Singular Chains on Topological Stacks

Abstract

The main objective of this thesis is to introduce the concept of `singular chains on topological stacks'. The idea is to functorially associate to a topological stack, a simplicial set which captures its homotopy type. This will allow us to compute the singular homology and cohomology of topological stacks. Noohi and Behrend have given several approaches to this problem, however all of these approaches rely on the choice of an atlas for a topological stack. We shall show that our new approach agrees with the existing approaches but has the advantage of being functorial. Noohi has introduced weak equivalences and brations of topological stacks. In analogy to the singular chains functor for topological spaces, we shall show that the functor Sing preserves the weak equivalences and brations de ned by Noohi under certain ` brancy conditions'. In the second part, we shall push the analogy with the topological singular chains further by considering the adjunction with the geometric realization and the associated counit. We develop a corresponding (but weaker) notion for topological stacks. We shall give a method for computing the homotopy type of a stack which has a groupoid presentation. Finally, we shall compute the homotopy type of certain mapping stacks and develop the totalization of a cosimplicial topological stack. We shall indicate how this (using the approach of Cohen and Jones) gives a method for computing the string topology of a topological stack.