ALGEBRAIC TOPOLOGY FOR THE STUDY OF MANIFOLDS

一个(3D)拓扑量子场论(TQFT)是一个对称的单形函子V，它介于2+1边值(可能有一些额外数据)的张量1-范畴和具有张量积的固定场上的张量1-范畴之间。边界论范畴中的对象是闭的二维曲面，它们之间的态射由三维流形给出，流形的边界是曲面的不相交并。张量积由曲面的不相交并集给出。这样的TQFT给出了闭的三维流形和映射类曲面群的不变量。实际上，将空集看作一个闭曲面，每个三维闭流形M都是从空集到它自己的态射。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是一个满足一些额外假设的有限(可能非半单)带状范畴。1995年，Lyubashenko证明了，在给定一个模张量范畴C的情况下，如何构造闭的3-流形LC的不变量以及曲面映射类群的一个不变量。于是，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。证明了对于非半单流形，不可能存在产生流形LC不变量的TQFT VC。的确，如果M是具有非零第一Betti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(SxS1)=0，因此Vc=0。然而，De Renzi，Gainutdinov，Geer，Patureau-Mirand和Runkel于2021年在模张量范畴C中构造了一个生成Lyubashenko不变量的TQFT，用于映射类群L‘C。显然，这样的TQFT也带有一个闭的三维流形的不变量。实际上，从它们的构造可以明显地看出，对于C的每个射影对象P，都有一个这样的不变量。回想一下，射影对象是这样的，对它来说函子Hom(P，-)：C-&gt；AB是精确的。就我们所知，这些不变量还没有被研究过。尤其有趣的是，知道它们有多微不足道，以及它们中是否有任何在实践中有用。同样有趣的是，将这些特征与C中的射影对象P的性质联系起来，给出每个不变量。该项目属于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.