Covering moves and Kirby calculus

Details Covering moves and Kirby calculus Covering moves and Kirby calculus

2004-07-02

arxiv.org/abs/math/0407032v4

Mathematics

Geometric Topology

Minor corrections. Added reference to the second part of our study of 4-dimensional 2-handlebodies (arXiv:math.GT/0612806). 68

Scientific paper

We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies 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 results 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 non-simple coverings of S^3 branched over embedded graphs. This work represents the first part of our study of 4-dimensional 2-handlebodies. In the second part (arXiv:math.GT/0612806), we factor such bijective correspondence between simple coverings of B^4 branched over ribbon surfaces and orientable 4-dimensional 2-handlebodies through a map onto the closed morphisms in a universal braided category freely generated by a Hopf algebra object.

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