Communications in Analysis and Geometry
Volume 30 (2022)
Morse functions to graphs and topological complexity for hyperbolic $3$-manifolds
Pages: 843 – 868
Scharlemann and Thompson define the width of a $3$-manifold $M$ as a notion of complexity based on the topology of $M$. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in which the adjacency relation on handles may give an arbitrary graph. We show that for closed hyperbolic $3$-manifolds, this is linearly related to a notion of metric complexity, based on the areas of level sets of Morse functions to graphs, which we call Gromov area.
Received 11 September 2017
Accepted 26 September 2019
Published 30 January 2023