Theory of topological groupoids

This thesis takes up the notion of topological and differentiable categories and groupoids and of their local triviality. Beginning with definitions of an (algebraic) category with object class X, and appropriate commutative diagrams. These are extended easily to the topological and differentiable cases in later chapters. For brevity here, suppose X is a Hausdorff space, path-connected (p.c.), locally path-connected (l.p.c.), and locally simply connected (l.s.c.). As important examples we consider X (the set of all paths) as a topological category, and πX (the fundamental groupoid) as a topological groupoid, over X, also πX is a covering space of X × X, and π_l(πX,.) is computed. The relation between connected groupoids and fibre bundles is studied. We show that every connected locally trivial (l.t.) groupoid over X has a bundle structure over X × X, and for each x ∈ X, St_Gx is a principal bundle over X with group G {x}. Also, every connected l.t. groupoid with discrete vertex groups over X is shown to be isomorphic to a quotient groupoid of πX; and if π₁X is abelian, then π₁(G, o_x) ≈π₁(St_Gx , o_x) ⊕ π₁(X, x).

The notion of topological covering morphism is introduced; if p: X → Y is a covering map of Hausdorff spaces, then P_∗: πX → πY is a covering morphism of topological groupoids. In case G is a connected l.t. Hausdorff groupoid, ∃ a 1-1 correspondence between the closed subgroups of its vertex group and its covering groupoids. If G is a connected l.t. groupoid with discrete vertex groups over X, then the universal covering space G̃ of G is a groupoid over X̃, and in case G = πX, G̃ is the universal covering groupoid of G.

 

Finally, we study some examples of Lie categories and groupoids.

University of Southampton

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