Morse-Novikov numbers, tunnel numbers, and handle numbers of sutured manifolds
Kenneth L. Baker, Fabiola Manjarrez-Gutiérrez · Journal of Differential Geometry · 2025
Developed from geometric arguments for bounding the MorseNovikov 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 $h: H_2(M, \partial M ; \mathbb{R}) \rightarrow \mathbb{N}$ 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.