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 <th style="vertical-align: top;">
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cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a href="aboutquasi.html">Previous</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Resolutions For Extensions 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutArithmetic.html">next</a> 
 </td>
 </tr>
 </tbody>
 </table>
 <big></big> 
 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Consider
a direct product of groups E=N×G. Given a free ZN-resolution R of
Z, and a free ZG-resolution S of Z, one can construct a free
ZE-resolution of Z by forming a tensor product of resolutions
R(×)S. This is a special case of a result by Eilenberg and
Zilber. In dimension n the tensor product has free generators
e(×)f for each i-dimensional generator e in R and each
j-dimensional generator f in S with i+j=n. The boundary homomorphism in
the tensor product is described by an explicit formula involving just
the
boundaries of R and S. 
 
The following commands use this tensor product to show that the direct
product A5×S5 of the alternating group of
degree 5 with the symmetric group of degree 5 has 6-dimensional
integral homology H6(A5×S5,Z)=(Z2)10
+Z6 . 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionFiniteGroup(AlternatingGroup(5),7);; 
gap> S:=ResolutionFiniteGroup(SymmetricGroup(5),7);; 
gap> RS:=ResolutionDirectProduct(R,S);; 
gap> TRS:=TensorWithIntegers(RS);; 
gap> Homology(TRS,6); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
tensor product of resolutions can be used to construct a (minimal)
resolution for any abelian group. For instance, the following commands
confirm the well-known fact that the free abelian group Z5
of rank 5 has H4(Z5,Z) = Z5. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionAbelianGroup([0,0,0,0,0],5);; 
gap> TR:=TensorWithIntegers(R);; 
gap> Homology(TR,4); 
[ 0, 0, 0, 0, 0 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">Consider
now the more general situation of a short exact sequence 1 → N
→ E → G → 1. In [C.T.C. Wall, "Resolutions for
extensions of groups", Proc. Cambridge Philos. Soc, 57 (1961), 251-255]
it was shown that in this case too there exists a free ZE-resolution
with the same number of generators as in the tensor product
R(×)S. The boundary homomorphism in the resolution is generally
more complicated than that in the tensor product. 
 
A number
of related results have been discovered and studied by various people
under the
general name of Perturbation Theory. One variation of Wall's result is
given in [G. Ellis, J. Harris &
E. Sköldberg, "Polytopal resolutions for finite groups", J. Reine
Angewandte Math.]. It concerns the more general case where N
is not necessarily normal in G and also provides explicit formulae for
the boundary in the "twisted tensor product". These formulae involve,
in addition to the boundaries of R and S, a contracting homotopy
on R (by which we mean a
family of abelian group homomorphisms hn:Rn→Rn+1
 related to the boundary homomorphisms dn:Rn→Rn-1
 via the formula dnhn-1dn=dn).
Formulae are also given for a contracting homotopy on the twisted
tensor product itself. These formulae can be used to build resolutions
and
compute homology, as
the following commands illustrate. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
P:=SylowSubgroup(MathieuGroup(23),2);; gens:=GeneratorsOfGroup(P);; 
 
gap> gensP:=[gens[4], gens[6], gens[2]];;
Order(P)=Order(Group(gensP)); 
true 
 
gap>
hom:=NaturalHomomorphismByNormalSubgroup(P,LowerCentralSeries(P)[3]);; 
 
gap> gensG:=List(GensP,x->Image(hom,x));; 
 
gap> S:=ResolutionfFiniteGroup(gensG,6);; 
 
gap> R:=ResolutionFiniteExtension(gensP,gensG,S,6);; 
 
gap> TR:=TensorWithIntegers(R);; 
 
gap> Homology(TR,5); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
function ResolutionFiniteExtension(gensE,gensG,S,n)
 inputs generators
for the group E, the images of these generators in a quotient group G,
at least n terms of a ZG-resolution S and a positive integer n. It
outputs n terms of a ZE-resolution R. Thus the fifth integral homology
of
the Sylow 2-subgroup P of M23 is H5(P,Z)=(Z2)14. 
 
