Nielsen equivalence and trisections

Abstract

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every $$k \ge 2$$ , we exhibit infinitely many manifolds with $$2^{k}-1$$ non-diffeomorphic (3k, k)-trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier’s spun trisections. The technique used to distinguish the trisections parallels a common setup for distinguishing Heegaard splittings. In particular, we show that the Nielsen classes of the generators of the fundamental group, obtained from spines of the 4-dimensional 1-handlebodies of the trisection, are isotopy invariants of the trisection. If we additionally consider the action of the automorphism group on the Nielsen classes we obtain diffeomorphism invariants. Once the invariance is established, we analyze the Nielsen classes of the spun trisections in terms of the Nielsen classes of the original Heegaard splitting, and leverage work of Lustig, Moriah, and Rosenberger on Nielsen classes of Fuchsian groups in order to distinguish the trisections.