On characteristic classes of exotic manifold bundles

Given a closed simply connected manifold M of dimension 2 n ≥ 6 , we compare the ring of characteristic classes of smooth oriented bundles with fibre M to the analogous ring resulting from replacing M by the connected sum M ♯ Σ with an exotic sphere Σ . We show that, after inverting the order of Σ in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of Σ is necessary.