This thesis consists of three articles in which we make several contributions to the study of characteristic classes of manifold bundles and closely related topics.
In Article A, we compare the ring of characteristic classes of smooth bundles with fibre a closed simply connected manifold M of dimension 2n ≠ 4 to the respective ring resulting from replacing M by the connected sum M #∑ with an exotic sphere ∑. We show that, after inverting the order of ∑ in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite familiesof examples witnessing that inverting the order of ∑ is necessary.In Article B, which is joint with Jens Reinhold, we study smooth bundles over surfaces with highly connected almost parallelisable fibre M of even dimension. We provide necessary conditions for a manifold to be bordant to the total space of such a bundle and show that, in most cases, these conditions are also suficient. Using this, we determine the characteristic numbers realised by total spaces of bundles of this type, deduce divisibility constraints on their signatures and Â-genera, and compute the second integral cohomology of BDiff⁺(M ) up to torsion in terms of generalised Miller–Morita–Mumford classes.In Article C, we introduce a framework to study homological stability properties of E₂-algebras and their modules, generalising work of Randal-Williams and Wahl in the case of discrete groups. As an application, we prove twisted homological stability results for various families of topological moduli spaces, such as configuration spaces and moduli spaces of manifolds, and explain how these results imply representation stability for related sequences of spaces.