Homology Iterated Groups Package - Readme

Victor Alvarez Solano, Jose Andres Armario Sampalo, Maria Dolores Frau Garcia and Pedro Real Jurado
Departamento de Matematica Aplicada I, University of Seville (Spain).
20 August 2006

valvarez@us.es

Determining the homology of a group at a desired dimension is in general a very difficult task, which is actually not feasible from a computational point of view. Taking into account the tools that the Homological Perturbation Theory provides, these calculations may be significantly improved with the use of homological models. This way, if a homological model hG for a group G is known, it suffices to compute the homology modules for hG instead of those for G,which is meaningfully simpler. This package provides a means of computing the homology of any iterated product of central extensions, semidirect products and direct products of finite abelian groups. Furthermore, explicit calculations of the maps involved in the definition of the homological model may be achieved with a slight modification of the notebook.

Loading the package HomologyIteratedGroups

The package `HomologyIteratedGroups` may be placed in folder:  "/Applications/Mathematica 5.2.app/AddOns/Applications"/.
I use the Get request and select initialization as a cell property such as:

<< HomologyIteratedGroups.m;

For questions or comments please email me at valvarez@us.es

Examples to use the package and possible tests

Let consider the iterated product (Z_t x_chi[2] Z_2) x_chi[1] Z_2, for the 2-cocycle chi[2]:Z_2 x Z_2 ---> Z_t given by chi[2][1,1]= Mod[1+ceiling(t/2),t] and 0 otherwise; and an the dihedral action chi[1]:Z_2 x (Z_t x_chi[2] Z_2)--->(Z_t x_chi[2]_2) given by chi[1][1,{a,g}]=opposite({a,g}) and {a,g} otherwise.

These maps should be pre-defined by the user, before calling the function HomologyIteratedGroup. For instance, for t=5, one could type

t = 5; chi[2][a_, g_] := 
      If[g*a == 1, Mod[Ceiling[t/2] + 1, t], 
        0];
chi[1][g_, {a_, b_}] := 
        If[g == 1, {Mod[-(a + chi[2][b, b]), t], b}, {0, 
            0}];

A tree-list representing the iterated group is given by {{2},{4,0}}. A list of cardinalities, from left to the right, is {5,2,2}. If we are interested in computing the homology groups from 0 to 10, we should type

HomologyIteratedGroups[{{2}, {4, 0}}, {5, 2, 2}, 0, chi]

and introduce 1 each time we are asked to going on computing the next homology group.