About HAP: Table Of Contents

- Introduction

- Algebraic definition of group cohomology

- Topological view of group cohomology

- Resolutions for extensions of groups

- Functorial properties of cohomology

- Second homotopy groups of presentations and a periodic resolution for a group not acting on a sphere

- Aspherical presentations

- Third homotopy groups of suspensions of classifying spaces

- A relative Schur multiplier and the capability of groups

- Resolutions for some infinite (Artin) groups

- Cup product and integral cohomology rings

- Cohomology with twisted coefficients

- Piecing things together - a space group example (work in progress)

- Explicit cocycles

- Groups acting on polytopes

- Computational performance of HAP

Future work:

- Implement contracting homotopies for the functions: ResolutionArtinGroup(), TwistedTensorProduct() and ResolutionPrimePowerGroup().
- Implement functions for the abelian extensions of groups which are classified by second cohomology. It might also be worth implementing Mac Lanes's third cohomology obstruction to the existence of nonabelian extensions.
- Introduce the data type "ZG-resolution with non-free G-action". Such a resolution R will be similar to a free resolution except that it will have a component R.stabilizer(e) which returns the stabilizer subgroup Ge in G of each generator e of R. We need to implement a function MakeFree(R) which, using the non-free resolution R and free resolutions Re for each stabilizer group, constructs a free ZG-resolution. The algorithm is explained in [G. Ellis, J. Harris & E. Sköldberg, "Polytopal resolutions for finite groups", J. Reine Angewandte Math., to appear].
- Apply the function MakeFree(R) in the construction of free ZG-resolutions for: (1) groups G acting on trees (such as amalgamated products, HNN extensions, graph products); (2) Coxeter groups G which act on the so-called Davis complex; (3) some infinite generalised triangle groups G which act on hyperbolic 3-space; finite groups G acting faithfully on Euclidean space (this will use Polymake software).

HAP main page