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big><
span
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"font-weight: bold;">About HAP: Homology of simplicial groups<
br>
</
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">This
page illustrates functions written by <
span style=
"font-weight: bold;">Le
Van Luyen</
span> for computing the integral homology of simplicial
groups.<
br>
</
td>
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<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">To
obtain an interesting example of a simplicial group let us consider the
dihedral group G of order 16, and the canonical homomorphism µ:G
--> Aut(G) from G to its automorphism group which sends an element g
to the inner automorphism x--> gxg<
sup>-1</
sup> . The homomorphism
µ is an example of a <
span style=
"font-style: italic;">crossed
module</
span> and, as such, is equivalent to a cat-1-group C. The
following commands construct this cat-1-group and show that C has order
512, has fundamental group C<
sub>2</
sub>xC<
sub>2</
sub> and second
homotopy group C<
sub>2</
sub>.<
br>
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<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
C:=AutomorphismGroupAsCatOneGroup(DihedralGroup(16));<
br>
Cat-1-group with underlying group Group( [ f1, f2, f3, f4, f5, f6, f7,
f8, f9<
br>
] ) .<
br>
<
br>
gap> Size(C);<
br>
512<
br>
<
br>
gap> Pi_1:=HomotopyGroup(C,1);;<
br>
gap> Pi_2:=HomotopyGroup(C,2);;<
br>
gap> StructureDescription(Pi_1);<
br>
"C2 x C2"<
br>
gap> StructureDescription(Pi_2);<
br>
"C2"<
br>
</
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<
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">A
morphism C->Q of cat-1-groups is called a <
span
style=
"font-style: italic;">quasi-isomorphism</
span> if it induces
isomorphisms of homotopy groups. Quasi-isomorphic cat-1-groups have the
same homology and cohomology.<
br>
<
br>
The following commands construct a quasi-isomorphic cat-1-group Q of
order 32.<
br>
</
td>
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Q:=QuasiIsomorph(C);<
br>
Cat-1-group with underlying group Group( [ f5, f9, f1, f2*f3, f8 ] ) .<
br>
<
br>
gap> Size(Q);<
br>
32<
br>
</
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tr>
<
td
style=
"background-color: rgb(255, 255, 255); vertical-align: top;">Now
a cat-1-group Q can be regarded as a category endowed with a
"compatible" group operation (i.e. it is a category
object in the
category of groups). The classical nerve of the category Q will be a
simplicial group. The following
command constructs the nerve of Q (up
to degree 6).<
br>
</
td>
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tr>
<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
N:=NerveOfCatOneGroup(Q,6);<
br>
Simplicial group of length 6<
br>
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style=
"vertical-align: top; background-color: rgb(255, 255, 255);">
<
div style=
"text-align: left;">By taking the nerve of each group N<
sub>n</
sub>
in the simplicial group N we obtain a bisimplicial set. The diagonal of
this bisimplicial set is a simplicial set which we denote by D. By
taking the free abelian group FD<
sub>n</
sub> on each set in D we obtain
a simplicial free abelian group FD. Let Chn(N) denote the standard
chain complex of this simplicial abelian group FD. (The chain
groups Chn(N)<
sub>n</
sub> are just the free abelian groups FD<
sub>n</
sub>;
the boundary homomorphisms in Chn(N) are got by taking alternating sums
of the homomorphisms in FD.)<
br>
</
div>
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<
td style=
"vertical-align: top;"><
span
style=
"font-weight: bold;">THEOREM</
span><
br>
<
div style=
"text-align: left;"><
br>
</
div>
The homology groups of the chain complex Chn(N) are isomorphic to the
homology groups of a connected CW-space X whose homotopy type is
represented by the simplicial group N. <
br>
</
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table>
<
div style=
"text-align: left;"><
br>
The chain complex Chn(N) is typically enormous. The following
command
(due to <
span style=
"font-weight: bold;">Le Van Luyen</
span>) returns
a vastly smaller chain complex K which is chain homotopic to Chn(N).
(The smaller chain complex is constructed using homological
perturbation theory.) <
br>
</
div>
</
td>
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tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
K:=ChainComplexOfSimplicialGroup(N);<
br>
Chain complex of length 6 in characteristic 0 .<
br>
</
td>
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<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands now yield, by definition, the low-dimensional
integral homology groups of our original crossed module µ .<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Homology(K,1);<
br>
[ 2, 2 ]<
br>
gap> Homology(K,2);<
br>
[ 2 ]<
br>
gap> Homology(K,3);<
br>
[ 2, 2, 2 ]<
br>
gap> Homology(K,4);<
br>
[ 2, 2, 2 ]<
br>
gap> Homology(K,5);<
br>
[ 2, 2, 2, 2, 2, 2 ]<
br>
</
td>
</
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<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">Other
examples of cat-1-groups and their nerves can be obtained from the XMOD
package for GAP. This package contains a library of all cat-1-groups
with underlying group C of
small order. For instance, the following
commands construct the nerve of a cat-1-group whose underlying
groups C has order 8. <
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
LoadPackage(
"xmod");;<
br>
------------------------------------------------------------<
br>
loading XMod 2.008 for GAP 4.4 - Murat Alp and Chris Wensley<
br>
------------------------------------------------------------<
br>
<
br>
gap> C:=Cat1Select(8,5,3);<
br>
[c2^3=>c2]<
br>
<
br>
gap> N:=NerveOfCatOneGroup(C,4);<
br>
Simplicial group of length 4<
br>
</
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<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">Other
examples of simplicial groups are the Eilenberg-MacLane simplicial
groups K(G,n) with (n-1)-st homotopy group equal to G and all other
homotopy groups equal to 0. The group G must be abelian when n>0. <
br>
<
br>
The following computes the first eleven terms of the chain complex CK
of the Eilenberg-MacLane space K(Z,2) and shows that in dimension d the
number of free generators of this complex is the d-th Fibonacci number.
