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arXiv · 2601.20828

TQFTs do not detect the Milnor sphere

Ben Gripaios·Oscar Randal-Williams
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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.

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Ben Gripaios, Oscar Randal-Williams. 2026-01-28. TQFTs do not detect the Milnor sphere. https://arxiv.org/abs/2601.20828

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