TQFT diffeomorphism invariants and skein modules

Abstract

There is a well-known correspondence between two-dimensional topological quantum field theories (2-D TQFTs) and commutative Frobenius algebras. Every 2-D TQFT also gives rise to a diffeomorphism invariant of closed, orientable twomanifolds, which may be investigated via the associated commutative Frobenius algebras. We investigate which such diffeomorphism invariants may arise from TQFTs, and in the process uncover a distinction between two fundamentally different types of commutative Frobenius algebras (“weak” Frobenius algebras and “strong” Frobenius algebras). These diffeomorphism invariants form the starting point for our investigation into marked cobordism categories, which generalize the local cobordism relations developed by Dror Bar-Natan during his investigation of Khovanov’s link homology. We subsequently examine the particular class of 2-D TQFTs known as “universal sl(n) TQFTs”. These TQFTs are at the algebraic core of the link invariants known as sl(n) link homology theories, as they provide the algebraic structure underlying the boundary maps in those homology theories. We also examine the 3-manifold diffeomorphism invariants known as skein modules, which were first introduced by Marta Asaeda and Charles Frohman. These 3-manifold invariants adapt Bar-Natan’s marked cobordism category (as induced by a specific 2-D TQFT) to embedded surfaces, and measure which such surfaces may be embedded within in 3-manifold (modulo Bar-Natan’s local cobordism relations). Our final results help to characterize the structure of such skein modules induced by universal sl(n) TQFTs.