Morse-Novikov numbers, tunnel numbers, and handle numbers of sutured manifolds

TL;DR

This paper establishes bounds on the handle number of sutured manifolds using geometric arguments related to Morse-Novikov and tunnel numbers, and introduces a handle number function with specific properties.

Contribution

It introduces bounds on the handle number of Heegaard splittings in sutured manifolds and develops a new handle number function with notable properties.

Findings

Bounds on handle number in terms of tunnel number

Definition of a bounded, ray-constant, locally maximal handle number function

Characterization of when the handle number is zero for integral classes

Abstract

Developed from geometric arguments for bounding the Morse-Novikov number of a link in terms of its tunnel number, we obtain upper and lower bounds on the handle number of a Heegaard splitting of a sutured manifold $(M,\gamma)$ in terms of the handle number of its decompositions along a surface representing a given 2nd homology class. Fixing the sutured structure $(M,\gamma)$, this leads us to develop the handle number function $h \colon H_2(M,\partial M;\mathbb{R}) \to \mathbb{N}$ which is bounded, constant on rays from the origin, and locally maximal. Furthermore, for an integral class $\xi$, $h(\xi)=0$ if and only if the decomposition of $(M,\gamma)$ along some surface representing $\xi$ is a product manifold.