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$4$–Manifold invariants from Hopf algebras

Julian Chaidez, Jordan Cotler and Shawn X Cui

Algebraic & Geometric Topology 22 (2022) 3747–3807

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.

$4$–manifolds, trisections, quantum topology, Hopf algebras
Mathematical Subject Classification
Primary: 16T05, 57K41
Received: 21 April 2020
Revised: 8 July 2021
Accepted: 23 July 2021
Published: 14 March 2023
Julian Chaidez
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Department of Mathematics
Princeton University
Princeton, NJ
United States
Jordan Cotler
Stanford Institute for Theoretical Physics
Stanford University
Stanford, CA
United States
Society of Fellows
Harvard University
Cambridge, MA
United States
Shawn X Cui
Department of Mathematics
Department of Physics and Astronomy
Purdue University
West Lafayette, IN
United States