<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
ZZ:=AbelianPcpGroup(1,[0]);; #ZZ is the infinite cyclic group<
br>
gap> K:=EilenbergMacLaneSimplicialGroup(ZZ,2,10);;<
br>
gap> CK:=ChainComplexOfSimplicialGroup(K);;<
br>
<
br>
gap> List([0..10],CK!.dimension);<
br>
[ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ]<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following produces a chain complex D which is quasi-isomorphic to CK
and which has just one generator in each odd dimension. (This
corresponds to the standard CW-structure on complex projective
space
involving just one cell in each even dimension.) <
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
D:=CoreducedChainComplex(CK);;<
br>
gap> List([0..9],D!.dimension);<
br>
[ 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 ]<
br>
</
td>
</
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<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following analogous commands produce a chain complex for K(Z/2Z,2) in
which the number of free generators in degree d+2 is the d-th Fibonacci
number.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
K:=EilenbergMacLaneSimplicialGroup(CyclicGroup(2),2,10);;<
br>
gap> CK:=ChainComplexOfSimplicialGroup(K);;<
br>
gap> List([0..10],CK!.dimension);<
br>
[ 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 ]<
br>
gap> D:=CoreducedChainComplex(CK);;<
br>
gap> List([0..9],D!.dimension);<
br>
[ 1, 0, 1, 1, 2, 3, 5, 8, 13, 21 ]<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">Many
homotopy 2-types are direct products of Eilenberg-MacLane spaces. For
instance, if a 2-type is represented by a cat-1-group whose underlying
group is abelian, then the 2-type is a product of Eilenberg-MacLane
spaces.<
br>
<
br>
For a non-ableian example let us consider the automorphism cat-1-group
C of the 5-
th small group of order 36. By definition the fundamental
group of C acts trivially on the second homotopy group. Moreover, the
following commands show that the order of C is equal to the product of
the orders of its homotopy groups. This implies that C is a direct
product of a K(G,1) and a K(A,2).<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SmallGroup(36,5);;<
br>
gap> C:=AutomorphismGroupAsCatOneGroup(G);;<
br>
gap> Order(HomotopyGroup(C,1))*;Order(HomotopyGroup(C,2)) = Size(C);<
br>
true<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands imply that C is quasi-isomorphic to the product of
Eilenberg-MacLane spaces K(Z/2Z,2) x K(Z/2Z,2) x K(Z/9Z,2) x K(C<
sub>6</
sub>,1)
x
K(S<
sub>3</
sub>,1) .<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
StructureDescription(HomotopyGroup(C,2));<
br>
"C18 x C2"<
br>
gap> StructureDescription(HomotopyGroup(C,1));<
br>
"C6 x S3"<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands compute the homology of the cat-1-group C in degree
4.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
C1:=TensorWithIntegers(ResolutionFiniteGroup(SymmetricGroup(3),5));;<
br>
gap>
C2:=TensorWithIntegers(ResolutionFiniteGroup(CyclicGroup(6),5));;<
br>
gap> G3:=EilenbergMacLaneSimplicialGroup(CyclicGroup(9),2,5);;<
br>
gap> C3:=ChainComplexOfSimplicialGroup(G3);;<
br>
gap> G4:=EilenbergMacLaneSimplicialGroup(CyclicGroup(2),2,5);;<
br>
gap> C4:=ChainComplexOfSimplicialGroup(G3);;<
br>
gap> C5:=C4;;<
br>
<
br>
gap> D1:=CoreducedChainComplex(C1);;<
br>
gap> D2:=CoreducedChainComplex(C2);;<
br>
gap> D3:=CoreducedChainComplex(C3);;<
br>
gap> D4:=CoreducedChainComplex(C4);;<
br>
gap> D5:=D4;;<
br>
<
br>
gap> T:=TensorProductOfChainComplexes(D1,D2);;<
br>
gap> T:=TensorProductOfChainComplexes(D3,T);;<
br>
gap> T:=TensorProductOfChainComplexes(D4,T);;<
br>
gap> T:=TensorProductOfChainComplexes(D5,T);;<
br>
<
br>
<
br>
gap> Homology(T,4);<
br>
[ 3, 3, 3, 3, 9, 9, 9, 9, 18, 18 ]<
br>
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