A universal invariant of four-dimensional 2-handlebodies and three-manifolds

Abstract

This paper is the second part of our work on 4-dimensional 2-handlebodies. In the first part (arXiv:math.GT/0407032) 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 4-dimensional 2-handlebodies up to 2-deformations (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 freely generated by a Hopf algebra object H. In this way we obtain a complete algebraic description of 4-dimensional 2-handlebodies. This result is then used to obtain an analogous description of the boundaries of such handlebodies, i.e. 3-dimensional manifolds, which resolves for closed manifolds the problem posed by Kerler in "Towards an algebraic characterization of 3-dimensional cobordisms", Contemporary Mathematics 318 (2003). (cf. Problem 8-16 (1) in T. Ohtsuki, "Problems on invariants of knots and 3-manifolds", Geom. Topol. Monogr. 4 (2002).