# Algebraic Topology

### Graphs of Groups : Word computations and free crossed resolutions

#### Summary:

We give an account of graphs of objects and total objects, where the objects are groups, groupoids, spaces and free crossed resolutions respectively.
Graphs of groups were used by Higgins who defines the fundamental groupoid of a graph of groups and gives a normal form theorem. We give full details of this construction and illustrative examples.
The new work generalises the notion of graphs of groups to graphs of groupoids, defines the fundamental groupoid of a graph of groupoids and gives a normal form theorem.
We also implement the structure of graphs of groups and graphs of groupoids as the first two parts of a share package $\mathsf{XRes}$ in $\mathsf{GAP4}$ to obtain normal forms computationally.
Scott and Wall generalise graphs of groups to graphs of spaces and define a total space of a graph of spaces. We define analogous new constructions - the total groupoid of a graph of groups and the total crossed complex of a graph of free crossed resolutions. The total groupoid is isomorphic to the fundamental groupoid of a graph of groups.
We also use a result of Scott and Wall on the asphericity of total spaces together with realisations of crossed complexes to give a result on the asphericity of total crossed complexes.
We construct graphs of free crossed resolutions over graphs of groups to give free crossed resolutions of total groupoids. We conclude this work with applications of the total crossed complex of graphs of free crossed resolutions.
We can use these free crossed resolutions to determine identities among relations and higher syzygies of finitely presented groups, obtained as vertex groups of total groupoids. We also extend presentations of free products with amalgamation and HNN-extensions obtained from reformulating the van Kampen theorem in terms of group presentations to give generatings set for modules of identities among relations. We also give non-abelian extensions of groups using morphisms of free crossed resolutions to automorphism crossed modules. We conclude by relating our work to picture methods which are used to determine identities among relations.

#### Published in:

U W Bangor PhD thesis (July 2001)

• gzipped postscript of the thesis: moore.ps.gz
• gzipped postscript file of the appendix: xres.ps.gz -
manual for the GAP crossed resolutions share package XRES.

### A homotopy double groupoid of a Hausdorff space

#### Abstract:

We associate to a Hausdorff space X a double groupoid \rho_2^{\square}(X), the homotopy double groupoid of X. The construction is based on the geometric notion of thin square. Under the equivalence of categories between small 2-categories and double categories with connection given in Brown & Mosa (TAC 5 (1999) 163-175), the homotopy double groupoid corresponds to the homotopy 2-groupoid, G_2(X), constructed in Hardie, Kamps & Kieboom (Appl. Cat. Structures 8 (2000) 209-234). The cubical nature of \rho_2^{\square}(X) as opposed to the globular nature of G_2(X) should provide a convenient tool when handling `local-to-global' problems as encountered in a generalised van Kampen theorem and dealing with tensor products and enrichments of the category of compactly generated Hausdorff spaces.

#### Published in:

Theory and Applications of Categories 10 (2002) 71-93.

### Holonomy and monodromy groupoids

#### Abstract:

We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.

#### Published in:

Lie Algebroids, Banach Center Publications, Vol.54,
Institute of Mathematics, Polish Academy of Sciences, Warszawa (2001) 9-20.