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.

A（3D）拓扑量子场理论（TQFT）是在2+1个bordism的张量1类（可能带有一些额外的数据）和配备有张量产品的固定场上的张量1类别的向量空间之间的对称单函数V。边界类别中的对象是封闭的二维表面，它们之间的形态是由三维歧管给出的，其边界是考虑到的表面的脱节结合。张量产品由表面的不交互结合给出。这样的TQFT给出了封闭的3维流形以及映射表面类群组的不变性。实际上，考虑到空的集合是封闭的表面，每个3维闭合的歧管M是从空集到自身的形态。因此，V（M）是V（空集）的线性内态性，因此产生了不变性。一个类似的参数会产生用于映射类组的不变性。模块化张量类别C是一种有限的（可能是非偏simimple）的功能区类别，可满足一些额外的假设。 1995年，Lyubashenko展示了如何在模块化张量C类C的情况下如何构建和不变的3个Manifolds LC，以及映射L'C表面类别组的不变性。然后是合理的，询问是否存在产生这种不变性的（非偏simimple）TQFT。事实证明，不能存在为c nonemisimple产生歧管LC不变的TQFT VC。 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张量C类C，TQFT产生Lyubashenko的不变型，用于映射l'C的班级组。显然，这样的TQFT还带有封闭的3个manifolds的不变。实际上，从他们的构造中可以明显看出，每个投影对象p都会使一个不变。回想一下，一个投影对象就是这样的函数hom（p， - ）：c-> ab是准确的。据我们所知，尚未研究这些不变的人。特别是，很有趣的是知道它们的频率是小时候，以及它们中的任何一个在实践中是否有用。将这种特征与C在C中的投影对象P的属性相关联，这也很有趣。该项目属于EPSRC基础和严格的治疗方法。