TQFTs do not detect the Milnor sphere

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.

💡 Research Summary

The paper proves that a very broad class of topological quantum field theories (TQFTs) cannot distinguish homotopy spheres that bound parallelizable manifolds, in particular Milnor’s exotic 7‑sphere. The authors work in the oriented bordism category Bord_{SO}^{4k‑1} and consider any symmetric monoidal functor F into a “well‑rounded” target category – a condition satisfied by vector spaces over a field, modules over a ring, graded modules, derived categories of vector spaces, quasicoherent sheaves, and even super‑vector spaces. The main theorem states that for any non‑empty (4k‑1)‑dimensional bordism M, the value of F on the connected sum M # Σ (where Σ is a homotopy sphere bounding a parallelizable 4k‑manifold) equals the value on M itself, regardless of the choice of gluing.

The proof proceeds by embedding a large handlebody V_g = ♮g S^{2k‑1} × D^{2k} into the interior of M, cutting it out to obtain M′, and then gluing V_g back via a diffeomorphism φ of its boundary. The diffeomorphism φ is chosen to realize the exotic sphere Σ (via the identification Θ{4k‑1} ≅ π₀Diff∂(D^{4k‑1})). For g ≥ 2, a theorem of Krannich‑Kupers‑Mezher (cited as