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arXiv · 1910.14662

4-Manifold Invariants From Hopf Algebras

Julian Chaidez·Jordan Cotler·Shawn X. Cui
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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.

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BibTeXRIS

Julian Chaidez, Jordan Cotler, Shawn X. Cui. 2021-07-26. 4-Manifold Invariants From Hopf Algebras. https://doi.org/10.2140/agt.2022.22.3747

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