We show that for any n ≥ 4 there exists an equivalence functor [Formula: see text] from the category [Formula: see text] of n-fold connected simple coverings of B3 × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category [Formula: see text] of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S3 branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S3 branched over embedded graphs. Then, we factor the functor above as [Formula: see text], where [Formula: see text] is an equivalence functor to a universal braided category [Formula: see text] freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category [Formula: see text]. From this we derive an analogous description of the category [Formula: see text] of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

Given a semisimple stable autonomous tensor category A over a field K, to any group presentation with finite number of generators we associate an element Q(P ) ∈ K invariant under the Andrews-Curtis moves. We show that in fact, this is the same invariant as the one produced by the algorithm introduced by Frank Quinn in [8]. The new definition allows us to present a relatively simple proof of the invariance and to evaluate Q(P ) for some presentations. On the basis of some numerical calculations over different Gelfand-Kazhdan categories, we make a conjecture which allows us to relate the value of Q(P ) for two different classes of presentations.

The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1-and 2-handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T 4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T 4 two subsets are identified : T 3 ⊂ T 4 , which produces invariants of 4-thickenings normalizable to invariants of the boundary, and T 2 ⊂ T 4 , which produces invariants of 4-thickenings depending only on the 2-dimensional spine and the second Whitney number of the 4-thickening. The case of the quantum sl(2) is studied in details. We conjecture that sl(2) leads to four HKR-type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center.

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Abstract. We study the sensibility of an invariant of 2-dimensional CW complexes in the case when it comes as a reduction (through a change of ring) of a modular invariant of 4-dimensional thickenings of such complexes: it is shown that if the Euler characteristic of the 2-complex is greater than or equal to 1, its invariant depends only on homology. To see what is happening when the Euler characteristic is smaller than 1, we use ideas of Kerler and construct, from any tortile category, an invariant of 4-thickenings which, if the category is modular, is a normalization of the Reshetikhin-Turaev invariant, and if the category is symmetric, depends only on the spine and the second Whitney number of the thickening. Then we show that the so(3) quantum invariant at a 5th root of unity has its reduction, and this reduction is able to distinguish complexes with the same homology groups when the Euler characteristic is smaller than 1.