In [Galatius-Szucs] the authors identify the homotopy type of the G-equivariant cobordism category of dimension d with that of the infinite loop-space associated to a certain spectrum, when G is a finite group. The unequivariant version of this theorem is known to help understand the moduli space BDiff(M) associated to one given manifold M, hence the authors raise the question of an equivariant analog of that: is it possible to use their result to compute the cohomology of BDiff^G(M), given a G-manifold M ?
We give conditions under which the answer is positive by showing an additivity theorem for equivariant cobordism categories, which also gives another proof of the main result of [Galatius-Szucs].
This is on-going rearch, soon to appear on the ArXiv.