4-Manifold Invariants From Hopf Algebras

TL;DR

This paper introduces a new method for constructing invariants of closed 4-manifolds using Hopf triplets, extending ideas from 3-manifold invariants and relating to known invariants like Crane-Yetter.

Contribution

It develops a novel framework for 4-manifold invariants based on Hopf triplets and trisection diagrams, generalizing existing invariants and opening avenues for richer topological invariants.

Findings

Every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds.

Special cases recover Crane-Yetter and dichromatic invariants.

Conjectural relation to Kashaev's invariant.

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 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. We speculate that relaxing semisimplicity will lead to even richer invariants.