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A universal invariant of fourdimensional 2handlebodies and threemanifolds, preprint ArXiv:math.GT/0612806
"... In [2] it is shown that up to certain set of local moves, connected simple coverings of B 4 branched over ribbon surfaces, bijectively represent connected orientable 4dimensional 2handlebodies up to 2deformations (handle slides and creations/cancellations of handles of index ≤ 2). We factor this ..."
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In [2] it is shown that up to certain set of local moves, connected simple coverings of B 4 branched over ribbon surfaces, bijectively represent connected orientable 4dimensional 2handlebodies up to 2deformations (handle slides and creations/cancellations of handles of index ≤ 2). We factor this bijective correspondence through a map onto the closed morphisms in a universal braided category H r freely generated by a Hopf algebra object H. In this way we obtain a complete algebraic description of 4dimensional 2handlebodies. This result is then used to obtain an analogous description of the boundaries of such handlebodies, i.e. 3dimensional manifolds, which resolves for closed manifolds the problem posed by Kerler in [13] (cf. [21, Problem 816 (1)]).
COVERING MOVES AND KIRBY CALCULUS
, 2004
"... We show that simple coverings of B 4 branched over ribbon surfaces up to certain local ribbon moves coincide with orientable 4dimensional 2handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S 3 b ..."
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We show that simple coverings of B 4 branched over ribbon surfaces up to certain local ribbon moves coincide with orientable 4dimensional 2handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S 3 branched over links, in terms of local moves. This result generalizes to coverings of any degree the ones by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of our equivalence theorem to possibly nonsimple coverings of S 3 branched over embedded graphs.