Combed Trisection Diagrams and Non-Semisimple 4-Manifold Invariants

TL;DR

This paper introduces a new invariant for 4-manifolds based on non-semisimple Hopf algebras, extending previous semisimple invariants and applicable to exotic Stein nuclei, with computational challenges noted.

Contribution

It develops a novel 4-manifold invariant using non-semisimple Hopf algebras and trisection diagrams, generalizing earlier semisimple invariants and exploring their applications.

Findings

Invariant generalizes earlier semisimple versions.

Calculated for Stein nuclei, exotic 4-manifolds with boundary.

Failed to find suitable non-semisimple Hopf triples up to dimension 11.

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 $\tau_H(X,\pi)$ 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 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.