Reducibility and nonbinding (Q1903600)

This paper gives equivalent descriptions of reducible and weakly reducible Heegaard splittings in terms of nonbinding. Let \((V,V')\) be a Heegaard splitting for a closed orientable 3-manifold. \((V,V')\) is said to be nonbinding if there is an associated H-diagram \((V;J_1,\dots,J_n)\) so that \(\Gamma= \{[J_1],\dots, [J_n]\}\) (contained in \(F_n= \pi_1(V)\)) does not bind the free group \(F_n\), and is said to be weakly nonbinding if a non-empty subset of \(\Gamma\) does not bind \(F_n\). The concepts of nonbindingness are purely algebraic. The author proved the following main results: a Heegaard splitting is reducible if and only if it is nonbinding, and a Heegaard splitting is weakly reducible if and only if it is weakly nonbinding. The interest of the paper is that the author characterizes the geometric reducibility by algebraic nonbinding conditions.