About HAP: Table Of Contents

HAP Home

- Introduction

- Algebraic definition of group cohomology

- Topological view of group cohomology

- Extensions and second cohomology

- Homotopy 2-types and third cohomology

- Resolutions for extensions of groups

- Functorial properties of cohomology

- Third integral homology of some superperfect groups

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

- Presentations and modules of identities for the nonabelian groups of order at most 30

- Aspherical presentations

- Third homotopy groups of suspensions of classifying spaces

- A relative Schur multiplier, Baer invariants and the capability of groups

- Resolutions for some infinite (Artin) groups

- Cup product and integral cohomology rings

- Mod p cohomology rings

- Poincare series for groups of order 32

- Poincare series for groups of order 64

- Tor and Ext of modules over a mod p group ring

- Cohomology with twisted coefficients

- Resolutions for graphs of groups, Fuchsian groups and Kleinian groups

- Piecing things together - a space group example

- Piecing things together - the Rosenberger monster

- Explicit cocycles

- Groups acting on polytopes

- Betti numbers for some orientable 7- dimensional Hantzsche-Wendt Manifolds (using HAPcryst)

- Fundamental domains for 3-dimensional flat manifolds (using HAPcryst)

- On a relationship between group homology and Lie algebra homology

- Lie covering homomorphisms

- Topological data analysis

- Parallel computation

- Programming homological diagrams in abelian categories

- Computational performance of HAP

Future work:

- Implement contracting homotopies for the function ResolutionArtinGroup() .
- Develop more functions for "ZG-resolutions with non-free G-action". Such a resolution R is similar to a free resolution except that it has 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) Coxeter groups G which act on the so-called Davis complex; (2) 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 Home