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Mathematics Preprints 1995

Algebraic Topology

95.03 : R. Brown & C.D. Wensley

On finite induced crossed modules, and the homotopy 2-type of mapping cones


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.

Published in:

Theory and Applications of Categories, 1(3) (1995) 54-71.

95.04 : R. Brown & C.D. Wensley

Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types


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.

Published in:

Theory and Applications of Categories, 2(1) (1996) 3-16.

95.08 : R. Brown, M. Golasinski, T. Porter, & A. Tonks

Spaces of maps into classifying spaces for equivariant crossed complexes


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.

AMS Subject Classification:

55P91, 55U10, 55U35

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Published in:

Indag. Math. 8 (1997) 157-172.

95.17 : BROWN, R. & PORTER, T.

On the Schreier theory of non abelian extensions: generalisations and calculations


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.

95.34 : BROWN, R. & DRECKMANN, W.

Domains of data and domains of terms in AXIOM


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.

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