ALGEBRAIC TOPOLOGY FOR THE STUDY OF MANIFOLDS

（3d）拓扑量子场论（TQFT）是一个在2+1阶的张量1-范畴（可能有一些额外的数据）和一个固定域上的向量空间的张量1-范畴（配备张量积）之间的对称monoidal函子V。边界范畴中的对象是闭的2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给出了封闭的三维流形以及映射类曲面群的不变量。事实上，把空集看作一个闭曲面，每个三维闭流形M都是空集到它自身的态射。因此，V（M）是V（空集）的线性自同态，从而产生一个不变量。一个模张量范畴C是一个有限的（可能是非半单的）带范畴，它满足一些额外的假设。在1995年，Lyubashenko展示了如何给定一个模张量范畴C，人们可以构造闭3-流形LC的不变量和曲面L 'C的映射类群的不变量。然后，我们有理由问是否存在一个（非半单的）TQFT产生这样的不变量。证明了对于C非半单，不存在产生流形LC不变量的TQFT VC。事实上，如果M是一个具有非零第一贝蒂数的三维闭定向三维流形，则LC（M）= 0。这意味着，给定一个闭曲面S，dim（VC（S））= LC（SxS 1）= 0，因此VC = 0.然而，de Renzi，Gainutdinov，Geer，Patureau-Mirand和伦克尔在2021年从模张量范畴C中构造了一个产生映射类群L 'C的Lyubashenko不变量的TQFT.显然，这样的TQFT也带有闭3-流形的不变量。实际上，从它们的构造中可以明显看出，对于C的每一个射影对象P，我们都得到一个这样的不变量。回想一下，投射对象是函子Hom（P，-）：C -> Ab是正合的对象。据我们所知，这些不变量还没有被研究。特别是，了解它们有多少次是微不足道的，以及它们中是否有任何一个在实践中是有用的，这是很有趣的。同样有趣的是，将这些特征与C中的投射对象P的属性联系起来，给出这些不变量中的每一个。本项目福尔斯属于EPSRC基础和严格处理范围。

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.