The differentiable stack cohomology associated to a regular and proper Lie groupoid

Abstract

This thesis establishes a model for the differentiable stack cohomology associated to a regular and proper Lie groupoid. <br/> The 2-category of differentiable stacks is equivalent to a 2-category of Lie groupoids, which means that definitions such as cohomology on the former can be transferred to a corresponding notion on the latter. This establishes the concept of the differentiable stack cohomology associated to a Lie groupoid, which has a well-known interpretation as a generalisation of equivariant cohomology. The search for models computing this differentiable stack cohomology can therefore be motivated by pre-existing results about similar objects for equivariant cohomology. Generalising their constructions from the setting of equivariant cohomology to describe a model for the differentiable stack cohomology associated to a general Lie groupoid has been an ongoing effort. A key object in previous research has been the Bott-Shulman-Stasheff bicomplex, which is an established non-infinitesimal model computing the differentiable stack cohomology associated to a Lie groupoid. <br/> In this thesis, it will be shown that in the case of a proper and regular groupoid, the cohomology of the columns of the Bott-Shulman-Stasheff bicomplex can be computed explicitly. For this, a notion of invariance is introduced through a construction using the first jet groupoid and the concept of a multiplicative Ehresmann connection with respect to the isotropy Lie algebroid. The first page of the spectral sequence associated to the induced filtration is computed by analysing the de Rham differential. Furthermore, it is proven that the associated spectral sequence collapses at the second page. It then can be concluded that we obtain a model for the differentiable stack cohomology associated to a proper and regular groupoid, which can be explicitly constructed and computed.