Combed Trisection Diagrams and NonSemisimple 4Manifold Invariants
Abstract
Given a triple $H$ of (possibly nonsemisimple) Hopf algebras equipped with pairings satisfying a set of properties, we describe a construction of an associated smooth, scalar invariant $\tau_H(X,\pi)$ of a simply connected, compact, oriented $4$manifold $X$ and an open book $\pi$ 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 4manifolds with boundary called Stein nuclei, introduced by Yasui. After investigating many lowdimensional Hopf algebras up to dimension 11, we have not been able to find nonsemisimple Hopf triples that satisfy the criteria for our invariant. Nonetheless, appropriate Hopf triples may exist outside the scope of our explorations.
 Publication:

arXiv eprints
 Pub Date:
 September 2023
 DOI:
 10.48550/arXiv.2309.08461
 arXiv:
 arXiv:2309.08461
 Bibcode:
 2023arXiv230908461C
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Geometric Topology
 EPrint:
 62 pages, many figures and diagrams