On characteristic classes of exotic manifold bundles

TL;DR

This paper compares characteristic classes of smooth manifold bundles with exotic spheres, showing isomorphisms after inverting the sphere's order and providing examples where this inversion is essential.

Contribution

It establishes conditions under which characteristic class rings are isomorphic when replacing a manifold with its connected sum with an exotic sphere.

Findings

Rings are isomorphic after inverting the order of the exotic sphere.

Constructs infinite families of examples demonstrating the necessity of inversion.

Provides a range of degrees where the isomorphism holds.

Abstract

Given a closed simply connected manifold $M$ of dimension $2n\ge6$, 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 $M\sharp\Sigma$ 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.