For manifolds $M$ of dimension $2n \neq 4$ we now have a good understanding of the cohomology of the classifying spaces $BDiff_\partial(M)$ in a range of degrees tending to infinity with the number of $S^n \times S^n$ connect-summands of $M$. In dimension $2n=2$ this is given by the combination of the Madsen--Weiss theorem and Harer's stability theorem, and in dimensions $2n \geq 6$ it is given by analogues of these results due to Galatius and myself.

In dimension $2n=4$ no homological stabilisation theorem is known. Nonetheless all other steps in my work with Galatius do go through in dimension 4, and one can still make a conclusion about the (co)homology of the direct limit $BDiff_\partial(M^4 \# g S^2 \times S^2)$ as $g \to \infty$. I will explain what these results are, and some consequences for characteristic classes of 4-manifold bundles.