4-Manifold Invariants From Hopf Algebras

Abstract

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperbergs 3manifold 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. We speculate that relaxing semisimplicity will lead to even richer invariants.

Semisimple four‐dimensional topological field theories cannot detect exotic smooth structure

“…Similar to the Crane-Yetter-Kauffman theory, the oriented 4-manifold invariants in [3,8,12,16] are either proven or expected to arise from once-extended topological field theories with values in one of the symmetric monoidal bicategories of the 'bestiary of 2-vector spaces' of [4, Appendix A] and should therefore be subject to our results.…”