Shawn Cui | June 22, 2022 | Trisection invariants of 4-manifolds from Hopf algebras

Speaker: Shawn Cui, Purdue University
Title: Trisection invariants of 4-manifolds from Hopf algebras
Abstract: The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite- dimensional Hopf algebras. Here we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. Time permitting, we also sketch an ongoing effort to generalize the invariant using non-semisimple Hopf algebras.