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).
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