Failure of Ordinary TQFTs to Distinguish Homotopy Type
Abstract
Work by David Reutter and Christopher Schommer-Pries has shown that ordinary smooth 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 π2 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 closed 6-manifolds was open. We consider TQFTs out of the topological and smooth bordism categories, as well as out of a new bordism category constructed in this paper called the formally smooth (FS) bordism category. We present a pair of simply connected closed 6-manifolds with infinite π2 that are not homotopy equivalent, yet are indistinguishable by ordinary TQFTs in all three categories. We extend this counter-example to show that ordinary TQFTs cannot distinguish the homotopy type of simply connected closed manifolds in all dimensions greater than or equal to 6 and non-simply connected closed 5-manifolds. We show that our class of counter-example pairs is spinnable and present a non-spinnable class of counter-examples by introducing a ℂP2 connected summand. Lastly, we use more highly connected analogues of our counter-examples to show that, for all n≥2 and m≥ 2n, there is a pair of (2n−1)-connected closed (4n+m)-manifolds that are not homotopy equivalent, yet cannot be distinguished by ordinary TQFTs.
Related
- On Left Adjoints Preserving Colimits in Homotopy Type Theory
- Defect TQFTs, the Generalized Cobordism Hypothesis, and Algebraic K-Theory
- Involutions on (∞,n)-categories of bordisms and reflection TQFTs
- On the Relation Between Non-Semisimple TQFTs
- Anomalies of TQFTs coupled to parameter spaces and constraints on ground-state degeneracies
BibTeX
@misc{novey2026failure,
title = {{Failure of Ordinary TQFTs to Distinguish Homotopy Type}},
author = {Katherine Novey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05480},
eprint = {2610.05480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05480},
}
RIS
TY - UNPB
TI - Failure of Ordinary TQFTs to Distinguish Homotopy Type
AU - Katherine Novey
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05480
UR - https://arxiv.org/abs/2610.05480
AB - Work by David Reutter and Christopher Schommer-Pries has shown that ordinary smooth 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 π_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 closed 6-manifolds was open. We consider TQFTs out of the topological and smooth bordism categories, as well as out of a new bordism category constructed in this paper called the formally smooth (FS) bordism category. We present a pair of simply connected closed 6-manifolds with infinite π_2 that are not homotopy equivalent, yet are indistinguishable by ordinary TQFTs in all three categories. We extend this counter-example to show that ordinary TQFTs cannot distinguish the homotopy type of simply connected closed manifolds in all dimensions greater than or equal to 6 and non-simply connected closed 5-manifolds. We show that our class of counter-example pairs is spinnable and present a non-spinnable class of counter-examples by introducing a ℂP^2 connected summand. Lastly, we use more highly connected analogues of our counter-examples to show that, for all n≥2 and m≥ 2n, there is a pair of (2n−1)-connected closed (4n+m)-manifolds that are not homotopy equivalent, yet cannot be distinguished by ordinary TQFTs.
ER -