Generalized Kuperberg invariants of 3-manifolds

TL;DR

This paper generalizes Kuperberg's 3-manifold invariants from involutory Hopf algebras over fields to those in symmetric monoidal categories with good pairs, broadening the scope of topological invariants.

Contribution

It introduces a new framework for constructing 3-manifold invariants using involutory Hopf algebras in symmetric monoidal categories with good pairs, extending previous algebraic approaches.

Findings

Constructed examples of good pairs for central involutory Hopf algebras.

Extended Kuperberg invariants to involutory super Hopf algebras.

Provided a new categorical perspective on 3-manifold invariants.

Abstract

In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting certain pairs of morphisms called good pairs. We construct examples of such good pairs for involutory Hopf algebras whose distinguished grouplike elements are central. The generalized construction is illustrated by an example of an involutory super Hopf algebra.