A universal bound for surfaces in 3-manifolds with a given Heegaard genus

Abstract

It is shown that for given positive integers $g$ and $b$, there is a number $\mathcal{C}\left(g,b\right)$, such that any orientable compact irreducible 3-manifold of Heegaard genus $g$ has at most $\mathcal{C}\left(g,b\right)$ disjoint, nonparallel incompressible surfaces with first Betti number $b_1<b$.