TQFTs do not detect the Milnor sphere

TL;DR

This paper proves that topological quantum field theories cannot distinguish certain exotic spheres, like Milnor's 7-dimensional sphere, from standard spheres under broad conditions.

Contribution

It demonstrates the limitations of TQFTs in detecting exotic smooth structures on homotopy spheres, extending previous understanding to general hypotheses and categories.

Findings

TQFTs cannot detect Milnor's exotic 7-sphere

Results hold for various target categories and tangential structures

Includes new results on mapping class groups of framed manifolds

Abstract

We show that, under very general hypotheses, topological quantum field theories (TQFTs) cannot detect homotopy spheres bounding parallelisable manifolds, such as Milnor's exotic 7-dimensional sphere. The result holds for a wide variety of target categories (or $(\infty,n)$-categories) and arbitrary tangential structures. An appendix contains results on the mapping class groups of (stably-) framed manifolds that may be of independent interest.