


There
is
a standard way of associating to any group presentation P = < x  r > a 2dimensional
connected CWspace K_{P} whose fundamental group is
isomorphic to the group G defined by the presentation. The space K_{P}
has a single 0cell, one
1cell for each generator in x,
and
one 2cell for each relator in r (attached to the
1skeleton of K_{P} in such a way that its boundary spells the
relator). Quite a number of papers have been written on the
problem of
calculating the second homotopy group Pi_{2}(K_{P}).
This homotopy group is a torsion free ZGmodule and is called the module of identities of P.
Good introductions to the topic are given in
For presentations P defining a small group G the structure of the module Pi_{2}(K_{P}) can be determined using the HAP function ResolutionSmallFpGroup(G,n) . This function inputs a finitely presented group G and integer n>2. It returns n terms of a ZGresolution R arising as the cellular chain complex of a space X where: X is contractible; X admits a free cellular action of G; the 2skeleton X^{2} is the universal cover of the space K_{P}. Standard properties of the universal cover and the Hurewicz Theorem yield ZGisomorphisms
The article by Bogley and Pride mentioned above explains how the theory of Igusa's pictures can be used to find generators for the module of identities of the standard presentation P = < x,y  x^{2}, y^{4}, xyxy > of the dihedral group D_{4}. This example can also be handled using the following commands. 

gap>
F:=FreeGroup("x",
"y");; x:=F.1;; y:=F.2;; gap> D_4:= F / [ x^2, y^4, (x*y)^2 ];; gap> R:=ResolutionSmallFpGroup(D_4,3);; gap> RankOfIdentityModuleOverZ:= ( R!.dimension(2)  R!.dimension(1) + R!.dimension(0) )*Order(D_4)  1; 15 gap> NumberOfGeneratorsOfIdentityModuleOverZG:=R!.dimension(3); 4 gap> #The four ZGmodule generators are: gap> for i in [1..4] do > Print("\n Generator ", i, "=\n "); PrintZGword(R!.boundary(3,i),R!.elts); >od; Generator 1= (  x*y^2 + y^2 )E1 Generator 2= (  x + x*y )E2 Generator 3= (  <identity ...> + x*y )E3 Generator 4= (  <identity ...>  x*y  y  x*y^2 )E1 + (  <identity ...>  x )E2 + ( + <identity ...> + x + x*y*x + y*x )E3 

The
generators for the module of identities are here expressed as elements
in the free ZGmodule R_{3} with basis E_{1}, E_{2},
E_{3}. An element in R_{3} of the form (1g)E_{i}
is called a dipole by Bogley
and Pride. As in their paper we see that the module of identities is
generated by three dipoles and a fourth more complicated element. The
fourth element can be viewed as a map from the 2sphere to the space K_{p}
using the following command. 

gap>
IdentityAmongRelatorsDisplay(R,4); 

The
function ResolutionSmallFpGroup(G,n)
is based on a fairly naive use of the standard LLL and Smith Normal
Form algorithms. It works only for groups of fairly low order but tends
to produce smaller resolutions than the function ResolutionFiniteGroup(gens,n).
The
method is described in [G. Ellis and I. Kholodna,
"Threedimensional presentations for the groups of order at most 30", LMS J. Math. Comp. Vol. 2 (1999),
93117] and the function was implemented by Irina Kholodna. The function certainly works for groups larger than D_{4}. For instance, the following commands take a couple of minutes to show that the standard presentation of the dihedral group D_{200} of order 400 also has a module of identities generated by four elements. (No doubt there's a theorem here!) 

gap>
F:=FreeGroup(2);;
x:=F.1;; y:=F.2;; gap> D_200:= F / [ x^2, y^200, (x*y)^2 ];; gap> R:=ResolutionSmallFpGroup(D_200,3);; gap> R!.dimension(3); 4 

This
approach
to modules of identities can yield interesting results for
some surprisingly small groups. We give two examples.


gap>
#EXAMPLE
1 gap> F:=FreeGroup(2);;x:=F.1;;y:=F.2;; gap> S_3:=F/[ x^2, y^3, (x*y)^2 ];; gap> R:=ResolutionSmallFpGroup(S_3,3);; gap> R!.dimension(3); 3 gap> #EXAMPLE 2 gap> F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;; gap> S_4:=F/[x^2, y^2, z^2, (x*z)^2, (y*z)^3, (x*y)^3];; gap> R:=ResolutionSmallFpGroup(S_4,3);; gap> R!.dimension(3); 6 

A
table listing presentations for the nonabelian groups of order at most
30, together with explicit generators for the corresponding modules of
identities, can be found here. 

