Homotopy Types of Topological Groupoids and Lusternik-Schnirelmann Category of Topological Stacks

The concept of a groupoid was first introduced in 1926 by H. Brandt in his fundamental paper [7]. The idea behind it is a small category in which every arrow is invertible. This notion of groupoid can be thought of as a generalisation of the notion of a group. Namely, a group is a groupoid with only one object. After the introduction of topological and differentiable groupoids by Ehresmann in 1950 in his paper on connections [19], the concept has been widely studied by many mathematicians in many areas of topology, geometry and physics. In this thesis, we deal with topological groupoids as the main object of study. We first develop the main concepts of homotopy theory of topological groupoids. Also, we study general versions of Morita equivalence between topological groupoids, which lead us to discuss topological stacks. The main objective of this thesis is then to develop and analyse a notion of Lusternik-Schnirelmann category for general topological groupoids and topological stacks, generalising the classical work by Lusternik and Schnirelmann for topological spaces and manifolds [30] and for orbifolds and Lie groupoids as introduced by Colman [11]. Fundamental in the classical definition of the LS-category of a smooth manifold or topological space is the concept of a categorical set. A subset of a space is said to be categorical if it is contractible in the space. The Lusternik-Schnirelmann category cat(X) of a topological space X is defined to be the least number of categorical open sets required to cover X, if that number is finite. Otherwise the category cat(X) is said to be infinite. Here using a generalised notion of categorical subgroupoid and substack, we generalise the notion of the Lusternik-Schnirelmann category to topological groupoids and topological stacks with the intention of providing a new useful tool and invariant to study homotopy types of topological groupoids and topological stacks, which will be important also to understand the geometry and Morse theory of Lie groupoids and differentiable stacks from a purely homotopical viewpoint.