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<td style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span style="font-weight: bold;">About HAP: Torsion Subcomplexes<br>
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<td style="vertical-align: top; background-color: rgb(255, 255, 255);"><big style="font-weight: bold;">Torsion Subcomplexes <br>
</big> Sub-package by Alexander D. Rahm and Bui Anh Tuan, version
2.1 </td>
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<td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">Consider
a cell complex with a cellular action of a discrete group G on it, and
consider a prime number p. The goal for the usage of this subpackage is
to compute the homological p-torsion of G, by which we mean the modulo
p homology of G (i.e with non-twisted Z/pZ coefficients) in degrees
above the virtual cohomological dimension, or the modulo p Farrell-Tate
cohomology of G.
<br>
For the computation of the homological p-torsion of G, only the p-<i>torsion
subcomplex</i> is relevant, consisting of all the cells the stabilizers
in G of which contain elements of order p.
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<td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">For
instance, let us input Soulé's cell complex for SL_3(Z). <br>
<divstyle="text-align: center;"><imgstyle="height: 323px;"
alt="" src="AboutTorsionSubcomplexes_files/truncatedCube.jpg"><br>
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<td style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
S:= ContractibleGcomplex("SL3Z"); <br>
Non-free resolution in characteristic 0 for matrix group with
65 generators. No contracting homotopy available.
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<td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">Rigid
Facets Subdivision allows us to recover (essentially) Soulé's
subdivision of the above truncated cube, which is a fundamental domain
for a cell complex for SL_3(Z) such that each cell stabilizer fixes its
cell pointwise. <br>
<divstyle="text-align: center;"><imgstyle="height: 323px;"
alt="" src="AboutTorsionSubcomplexes_files/subdividedCube.jpg"><br>
</div>
[The above two pictures are shown here with the kind permission of
Ruben Sanchez-Garcia, who has reconstructed them from Soulé's paper.]
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<td style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
R := RigidFacetsSubdivision(S); <br>
Non-free resolution in characteristic 0 for matrix group with
65 generators.
No contracting homotopy available. <br>
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<td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">Now
that the cell stabilizers are "small" enough, it becomes useful to
extract the 2-torsion subcomplex. <br>
<divstyle="text-align: center;"><imgstyle="width: 523px;" alt=""
src="AboutTorsionSubcomplexes_files/SL3Z.jpg"><br>
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<td style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
TorsionSubcomplex(R,2); <br>
<br>
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<td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">To
this 2-torsion subcomplex, we can apply the torsion subcomplexes
reduction technique.
<br>
In fact, every time that two adjacent edges and their joining vertex
satisfy the following conditions on their stabilizers, we can merge
them without changing the equivariant modulo p Farrell homology of the
p-torsion subcomplex [see the paper <a
href="http://hal.archives-ouvertes.fr/hal-00618167">"Accessing the
Farrell-Tate cohomology of discrete groups"</a> on how torsion
subcomplex reduction works in detail].
<br>
One of the sufficient conditions reads as follows.
Let G_1 and G_2 be the stabilizers of the two adjacent edges,
and let S be the stabilizer of their joining vertex.
Then we require G_1 and G_2 to be isomorphic and <br>
either G_1 to be isomorphic to S <br>
or S to be p-normal and G_1 to be isomorphic to the normaliser in S of
the center of a Sylow-p-subgroup of S.
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<td style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
ReduceTorsionSubcomplex(R,2); <br>
<br>
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<td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">Then
we obtain the reduced system of stabilizer inclusion displayed in
Soulé's paper. <br>
<divstyle="text-align: center;"><imgstyle="width: 523px;" alt=""
src="AboutTorsionSubcomplexes_files/reduced2torsionsubcomplex.jpg"><br>
</div>
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<td style="background-color: rgb(255, 255, 204); vertical-align: top;">Download
of the Torsion Subcomplexes Subpackage at:
<a
href="http://math.uni.lu/%7Erahm/subpackage-documentation/TorsionSubcomplexesSubpackage.tar.gz">http://math.uni.lu/~rahm/subpackage-documentation/
TorsionSubcomplexesSubpackage.tar.gz</a> <br>
or
<br>
<a
href="http://math.uni.lu/%7Erahm/subpackage-documentation/TorsionSubcomplexesSubpackage.zip">http://math.uni.lu/~rahm/subpackage-documentation/
TorsionSubcomplexesSubpackage.zip</a> <br>
<br>
Documentation of the functions in the Torsion Subcomplexes Subpackage
at:
<a href="http://hamilton.nuigalway.ie/Hap/doc/chap27.html">http://hamilton.nuigalway.ie/Hap/doc/chap27.html</a>
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