</tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">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 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">This
approach
to modules of identities can yield interesting results for
some surprisingly small groups. We give two examples. 
 <ol>
 <li>The presentation P = < x,y | x2, y3,
(xy)2 > of the symmetric group S3 was
considered in [R. Brown and A. Razak Salleh, "Free crossed resolutions
of groups and presentations of modules of identities among relations",
style="font-style: italic;">LMS J. Comput. Math., (1999),
28-61] where covering groupoid techniques were used to show that the
ZG-module Pi2(KP) can be generated by four
elements. The following commands show that, in fact, it can be
generated by three elements. (Subsequent commands will show that
the module can even be generated by two elements.) 
 </li>
 <li>A conjecture of Salvetti (mentioned on a <a
href="aboutTopology.html">previous page</a>) would imply that the
module of identities Pi2(KP) for the Coxeter
presentation P = < x,y,z, | x2, y2, z2,
(xy)3, (yz)3, (xz)2 > of S4
can be generated by no fewer than 10 elements. The following commands
show that, in fact, it can be generated by six elements.</li>
 </ol>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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 <a href="table/help.html">here</a>. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Let
us now reconsider the symmetric group S3 with the unusual
presentation P'=< x, y | x², xyx^-1y^-2
>. The following
commands establish the existence of an infinite periodic ZS3-resolution
R
of period 4 (meaning Rn = Rn+4 for all n)
which, in any given dimension, has either one or two
generators. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
existence of such a periodic ZS3-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), 267-291]. Swan proved that if a group G acts fixed-point
freely on a sphere then there is a periodic ZG-resolution of Z. The
interest in the group S3 is that it does not act freely on a
sphere. 
 
The periodic ZS3-resolution arises as the cellular chain
complex of a contractible CW-space X on which S3 acts
freely. An analysis of the boundary maps in the resolution yield the
following explicit description of the orbit space B=X/S3 in
low dimensions. 
 
This classifying space B has a unique 0-cell and two 1-cells which we
label by x and y. It has two 2-cells which are attached to the
1-skeleton according to the following pictures. 
 <div style="text-align: center;"><img alt="" src="relators.jpg"
style="width: 479px; height: 194px;"> 
 <div style="text-align: left;"> 
The space B has one 3-cell attached to the 2-skeleton according to the
following picture. 
 
 <div style="text-align: center;"><img alt="" src="syzygy.jpg"
style="width: 277px; height: 277px;"> 
 <div style="text-align: left;"> 
This last picture corresponds to an identity
between the relators r:=x2 and s:=xyx-1y-2
and represents a map S2 → B2 from the 2-sphere to
the 2-skeleton of B. This kind of map is called a homotopical syzygy in the paper
[J.-L. Loday, "Homotopical sysygies", Contemp.
Math. 265 (2000), 99-127]. 
 
Note that S3 can be regarded as the dihedral group D3.
It
is well known that the dihedral groups Dn admit periodic
ZDn-resolutions if and only if n is odd, say=2k+1 . The
period is always 4. One could attempt to construct periodic ZD2k+1-resolutions
using
a presentation such as < x,y | x2, y-k-1xyx-1>
for
D2k+1. For example, with k=20 the following commands can
be used to construct an infinite periodic resolution for the diherdral
group D41 of order 82. 
 </div>
 </div>
 </div>
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
F:=FreeGroup(2);;x:=F.1;;
y:=F.2;; 
 
gap> k:=20;; G:=F/[x^2,y^(-k-1)*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</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Let
us now return to the module of identities of the presentation P = <
x,y | x2, y3, (xy)2 >. This defines
the same group as the presentation P' = < x, y | x²,
xyx-1y-2
> and we have seen that the module Pi2(KP') is
generated by a single element. It is not difficult to
establish (by an easy theoretical argument) that Pi2(KP)
=
ZS3 + Pi2(KP') and that the Z-rank
of Pi2(KP') is equal to 1. In particular, Pi₂(K_P)
is
generated by just two elements. </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">
 <table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutSuperperfect.html">Previous
Page</a> 
 </td>
 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
 </td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutAspherical.html">Next
page 
 </a> </td>
 </tr>
 </tbody>
 </table>
 <a href="aboutTopology.html"> 
 </a> </td>
 </tr>
 </tbody>
</table>
 
 
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Second Homotopy Groups of
Presentations And A Periodic Resolution 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutAspherical.html">next</a> 
 </td>
 </tr>
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 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">There
is
a standard way of associating to any group presentation P = < x | r > a 2-dimensional
connected CW-space KP whose fundamental group is
isomorphic to the group G defined by the presentation. The space KP
has a single 0-cell, one
1-cell for each generator in x,
and
one 2-cell for each relator in r (attached to the
1-skeleton of KP 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 Pi2(KP).
This homotopy group is a torsion free ZG-module and is called the module of identities of P.
Good introductions to the topic are given in 
 <ul>
 <li>W. Bogley and S.J. Pride, "Calculating generators of Pi₂",
in
Two-dimensional homotopy and
combinatorial group theory (edited by C. Hog-Angeloni, W. Metzler and
A.J. Sieradski), LMS Lecture Note Series 197 (1993), 157-188. 
 </li>
 <li>R. Brown and J. Huebschmann, "Identities among relations",
in Low dimensional topology (edited
by
R. Brown and T.L. Thickstun), LMS Lecture Note Series 48
(1982), 153-202.</li>
 </ul>
 
For presentations P defining a small group G the structure of the
module Pi2(KP) 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 ZG-resolution R arising as the cellular chain
complex of a space X where: X is contractible; X admits a free
cellular action of G; the 2-skeleton X2 is the universal
cover of the space KP. Standard properties of the universal
cover and the Hurewicz Theorem yield ZG-isomorphisms 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td
style="text-align: center; vertical-align: top; background-color: rgb(204, 255, 255);"> 
Pi2(KP) = Pi2(X2)
=
H2(X2,Z) = Image( R3 → R2
). 
 
 </td>
 </tr>
 </tbody>
 </table>
 
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 | x2, y4,
xyxy
> of the dihedral group D4. This example can also be
handled using the following commands. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">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 ZG-module 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 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
generators for the module of identities are here expressed as elements
in the free ZG-module R3 with basis E1, E2,
E3. An element in R3 of the form (1-g)Ei
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 2-sphere to the space Kp
using the following command. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
IdentityAmongRelatorsDisplay(R,4); 
 
 <div style="text-align: center;"><img
style="width: 328px; height: 225px;" alt="" src="d4pic.gif"> 
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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,
"Three--dimensional presentations for the groups of order at most 30", LMS J. Math. Comp. Vol. 2 (1999),
93-117] and the function was implemented by Irina Kholodna. 
 
The function certainly works for groups larger than D4. For
instance, the following commands take a couple of minutes to show that
the standard presentation of the dihedral group D200 of
order 400 also has a module of identities generated by four elements.
(No doubt there's a theorem here!)

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