The drawback to this divide-and-conquer technique is that it tends to
substantially increase the number of generators in the resolution for
E. Nevertheless, in low dimensions is can be repeatedly applied to
large groups. One function for doing this is ResolutionNormalSeries([L1,
... , Lk],n). The
input is an integer n and a chain of
subgroups L1 > L2 > ... > Lk
each of which is normal in L1. It outputs n terms of a
resolution for the group L1/Lk. 
 
The
following commands show that the biggest Mathieu Group M24
has Sylow
2-subgroup P with sixth mod 2 homology H6(P,Z2)=(Z2)143. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
P:=SylowSubgroup(MathieuGroup(24),2);; 
 
gap> R:=ResolutionNormalSeries(LowerCentraSeries(P),7);; 
 
gap> TR:=TensorWithIntegersModP(R,2); 
 
gap> Homology(R,6); 
143 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;"><a
name="Adem"></a>For
an example involving a matrix group consider the group K arising as the
kernel of the canonical homomorphism SL2(Zp3)
--> SL2(Zp) with p an odd prime. This group
has order p6 and exponent p2. It was show in
[J. Pakianathan, "Exponents and the cohomology of finite groups, Proc.
American Math. Soc., Vol. 128, No. 7 (1999), 1893-1897] that the
homology groups Hn(K,Z) have
exponent less than or equal to p2 for all but at most
finitely many n. In an earlier and longer paper [W. Browder & J.
Pakianathan,
"Cohomology of uniformly powerful p-groups", Trans. American Math.
Soc., Vol. 352, No. 6 (1999), 2659-2688] it had been proved that H3(K,Z)
has exponent p3. The group K is therefore an example of a
p-group for which the maximal exponent of the homology groups Hn(K,Z)
occurs only finitely many times, thus disproving a conjecture of A.
Adem. 
 
The exponent of the low dimensional homology groups Hn(K,Z),
for say p=3, can be computed using the following commands. (A quick way
to construct K is as a maximal subgroup of the Sylow 3-subgroup of SL2(2,Z27).)
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
P:=SylowSubgroup(SL(2,Integers mod 27),3);; 
 
gap> K:=MaximalSubgroups(P)[2]; 
<group of 2x2 matrices of size 729 in characteristic 27> 
 
gap> L:=LowerCentralSeries(K);; 
 
gap>
R:=ResolutionNormalSeries([L[1],L[2],Group(Identity(K))],6);; 
 
gap> TR:=TensorWithIntegers(R);; 
 
gap> Homology(TR,1); 
[ 3, 3, 3 ] 
 
gap> Homology(TR,2); 
[3, 3, 3] 
 
gap> Homology(TR,3); 
[3, 3, 3, 3, 3, 3, 27] 
 
gap> Homology(TR,4); 
[ 3, 3, 3, 3, 3, 3, 3, 3 ] 
 
gap> Homology(TR,5); 
[ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 9, 9, 9 ] 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
following commands show that the finitely presented dihedral group D100
of order 200 has integral homology H99(D100,Z)=(Z2)50+Z100
in dimension 99. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
F:=FreeGroup(2);;x:=F.1;;y:=F.2;; D:=F/[x^100,y^2,(x*y)^2];; 
 
gap> D_100:=Image(IsomorphismPermGroup(D));; 
 
gap> gens:=GeneratorsOfGroup(D_100);; 
 
gap> R:=ResolutionNormalSeries([gens, gens[1],
Identity(D_100)],100);; 
 
gap> TR:=TensorWithIntegers(R);; 
 
gap> Homology(R,99); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 100 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
Sylow 2-subgroup of the Symmetric group S20 has order 218.
The following commands show that this group G has third homology H3(G,Z)
= (Z2)56+(Z4)7. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SylowSubgroup(SymmetricGroup(20));; 
 
gap> R:=ResolutionNormalSeries(BigStepLCS(G,64),4);; 
 
gap> TR:=TensorWithIntegers(R);; 
 
gap> Homology(R,3); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4 ] 
 </td>
 </tr>
 <tr>
 <td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">Using
the function TwistedTensorProduct()
this divide-and-conquer technique
can be applied to any
group G with a subnormal series G=L1 > L2 >
... > Lk . The only requirements are that we have
resolutions (with contracting homotopies) for the group Lk
and for each quotient Li/Li+1. In particular,
resolutions for polycyclic groups can be
found using the function ResolutionSubnormalSeries(L,n)
 . 
 
