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arXiv · 2309.08461

Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants

Julian Chaidez·Jordan Cotler·Shawn X. Cui
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Abstract

Given a triple $H$ of (possibly non-semisimple) Hopf algebras equipped with pairings satisfying a set of properties, we describe a construction of an associated smooth, scalar invariant $τ_H(X,π)$ of a simply connected, compact, oriented $4$-manifold $X$ and an open book $π$ on its boundary. This invariant generalizes an earlier semisimple version and is calculated using a trisection diagram $T$ for $X$ and a certain type of combing of the trisection surface. We explain a general calculation of this invariant for a family of exotic 4-manifolds with boundary called Stein nuclei, introduced by Yasui. After investigating many low-dimensional Hopf algebras up to dimension 11, we have not been able to find non-semisimple Hopf triples that satisfy the criteria for our invariant. Nonetheless, appropriate Hopf triples may exist outside the scope of our explorations.

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Julian Chaidez, Jordan Cotler, Shawn X. Cui. 2024-11-25. Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants. https://arxiv.org/abs/2309.08461

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