The homology of groupnets

Abstract

In this thesis I use the theory of groupnets (Brandt groupoids) to investigate the homology of mapping cylinder groupnets; that is, groupnets G which are the homotopy colimits of diagrams (V, A) of groupnets. When the edge morphisms of (V, A) are all monomorphisms, G is known as a graph product. The principal result of the thesis is the construction of a G-complex with universal properties - the G-mapping cylinder - from a diagram of complexes corresponding to (V, A) , and the subsequent proof that if G is a graph product and the vertex complexes are all free resolutions of their respective trivial modules, then the G-mapping cylinder is a free resolution of its trivial module . An extension of the categorical approach to rings and modules is developed in order to provide the general result. The notion of chain homotopy is also extended to a form strongly motivated by the topological definition of homotopy. The mapping cylinder complex determines MayerVietoris sequences for the homology of graph products, which in turn may be used to extend several results on duality groups. For each group in a certain class of groupnets with cohomological dimension two (including torsion-free one-relater groups and tree products of free groups ), the mapping cylinder may be employed to evaluate a comultiplication which gives a coring structure to the integral homology module of the group. This comultiplication is in turn analysed (though not in full generality) to provide further information about the group.