On characteristic classes of exotic manifold bundles

Abstract

AbstractGiven a closed simply connected manifold M of dimension $$2n\ge 6$$ 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 $$M\sharp \Sigma $$ M ♯ Σ with an exotic sphere $$\Sigma $$ Σ . We show that, after inverting the order of $$\Sigma $$ 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 $$\Sigma $$ Σ is necessary.