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TQFTs do not detect the Milnor sphere

Ben Gripaios, Oscar Randal-Williams

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.

TQFTs do not detect the Milnor sphere

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 -categories) and arbitrary tangential structures. An appendix contains results on the mapping class groups of (stably-) framed manifolds that may be of independent interest.
Paper Structure (6 sections, 8 theorems, 27 equations)

This paper contains 6 sections, 8 theorems, 27 equations.

Table of Contents

  1. Other target categories
  2. Other tangential structures
  3. Extended TQFTs

Key Result

Theorem 1

Let $\Sigma$ be a $(4k-1)$-dimensional oriented homotopy sphere that bounds a parallelisable $4k$-dimensional manifold, and let $F : \mathsf{Bord}^{SO}_{4k-1} \to \mathsf{Vect}_\Bbbk$ be an oriented TQFT taking values in the category of vector spaces over a field $\Bbbk$. Then for any nonempty $(4k- for any choice of connected-sum.

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem:destab']}
  • Theorem 5
  • Lemma 6
  • proof
  • ...and 4 more