 <a name="Lambe"></a>The group G need not be finite. For example,
the following commands use
the divide-and-conquer technique to compute the
integral homology of the free nilpotent group G of class two on four
generators. The homology is trivial in dimensions greater than 10, and
there is 3-torsion in dimensions 4 and 5. This homology, and also that
of the free nilpotent group of class two on five generators, was first
calculated in [Larry Lambe, "Cohomology of principal G-bundles over a
torus when H*(BG,R) is polynomial",
style="font-style: italic;">Bulletin Soc. Math. de Belgium, 38
(1986), 247-264]. 
 
(The integral cohomology ring structure for the free nilpotent group of
class two on three generators is investigated on a <a
href="aboutCohomologyRings.html#rankthree">subsequent page</a>. ) 
 
The group G is created using the GAP package "nq" by Werner Nickel. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
F:=FreeGroup(4);; 
gap> G:=NilpotentQuotient(F,2);; 
 
gap> R:=ResolutionNilpotentGroup(G,12);; 
gap> TR:=TensorWithIntegers(R);; 
gap> for n in [1..11] do 
> Print("Homology in dimension ", n," is \n",Homology(TR,n),"\n"); 
> od; 
Homology in dimension 1 is 
[ 0, 0, 0, 0 ] 
Homology in dimension 2 is 
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
Homology in dimension 3 is 
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0 ] 
Homology in dimension 4 is 
[ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
Homology in dimension 5 is 
[ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
Homology in dimension 6 is 
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
Homology in dimension 7 is 
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0 ] 
Homology in dimension 8 is 
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
Homology in dimension 9 is 
[ 0, 0, 0, 0 ] 
Homology in dimension 10 is 
[ 0 ] 
Homology in dimension 11 is 
[ ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">By
contrast, the following commands show that the integral homology of the
free nilpotent group of class three on three generators has 2-torsion
in dimension 4. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
F:=FreeGroup(3);; 
gap> G:=NilpotentQuotient(F,3);; 
 
gap> R:=ResolutionNilpotentGroup(G,5);; 
gap> TR:=TensorWithIntegers(R);; 
gap> Homology(TR,4); time; 
[ 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0 ] 
46521881 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><a
name="time"></a>Note
the time taken by GAP's Smith Normal Form
algorithm to compute the homology in the last example! The 2-torsion
can be detected more quickly using the universal coefficient exact
sequence for a trivial ZG-module A (where (×) denotes the tensor
product over Z). 
 
 <table
style="width: 80%; text-align: left; margin-left: auto; margin-right: auto;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td
style="text-align: center; vertical-align: top; background-color: rgb(204, 255, 255);"> 
1
→ Hn(G,Z) (×) A → Hn(G,A)
→ Tor1(Hn-1(G,Z),A) → 1 
 
 </td>
 </tr>
 </tbody>
 </table>
 
The third integral homology of the group G is quickly calculated and
seen to be torsion free. So H4(G,A) is isomorphic to H4(G,Z)
(×) A. The following commands now establish
2-torsion in the fourth integral homology. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
F:=FreeGroup(3);; 
gap> G:=NilpotentQuotient(F,3);; 
 
gap> R:=ResolutionNilpotentGroup(G,5);; 
gap> TRrational:=TensorWithRationals(R);; 
gap> TRmod2:=TensorWithIntegersModP(R,2);; 
 
gap> Homology(TRrational,4);time; 
171 
320702 
 
gap> Homology(TRmod2,4);time; 
174 
36705 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Some
of the
most interesting groups have no normal subgroup and so the above
techniques can't be applied to them. We need additional
techniques. 
 </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="aboutquasi.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="aboutArithmetic.html">Next
page 
 </a> </td>
 </tr>
 </tbody>
 </table>
 <a href="aboutTopology.html"> 
 </a> </td>
 </tr>
 </tbody>
</table>
 
 
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