# Algebraic topology

### Calculations with simplicial and cubical groups in AXIOM

#### Abstract:

Work on calculations with simplicial and cubical groups in AXIOM was carried out using loan equipment and software from IBM UJ and guidance from L. Lambe. We report on the results of this work, and present the AXIOM code written by the second author during this period. This includes an implementation of the monoids which model cubes and simplices, together with a new AXIOM category of near-rings with which to carry out non-abelian calculations. Examples of the use of this code in interactive AXIOM sessions are also given.

#### Published in:

J. Symbolic Comp., 17 (1994) 159-179.

### Covering groups of non-connected topological groups revisited

#### Abstract:

The purpose of this paper is to relate Taylor's result on the obstruction class of a topological group to modern work on coverings of groupoids and the equivalence between group-groupoids and crossed modules.
This leads to a clear relation of the theory with the classical theory of abstract kernels and with the theory of extensions of the type of a crossed module.
The results of this paper form Part 1 of Mucuk's Ph.D. thesis.

### The monodromy groupoid of a Lie groupoid

#### Abstract:

We show that under general circumstances, the disjoint union of the universal covers of the stars of a Lie groupoid admits the structure of a Lie groupoid, such that the projection has a monodromy property on the extension of local smooth morphisms. This completes a detailed account of results announced by J Pradines.

#### Published in:

Cahiers de Topologie Geometrie Differentielle categoriques 36 (1995) 345-369.

monod5.ps.gz

### Foliations, locally Lie groupoids, and holonomy

#### Abstract:

We show that a paracompact foliated manifold determines a locally Lie groupoid (or piece of a differentiable groupoid, in the sense of Pradines). This allows for the construction of holonomy and monodromy groupoids of a foliation to be seen as particular cases of constructions for locally Lie groupoids.

#### Published in:

Cahiers de Topologie Geometrie Differentielle categoriques, 37 (1996) 61-71.

fol-lie7.ps.gz

### Theory and applications of crossed complexes

#### Abstract:

We prove a slightly non-abelian' version of the classical Eilenberg-Zilber theorem: if K,L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K x L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various side conditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K_0,...,K_r, we discuss the r-cube of homotopies induced on \pi(K_0 x ... x K_r) and show these form a coherent system.
We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the total complex of a certain twisted' simplicial crossed complex, analagous to Bousfield and Kan's definition of simplicial homotopy colimits as the diagonal of a certain bisimplicial set. Using the Eilenberg-Zilber theorem we show that the fundamental crossed complex functor preserves these homotopy colimits up to a strong deformation retraction. This is applied to give a small crossed resolution of a semidirect product of groups.
We consider a simplicial enrichment of the category of crossed complexes, and investigate the coherent homotopy structure up to which a simplicial enrichment may be given to the fundamental crossed complex functor.
We end with a definition of homotopy coherent functors from a small category to the category of crossed complexes, and suggest a definition of homotopy colimits of such functors and of a small crossed resolution of an arbitrary group extension.

#### Published in:

University of Wales, Bangor, Ph.D. Thesis, 1994.

tonks.ps.gz

### Representation and computation for crossed modules

#### Abstract:

This paper discusses the notion of structural computation'', and illustrates it with the problem of translating a notion of tensor product between several equivalent categories, in this case crossed modules, cat-1-groups, double groupoids with connection, 2-groupoids. In this case, the easy definition of tensor product is for double groupoids with connection.

#### Published in:

Proceedings Cat\'egories, Algebres, Esquisses, Neo-esquisses, Caen, 1994.

caen.ps.gz

### Higher order symmetry of graphs

#### Abstract:

This paper advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. The basic idea is that the symmetry object' of a directed graph should be both a group and a directed graph. From this is obtained the notion of inner automorphism' of a directed graph. The work of J. Shrimpton completely describes these.

#### Published in:

Bulletin Irish Math. Soc., 32 (1994) 46-59.

irishcon.ps.gz