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- Title
ON 4-DIMENSIONAL 2-HANDLEBODIES AND 3-MANIFOLDS.
- Authors
BOBTCHEVA, IVELINA; PIERGALLINI, RICCARDO
- Abstract
We show that for any n ≥ 4 there exists an equivalence functor from the category of n-fold connected simple coverings of B3 × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S3 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 S3 branched over embedded graphs. Then, we factor the functor above as , where is an equivalence functor to a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category . From this we derive an analogous description of the category of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.
- Subjects
HANDLEBODIES; MANIFOLDS (Mathematics); DIMENSION theory (Topology); COBORDISM theory; CATEGORIES (Mathematics); HOPF algebras; QUANTUM theory
- Publication
Journal of Knot Theory & Its Ramifications, 2012, Vol 21, Issue 12, p-1
- ISSN
0218-2165
- Publication type
Article
- DOI
10.1142/S0218216512501106