ALGEBRAIC TOPOLOGY FOR THE STUDY OF MANIFOLDS

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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.