ALGEBRAIC TOPOLOGY FOR THE STUDY OF MANIFOLDS

研究流形的代数拓扑

• 批准号: 2747348
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2747348
• 关键词: ALGEBRAIC; TOPOLOGY; STUDY; MANIFOLDS

A (3d) topological quantum field theory (TQFT) is a symmetric monoidal functor V between the tensor 1-category of 2+1 bordisms (possibly with some extra data) and the tensor 1-category of vector spaces over a fixed field equipped with the tensor product. The objects in the category of bordisms are closed 2-dimensional surfaces, and the morphisms between them are given by 3-dimensional manifolds whose boundary is the disjoint union of the surfaces in consideration. The tensor product is given by the disjoint union of surfaces. Such a TQFT gives invariants of closed 3-dimensional manifolds as well as of mapping class groups of surfaces. Indeed, considering the empty set as a closed surface, every 3-dimensional closed manifold M is a morphism from the empty set to itself. Therefore, V(M) is a linear endomorphism of V(empty set), thus yielding an invariant. A similar argument produces the invariant for mapping class groups.A modular tensor category C is a finite (possibly non-semisimple) ribbon category satisfying some extra hypotheses. In 1995, Lyubashenko showed how, given a modular tensor category C, one could construct and invariant of closed 3-manifolds LC as well as an invariant of mapping class groups of surfaces L'C. It is then reasonable to ask whether there exists a (non-semisimple) TQFT producing such invariants. It turns out that there cannot exist a TQFT VC producing the invariant of manifolds LC for C non-semisimple. Indeed, if M is a 3-dimensional closed oriented 3-manifold with non-zero first Betti number, then LC(M) = 0. This implies that, given a closed surface S, dim(VC(S)) = LC(SxS1) = 0, and so VC = 0.However, de Renzi, Gainutdinov, Geer, Patureau-Mirand and Runkel constructed in 2021, out of a modular tensor category C, a TQFT producing Lyubashenko's invariant for mapping class groups L'C. Obviously, such a TQFT also carries an invariant of closed 3-manifolds. Actually, from their construction it is apparent that one gets one such invariant for every projective object P of C. Recall that a projective object is such for which the functor Hom(P,-): C -> Ab is exact. To the best of our knowledge, these invariants have not been studied yet. In particular, it is interesting to know how often they are trivial and whether any of them is useful in practice. It is also interesting to relate such characteristics to the properties of the projective object P in C giving each of these invariants. This project falls within the EPSRC foundations and rigorous treatments.

(3d) 拓扑量子场论 (TQFT) 是 2+1 个边数的张量 1-类别（可能带有一些额外数据）和固定场上的向量空间的张量 1-类别之间的对称幺半群函子 V，配备有张量积。棱射范畴中的对象是封闭的二维曲面，它们之间的态射由 3 维流形给出，其边界是所考虑的曲面的不相交并。张量积由曲面的不相交并给出。这样的 TQFT 给出了封闭 3 维流形以及曲面的映射类组的不变量。事实上，将空集视为一个闭曲面，每个 3 维闭流形 M 都是从空集到其自身的态射。因此，V(M) 是 V(空集) 的线性自同态，从而产生不变量。类似的论证产生映射类组的不变量。模张量类别 C 是满足一些额外假设的有限（可能非半简单）带状类别。 1995 年，Lyubashenko 展示了如何在给定模张量类别 C 的情况下构造闭合 3 流形 LC 的不变量以及曲面 L'C 的映射类组的不变量。那么询问是否存在产生此类不变量的（非半简单）TQFT 是合理的。事实证明，不存在为非半单 C 产生流形 LC 不变量的 TQFT VC。事实上，如果 M 是第一个 Betti 数非零的 3 维闭向 3 流形，则 LC(M) = 0。这意味着，给定一个闭合曲面 S，dim(VC(S)) = LC( SxS1) = 0，因此 VC = 0。然而，de Renzi、Gainutdinov、Geer、Patureau-Mirand 和 Runkel 于 2021 年构建，属于模张量类别C，一个 TQFT，产生用于映射类组 L'C 的 Lyubashenko 不变量。显然，这样的 TQFT 还带有闭 3 流形不变量。实际上，从它们的构造中可以明显看出，对于 C 的每个射影对象 P，我们都会得到一个这样的不变量。回想一下，射影对象对于函子 Hom(P,-): C -> Ab 是精确的。据我们所知，这些不变量尚未被研究。特别是，了解它们的重要性以及它们在实践中是否有用是很有趣的。将这些特征与 C 中的投影对象 P 的属性联系起来也很有趣，给出了这些不变量中的每一个。该项目属于 EPSRC 的基础和严格的处理范围。