__|____4_2___|_|
Table 1
13
An interesting problem is to obtain a clearer understanding of the geometric*
* significance
of these tables.
References
[1]Alp, M., GAP, Crossed modules, Cat1-groups: Applications of computational g*
*roup theory
Ph.D. thesis, University of Wales, Bangor, (1997),
http://www.informatics.bangor.ac.uk/public/math/research/ftp/theses/alp.ps.*
*gz.
[2]Alp, M. and Wensley, C.D., `Enumeration of cat1-groups of low order', Int. *
*J. Algebra
and Computation 10 (2000) 407-424.
[3]Ashley, N.K., Simplicial T -complexes, Ph.D. Thesis, University of Wales, B*
*angor, (1978).
Published as `Simplicial T-complexes: a non-abelian version of a theorem of*
* Dold-Kan', Diss.
Math. 265 (1988) 11-58.
[4]Atiyah, M., `Mathematics in the 20th Century', Bull. London Math. Soc. 34 (*
*2002) 1-15.
[5]Breen, L., `Th'eorie de Schreier sup'erieure', Ann. Sci. 'Ecol. Norm. Sup. *
*25 (1992) 465-514.
[6]Brown, K.S., Cohomology of groups, Graduate texts in Mathematics 87, Spring*
*er-Verlag,
New York (1982).
[7]Brown, R., `Higher dimensional group theory', in Low-dimensional topology, *
*ed. R.Brown
and T.L.Thickstun, London Math. Soc. Lecture Note Series 46, Cambridge Univ*
*ersity
Press (1982) 215-238.
[8]Brown, R., `Homotopy theory, and change of base for groupoids and multiple *
*groupoids',
Applied categorical structures, 4 (1996) 175-193.
[9]Brown, R. and Higgins, P.J., `On the connection between the second relative*
* homotopy
groups of some related spaces', Proc. London Math. Soc., (3) 36 (1978) 193-*
*212.
[10]Brown, R. and Higgins, P.J., `The classifying space of a crossed complex', *
*Math. Proc.
Camb. Phil. Soc. 110 (1991) 95-120.
[11]Brown, R. and Huebschmann, J., `Identities among relations', in Low-dimensi*
*onal topol-
ogy, ed. R.Brown and T.L.Thickstun, London Math. Soc. Lecture Note Series 4*
*6, Cam-
bridge University Press (1982) 153-202.
[12]Brown, R. and Spencer, C.B., `Double groupoids and crossed modules', Cah. T*
*op. G'eom.
Diff., 17 (1976) 343-362.
[13]Brown, R. and Spencer, C.B., `G-groupoids, crossed modules and the fundamen*
*tal
groupoid of a topological group', Proc. Kon. Ned. Akad. v. Wet., 79 (1976) *
*296-302.
[14]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 (1995) 5*
*1-74.
[15]Brown, R. and Wensley, C.D., `Computing crossed modules induced by an inclu*
*sion
of a normal subgroup, with applications to homotopy theory', Theory and app*
*lications of
categories, 2 (1996) 3-16.
[16]K. Dakin, Multiple compositions for higher dimensional groupoids, Ph.D. The*
*sis, University
of Wales, Bangor, (1977).
[17]Ellis, G. and Steiner, R., `Higher dimensional crossed modules and the homo*
*topy groups
of (n + 1)-ads', J. Pure Appl. Algebra, 46 (1987) 117-136.
14
[18]The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.3 (2002*
*),
http://www.gap-system.org.
[19]Heyworth, A. and Wensley, C.D., `Logged rewriting and identities among rela*
*tors', in
Proceedings of Groups St Andrews 2001 in Oxford, (to appear).
[20]Higher-dimensional discrete algebra,
http://www.informatics.bangor.ac.uk/public/math/research/hdda/.
[21]Hog-Angeloni, C., Metzler, W. and Sieradski, A.J. (Editors), Two-dimensional
homotopy and combinatorial group theory, London Math. Soc. Lecture Note Ser*
*ies 197, Cam-
bridge University Press (1993).
[22]Jones, D.W., Polyhedral T -complexes, Ph.D. thesis, University of Wales, Ba*
*ngor, (1984).
Published as `A general theory of polyhedral sets and their corresponding T*
* -complexes', Diss.
Math. 266 (1988).
[23]Loday, J.-L., `Spaces with finitely many non-trivial homotopy groups', J. P*
*ure Appl. Algebra,
24 (1982) 179-202.
[24]Mac Lane, S. and Whitehead, J.H.C., `On the 3-type of a complex', Proc. Nat*
*. Acad.
Sci. (1950) 41-48.
[25]Moore, E.J., Graphs of groups: word computations and free crossed resoluti*
*ons, Ph.D.
thesis, University of Wales, Bangor, (2001),
http://www.informatics.bangor.ac.uk/public/math/research/ftp/theses/moore.p*
*s.gz.
[26]Nan Tie, G., `A Dold-Kan theorem for crossed complexes', J. Pure Appl. Alge*
*bra, 56 (1989)
177-194.
[27]Pride, S.J. `Identities among relations', in Proc. Workshop on Group Theory*
* from a Geo-
metrical Viewpoint, International Centre of Theoretical Physics, Trieste, 1*
*990, ed. E. Ghys,
A. Haefliger and A. Verjodsky, World Scientific (1991) 687-716.
[28]Schönert, M. et al, GAP-Groups, Algorithms, and Programming, Lehrstuhl D fü*
*r Mathe-
matik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, sixth*
* edition, 1997.
[29]Whitehead, J.H.C., `On adding relations to homotopy groups', Annals of Math*
*. 41 (1941)
806-810.
[30]Whitehead, J.H.C., `Note on a previous paper entitled Ö n adding relations *
*to homotopy
groups"', Annals of Math. 47 (1946) 806-810.
[31]Whitehead, J.H.C., `Combinatorial homotopy II', Bull. Amer. Math. Soc. 55 (*
*1949) 453-
496.
15