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$4$–Manifold invariants
from Hopf algebras
Julian Chaidez, Jordan Cotler and Shawn X Cui |
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Algebraic & Geometric Topology 22 (2022) 3747–3807
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Abstract |
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The Kuperberg invariant is a topological invariant of closed –manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing –manifold invariants in the spirit of Kuperberg’s –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 –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 –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. |
Keywords
$4$–manifolds, trisections, quantum topology, Hopf algebras
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Mathematical Subject Classification
Primary: 16T05, 57K41
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References |
Publication
Received: 21 April 2020
Revised: 8 July 2021
Accepted: 23 July 2021
Published: 14 March 2023
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Authors |
| 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 |
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