Failure of Ordinary TQFTs to Detect Homotopy TypeMPIM

by Katherine Novey (MPIM)

Monday Oct 26, 2026, 10:30 AM → 12:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Quantum Topology Seminar 

Topological Quantum Field Theories (TQFTs) provide manifold invariants. There has been significant work done over the past couple of decades investigating the sensitivity of these invariants and which manifolds they distinguish, but the question remains largely open. Work by David Reutter and Christopher Schommer-Pries has shown that ordinary TQFTs can distinguish (stable) diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected, closed, smooth $6$-manifolds with finite $\pi_2$ are diffeomorphic if and only if they cannot be distinguished by ordinary TQFTs. The question of whether or not this result holds for all simply connected $6$-manifolds was open. I will show that the answer to this question is no by presenting a pair of simply connected, non-homotopy equivalent $6$-manifolds with infinite $\pi_2$ that are indistinguishable by ordinary TQFTs. This counter-example can be extended to show that ordinary TQFTs cannot distinguish the homotopy type of simply connected closed manifolds in all dimensions $\geq 6$ and non-simply connected closed $5$-manifolds.