Reducing weak H-systems (Q5939831)

Let \(F\) be a connected closed surface of genus \(m\). A complete system \(X=\{x_1, \dots,x_m\}\) on \(F\) is a collection of pairwise disjoint simple closed curves on \(F\) such that the surface obtained by cutting \(F\) along \(X\) is a \(2m\)-punctured 2-sphere. A Heegaard diagram is a closed surface \(S\) with two complete systems \(X\) and \(Y\) on it. Let \(S(X)\) be the handlebody constructed by attaching \(m\) 2-handles to \(S\times I\) along \(X\times 1\), then cap off the resulting 2-sphere with a 3-ball. The Heegaard diagram \((S:X,Y)\) defines a closed 3-manifold \(M=S(X)\cup_SS(Y)\), and the pair \((S(X),S(Y))\) of handlebodies is said to be a Heegaard splitting for \(M\). The pair \((S(X);Y)\) (resp. \((S(Y); X))\) is also called a Heegaard diagram of \(M\) arising from the splitting \((S(X), S(Y))\). As is is well known, Heegaard diagrams together with the Singer moves give a combinatorial representation of closed connected 3-manifolds modulo homeomorphisms. Let \(T_m\) be the standard handlebody of genus \(m\) embedded in the 3-sphere \(S^3\). A complete system \(\{x_1,\dots, x_n,y_{n+1}, \dots,y_m\}\) on \(S=\partial T_m\) is called an \(H\)-system if \(x_1,\dots,x_n\) bound pairwise disjoint surfaces \(F_1,\dots,F_n\) in the complement of \(T_m\) in \(S^3\), and \(y_{n+1}, \dots,y_m\) bound \(m-n\) pairwise disjoint disks in \(T_m\) [\textit{E. Rego} and \textit{C. Rourke}, Topology 27, No. 2, 137-143 (1988; Zbl 0646.57006)]. For an \(H\)-system as above, let \(V\) be the handlebody obtained by cutting \(T_m\) along the disks in \(T_m\) bounded by \(y_{n+1},\dots,y_m\). Then the pair \((V; \{x_1, \dots, x_n\})\) is an Heegaard diagram, which is said to be associated to the \(H\)-system. Rego and Rourke proved that a Heegaard diagram defines a homotopy 3-sphere if and only if it is associated to an \(H\)-system. In the paper under review, the author introduces the concept of weak \(H\)-system and obtains a theorem of Rego-Rourke's type.