  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.

Geometric Topology