On four-dimensional 2-handlebodies and three-manifolds

TL;DR

This paper establishes an equivalence between categories of branched coverings of 3-manifolds and 4-dimensional handlebody cobordisms, providing a complete algebraic description and solving longstanding problems in 3-manifold topology.

Contribution

It introduces a new functorial equivalence linking branched coverings to 4D handlebody cobordisms, generalizing previous results and extending to non-simple coverings.

Findings

Provides a complete solution to the Fox-Montesinos covering moves problem.

Establishes an algebraic description of the category of 4D handlebody cobordisms.

Resolves a problem posed by Kerler regarding 3D cobordisms.

Abstract

We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+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 S^3 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 S^3 branched over embedded graphs. Then, we factor the functor above through an equivalence functor from H^r to Chb^{3+1}, where H^r 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 Chb^{3+1}. From this we derive an analogous description of the category Cob^{2+1} of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.