On characteristic classes of exotic manifold bundles

Abstract

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

16 pages, to appear in Mathematische Annalen

References in corpus (2)

• Mapping class groups of highly connected $(4k+2)$-manifolds
• Tautological rings for high dimensional manifolds

Cited by in corpus (2)

• On the high-dimensional geography problem
• Mapping class groups of $h$-cobordant manifolds