On characteristic classes of exotic manifold bundles

Abstract

Abstract: Given a closed simply connected manifold M of dimension 2n≥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.