A finiteness result for Heegaard splittings (Q1879804)

The authors show that for any closed orientable 3-manifold \(M\) and any Heegaard splitting \(M=H_1\cup H_2\) of genus \(g\geq 2\) there are finitely many pairs of decomposing systems \(D_i\subset H_i(i=1,2)\) of \(3g-3\) disjoint essential disks, which satisfy the double rectangle condition. The double rectangle condition is given by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to the fact that in general there can be infinitely many such systems of disks which satisfy just the Casson-Gordon rectangle condition.