2011/08/12 by Piergallini, Riccardo, Bobtcheva, Ivelina
#16W30 #17B37 #18D35 #57M12 #57M27 #57N13 #57R56 #57R65 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1108.2717
We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb3+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 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 through an equivalence functor from Hr to Chb3+1, where Hr 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 Chb3+1. From this we derive an analogous description of the category Cob2+1 of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.