Let
us now reconsider the symmetric group S_{3} with the unusual
presentation P'=< x, y  x^{2}, xyx^{1}y^{2}
>. The following
commands establish the existence of an infinite periodic ZS_{3}resolution
R
of period 4 (meaning R_{n} = R_{n+4} for all n)
which, in any given dimension, has either one or two
generators. 

gap>
F:=FreeGroup(2);;
x:=F.1;; y:=F.2;; gap> G:=F/[ x^2, x*y*x^1*y^2 ];; gap> Order(G); IsAbelian(G); #Test that G really is isomorphic to S_3. 6 false gap> R:=ResolutionSmallFpGroup(G,9);; gap> for i in [1..9] do Print(R!.dimension(i),"\n"); od; 2 2 1 1 2 2 1 1 2 gap> R!.boundary(5,1)=R!.boundary(9,1); true gap> R!.boundary(5,2)=R!.boundary(9,2); true 

The
existence of such a periodic ZS_{3}resolution of Z
is originally due to J. Milnor and was first published as an appendix
to the paper [R.G. Swan, "Periodic resolutions for finite
groups", Annals of Mathematics (2),
72
(1960), 267291]. Swan proved that if a group G acts fixedpoint
freely on a sphere then there is a periodic ZGresolution of Z. The
interest in the group S_{3} is that it does not act freely on a
sphere. The periodic ZS_{3}resolution arises as the cellular chain complex of a contractible CWspace X on which S_{3} acts freely. An analysis of the boundary maps in the resolution yield the following explicit description of the orbit space B=X/S_{3} in low dimensions. This classifying space B has a unique 0cell and two 1cells which we label by x and y. It has two 2cells which are attached to the 1skeleton according to the following pictures. The space B has one 3cell attached to the 2skeleton according to the following picture. This last picture corresponds to an identity between the relators r:=x^{2} and s:=xyx^{1}y^{2} and represents a map S^{2} → B^{2} from the 2sphere to the 2skeleton of B. This kind of map is called a homotopical syzygy in the paper [J.L. Loday, "Homotopical sysygies", Contemp. Math. 265 (2000), 99127]. Note that S_{3} can be regarded as the dihedral group D_{3}. It is well known that the dihedral groups D_{n} admit periodic ZD_{n}resolutions if and only if n is odd, say=2k+1 . The period is always 4. One could attempt to construct periodic ZD_{2k+1}resolutions using a presentation such as < x,y  x^{2}, y^{k1}xyx^{1}> for D_{2k+1}. For example, with k=20 the following commands can be used to construct an infinite periodic resolution for the diherdral group D_{41} of order 82. 

gap>
F:=FreeGroup(2);;x:=F.1;;
y:=F.2;; gap> k:=20;; G:=F/[x^2,y^(k1)*x*y^k*x^1];; gap> R:=ResolutionSmallFpGroup(G,5);; gap> for i in [1..5] do; Print(R!.dimension(i),"\n"); od; 2 2 1 1 2 

Let us now return to the module of identities of the presentation P = < x,y  x^{2}, y^{3}, (xy)^{2} >. This defines the same group as the presentation P' = < x, y  x^{2}, xyx^{1}y^{2} > and we have seen that the module Pi_{2}(K_{P'}) is generated by a single element. It is not difficult to establish (by an easy theoretical argument) that Pi_{2}(K_{P}) = ZS_{3} + Pi_{2}(K_{P'}) and that the Zrank of Pi_{2}(K_{P'}) is equal to 1. In particular, Pi_{2}(K_{P}) is generated by just two elements.  
