Geometry and Topology Seminar

It is not known whether two diffeomorphic 4-dimensional 2-handlebodies are always equivalent through handle moves of index smaller than 2 (i.e. 2-equivalent). A result of Beliakova, Bobtcheva, de Renzi and Piergallini implies that any so-called BP Hopf algebra in a braided monoidal category yields an invariant of 4-dimensional 2-handlebodies up to 2-equivalence. Unfortunately, it is hard to exhibit such algebras, since we want the category to be non-semisimple and non-factorizable. After explaining all of this in detail, I will discuss a series of such examples, both well-known and newly constructed. We proved some of them degenerate to the invariants boundary of the handle body, and some of them to the spine of the 2-handlebodies. (Joint work in progress with A. Beliakova and Q. Faes)