Equivariant cobordism categories and the homology of moduli spaces of equivariant manifolds

The goal of this thesis is to study the moduli space MG(M ) associated to a smooth compact manifold M equipped with an action of a finite group G. This space is homotopy equivalent to the classifying space of DiffG(M ) the topological group of equivariant diffeomorphisms of M . We prove that under some connectivity conditions, its homology is often given by that of an infinite loop space in the stable range, answering a question raised by Galatius-Szucs in [GS21]. We strongly rely on the work of Galatius-Randal-Williams ([GR17a],[GR17b]) on the homology of moduli spaces of high dimensional manifolds, which gave such a stable computation in the non equivariant setting. Our proof relies on the existence of an isotropy separation sequence at the level of equivariant cobordism categories `a la Steimle. As a by-product, we give a new proof of the main result of [GS21].