Abstract
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 derivative M;R)-> 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.