  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.
