We show that for any n > 3 there exists an equivalence functor from the
category of n-fold connected simple coverings of B^3 x [0, 1] branched over
ribbon surface tangles up to certain local ribbon moves, to the category
Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to
2-deformations. As a consequence, we obtain an equivalence theorem for simple
coverings of S^3 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 S^3 branched over embedded graphs. Then, we factor the functor
above through an equivalence functor from H^r to Chb^{3+1}, where H^r is a
universal braided category freely generated by a Hopf algebra object H. In this
way, we get a complete algebraic description of the category Chb^{3+1}. From
this we derive an analogous description of the category Cob^{2+1} of 2-framed
relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.Comment: 213 pages, 272 figures, 15 table
