Tangle-Categories and Presentation of Cobordisms

Abstract

All known rigorous constructions of quantum invariants of 3-manifolds and TQFT’s proceed by, first, presenting a given manifold in a finite combinatorial way, and then applying algebraic functors to this combinatorial data. The most basic example of such is a presentation of a manifold as a simplicial. complex, taken modulo so called Alexander or Pachner subdivision moves. It is not hard to imagine that these types of presentation become quite complicated when considering cobordisms between closed surfaces and even more cumbersome when we also want to describe relative cobordisms with corners. Among other things, the latter would imply that we simplicially encode the framed braid groups.