Towards an Algebraic Characterization of 3-dimensional Cobordisms

Abstract : The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3-dimensional topology. The first is the category Cob of connected surfaces with one boundary component and 3-dimensional relative cobordisms, the second is a category Tgl of tangles with relations, and the third is a natural algebraic category Alg freely generated by a Hopf algebra object. From previous work we know that Tgl and Cob are equivalent. We use this fact and the idea of Heegaard splittings to construct a surjective functor from Alg onto Cob . We also find a map that associates to the generators of the mapping class group in Cob preimages in Alg . The single block relations in the mapping class group are verified for these expressions. We propose to find a version of Alg with possibly additional relations to obatin isomorphic algebraic presentations of the mapping class groups and eventually of Cob . 1