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 <td style="vertical-align: top;"><a href="aboutIntro.html">Previous</a> 
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Some Definitions</big></td>
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href="aboutTopology.html">next</a> 
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 <big> </big> 
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style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">The
cohomology of a group G can be defined in terms of resolutions. Let Z
be the group of integers considered as a trivial ZG-module. A free ZG-resolution of Z is a
sequence of ZG-module homomorphisms 
 
 <div style="text-align: center;">... → Mn → Mn-1
→ ... → M1 → M0 
 
 <div style="text-align: left;">satisfying: 
 <ul>
 <li>(Freeness) Each Mn is a free ZG-module. 
 </li>
 <li>(Exactness) The image of Mn+1 → Mn equals
the kernel of Mn → Mn-1 for all n>0. 
 </li>
 <li>(Augmentation) The cokernel of M1 → M0
is isomorphic to the trivial ZG-module Z.</li>
 </ul>
The maps Mn → Mn-1 are referred to as boundary homomorphisms. Setting TMn
 equal
to the abelian group Mn/G obtained from Mn by
killing the G-action, we get an induced sequence of abelian group
homomorphisms 
 
 <div style="text-align: center;">... → TMn → TMn-1
→ ... → TM1 → TM0 . 
 
 <div style="text-align: left;">This sequence will generally not
satisfy the above exactness condition, and one defines the integral homology of
G to be 
 
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 Hn(G,Z) = Kernel(TMn → TMn-1)
/ Image(TMn+1 → TMn) 
 
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 <div style="text-align: center;">
 <div style="text-align: left;">for all n>0. By changing the
definition of TMn
one arrives at the definition of homology Hn(G,A) and
cohomology Hn(G,A) with coefficients in a ZG-module A.
Needless to say, homology and cohomology are invariants of G, and do
not depend on the particular choice of free ZG-resolution. 
 
(See David Joyners <a href="http://arxiv.org/abs/0706.0549">intoduction</a>
to group cohomology for more details on this definition.) 
 </div>
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">There
are two steps to computing group
homology: 
 <ul>
 <li>Construct a free ZG-resolution R = {Mn}.</li>
 <li>Calculate the homology from TR = {TMn} using
some
version of the Smith Normal Form algorithm.</li>
 </ul>
For example, the 25th integral homology of the Dihedral group D512
 of order 1024 can be calculated using the following
commands. 
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style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
F:=FreeGroup(2);;x:=F.1;;y:=F.2;; 
gap> G:=F/[x^2,y^512,(x*y)^2];;
D_512:=Image(IsomorphismPermGroup(G));; 
 
gap> R:=ResolutionFiniteGroup(D_512,26); 
Resolution of length 26 in characteristic 0 for <permutation group
of size 1024 with 2 generators> . 
 
gap> time; #The time, in milliseconds, for constructing R. 
265262 
 
gap> TR:=TensorWithIntegers(R);; 
Chain complex of length 26 in characteristic 0 . 
 
gap> Homology(TR,25); 
[ 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;">We
see that H25(D512,Z) = (Z2)14.
(Quicker methods for computing this are given on subsequent pages!) 
 
Homology with other coefficients can also be calculated from the
resolution R. For
instance, the following additional command shows that the 25th
homology of D512 with coefficients in the trivial module Z2
is the vector space H25(D512,Z2) = (Z2)26.
 
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 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
T2R:=TensorWithIntegersModP(R,2);; 
Chain complex of length 26 in characteristic 2 . 
 
gap> Homology(T2R,25); 
26 
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 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">(At
this point we should note that there exist more efficient methods for
computing the homology of a finite group G over the finite field Zp
with trivial action. 
 
One approach, which has been used very succesfully by Jon Carlson on
small p-groups, is to regard the group ring ZpG as a vector
space of rank |G|. Each term in a ZpG-resolution R is then
also a vector space, and one can use linear algebra techniques to
construct R. Details of Carlson's computations can be found
href="http://www.math.uga.edu/%7Elvalero/cohointro.html">here</a>. 
 
The HAP function ResolutionPrimePowerGroup(G,n)
uses this idea to compute a free ZpG-resolution
for a p-group G. The resolution is minimal in the sense that the number
of free generators in degree n is equal to the rank of the vector space
Hn(G,Zp) . For example, the following commands
compute the ranks of the first 25 mod 2 homology groups of the Sylow
2-subgroup of the Mathieu group M12 . 
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style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SmallGroup(64,134);; #This is the Sylow 2-subgroup
of MathieuGroup(12). 
gap> R:=ResolutionPrimePowerGroup(G,25);; time; 
436107 
 
gap> #The dimensions of the first twenty-five mod 2 homology groups
of G
are: 
gap> List([0..25],n->Dimension(R)(n)); 
[ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153,
171, 
 190, 210, 231, 253, 276, 300, 325, 351 ] 
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
linear algebra approach to mod p cohomology will not work so well on
large
groups G since one ends up having to compute null-spaces of large
matrices. Consider for instance the above computation for G=D512.
Any resolution is going to have at least 26 Z2G-generators
in dimension 25, and 25 generators in dimension 24. So to obtain the
26th term one would have
to compute the nullspace of a matrix with (1024)2×26×25
= 681574400 entries. 
 
An efficient method for mod p cohomology of larger groups has been
developed by David Green. It involves non-abelian
Gröbner basis techniques and is described in the book [D.J.
Green, Gröbner bases and the
computation of group cohomology, Lecture Notes in Math., No.
1828 (Springer, 2003)] and on the corresponding <a
style="color: rgb(0, 0, 102);"
href="http://www.math.uni-wuppertal.de/%7Egreen/Coho/index.html">web
page</a>.) 
 </td>
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 <tr>
 <td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">For
integral (or mod p) computations to succeed the ZG-rank of the
resolution R
should not
be too large in any given dimension. The construction of small
ZG-resolutions is an interesting problem and at first glance might seem
to
be a purely algebraic one. However, we can benefit from the advice
of Sir Michael Atiyah. 
 
 
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style="background-color: rgb(204, 255, 255); vertical-align: top;"> 
 <div style="text-align: left;">
 <div style="margin-left: 40px; color: rgb(0, 0, 102);">"Fundamentally
the purpose of
algebra always was to produce a formula
which one could put into a machine, turn a handle and get an answer.
You took something that had a meaning; you converted it into a formula,
and you got the answer. In this process you do not
need to think any more about what
the different stages in the algebra correspond to in the geometry. You
lose the insights, and this can be important at different
stages.
You must not give up the insight altogether! You might want to come
back
to it later on."

 
 M. Atiyah, "Mathematics in the 20th century",
Bulletin London Math. Soc. 34 (2002), 1-15. 
 </div>
 
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