| Résume | A 4-dimensional 2-handlebody is a
4-manifold obtained from the 4-ball by attaching a finite number of
1-handles and 2-handles. A 2-deformation is a diffeomorphism
implemented by a finite sequence of handle moves that never introduce
3-handles and 4-handles. Whether there exist diffeomorphisms that are
not 2-deformations remains an open question, mainly due to the lack of
invariants for detecting them. I will explain how to construct quantum
invariants of 4-dimensional 2-handlebodies up to 2-deformation using
unimodular ribbon categories, such as the category of representations
of a unimodular ribbon Hopf algebra. In the case of factorizable
ribbon categories, the invariant depends exclusively on the
boundary. I will also discuss how this construction relates to a
famous question in combinatorial group theory known as the
Andrews–Curtis conjecture. Joint work with Anna Beliakova. |
