Commutative Frobenius Algebras and 3-Dimensional Compression Bordisms

Given a commutative Frobenius algebra V over a commutative ring R with 1 we construct certain V j -module categories V [j] for j ≥ 0. Let (M, α) be an oriented 3-manifold with a closed oriented 1-manifold α in its boundary. Then there are defined natural functors from a category of oriented surfaces in M bounding α and morphisms defined by compression bordisms in M × I, taking values in V [|α|]. (Here a compression bordism S1 → S2 is a 3-dimensional manifold with corners, properly embedded in M × I, which is a product over α, and with only embedded 2-handles and 3-handles attached to S1 × I, considered up to isotopy through those bordisms). The colimit of this functor is the Bar-Natan skein module defined for (M, α) and the Frobenius algebra V . Moreover, a glueing theorem can be proven for this functor. The above construction can be twisted with a (3 + 1)-dimensional TQFT over R to define functors on a category with the morphisms embedded in oriented 4-manifolds. We discuss the above constructions and some conjectures related to it.