普通TQFT无法区分同伦型的失败
Failure of Ordinary TQFTs to Distinguish Homotopy Type
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中文总结 AI 辅助
本文构造反例表明，普通TQFT无法区分某些具有无限π2的单连通闭6维流形，并推广到所有高维及非单连通5维流形，证明普通TQFT在区分同伦型上存在根本局限。
中文摘要 AI 辅助
David Reutter和Christopher Schommer-Pries的工作表明，在满足某些有限性条件下，普通光滑TQFT可以区分闭连通偶数维流形的（稳定）微分同胚类。特别地，具有有限π2的单连通闭光滑6维流形，当且仅当它们无法被普通TQFT区分时，它们才是微分同胚的。该结果是否对所有单连通闭6维流形成立的问题此前是开放的。我们考虑从拓扑和光滑配边范畴以及本文构造的新配边范畴（称为形式光滑（FS）配边范畴）出发的TQFT。我们给出了一对具有无限π2的单连通闭6维流形，它们不是同伦等价的，但在所有三个范畴中都无法被普通TQFT区分。我们扩展了这一反例，以表明普通TQFT无法区分所有维数大于等于6的单连通闭流形以及非单连通闭5维流形的同伦型。我们证明了我们的反例对类是可自旋的，并通过引入一个CP2连通和项，给出了一个非自旋的反例类。最后，我们使用反例的更高连通类比，表明对于所有n≥2和m≥2n，存在一对(2n-1)-连通的闭(4n+m)维流形，它们不是同伦等价的，但无法被普通TQFT区分。
英文摘要
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 $\mathbb{C}P^2$ connected summand. Lastly, we use more highly connected analogues of our counter-examples to show that, for all $n\geq2$ and $m\geq 2n$, there is a pair of $(2n-1)$-connected closed $(4n+m)$-manifolds that are not homotopy equivalent, yet cannot be distinguished by ordinary TQFTs.
