On finite induced crossed modules,
and the homotopy 2-type of mapping cones
Abstract:
Results on the finiteness of induced crossed modules are proved
both algebraically and topologically. Using the Van Kampen type
theorem for the fundamental crossed module, applications are given
to the 2-types of mapping cones of classifying spaces of groups.
Calculations of the cohomology classes of some finite crossed
modules are given, using crossed complex methods.
Computing crossed modules induced by an inclusion of a normal subgroup,
with applications to homotopy 2-types
Abstract:
We obtain some explicit calculations of crossed Q-modules
induced from a crossed module over a normal subgroup P of Q.
By virtue of theorems of Brown and Higgins, this enables the
computation of the homotopy 2-types and second homotopy
modules of certain homotopy pushouts of maps of classifying
spaces of discrete groups.
95.08 : R. Brown, M. Golasinski, T. Porter, & A. Tonks
Spaces of maps into classifying spaces
for equivariant crossed complexes
Abstract:
We give an equivariant version of the homotopy
theory of crossed complexes.
The applications generalise work on equivariant Ellenberg-MacLane spaces,
including the non-abelian case of dimension 1, and on local systems.
It also generalises the theory of equivariant 2-types,
due to Moerdyk and Svensson.
Further we give results not just on the homotopy classification of maps
but also on the homotopy types of certain equivariant function spaces.
On the Schreier theory of non abelian extensions:
generalisations and calculations
Abstract:
We use presentations and identities among relations to give a generalisation
of the Schreier theory of nonabelian extensions of groups. This replaces
the usual multiplication table for the extension group by more efficient,
and often geometric, data. The methods utilise crossed modules and crossed
resolutions. This work is related to work of Turing in 1938.
Published in:
Proceedings Royal Irish Acad., 96A (1996) 213-227.
This paper discusses the advantages of the AXIOM symbolic computation system,
and illustrates them with some AXIOM2.0 code for directed graphs and free
categories and groupoids on directed graphs. In order to implement the latter,
we have to make a distinction between domains of data and domains of terms,
where, for example, the first gives the data for a finite directed graph,
whereas the latter converts this data into an object of Axiom category
DirectedGraphCategory, where the terms range over the objects and arrows
of the directed graph.
