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.
gzipped postscript file of the appendix:
xres.ps.gz -
manual for the GAP crossed resolutions share package XRES.
01.21 :
BROWN, R., HARDIE, K.A., KAMPS, K.H. & PORTER, T.
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.
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.