Haken spheres in genus 2 Heegaard splittings of nonprime 3-manifolds (Q1888465)

Let \(M\) be a closed orientable 3-manifold with a Heegaard splitting \(M=V \cup_F W\), and \(S\) a separating \(2\)-sphere \(S\) embedded in \(M\). We call \(S\) a Haken sphere if it intersects the splitting surface \(F\) only in an essential circle \(C\) in \(F\). Suppose genus\((F)=2\) and \(M\) is not prime. It is shown that, for every pair of two Haken spheres \(S, S'\) in \(M\), there is a sequence of Haken spheres \(S=S_0, S_1, \dots, S_n=S'\) such that \(S_k\) and \(S_{k+1}\) are \((i,j)\)-related as below, where \((i,j)=(1,1), (1,2), (2,1)\) or \((2,2)\). \(S \cap V\) (resp. \(S \cap W\)) is a disk \(D\) (resp. \(E\)) which separates \(V\) (resp. \(W\)) into two solid tori \(V_1\) and \(V_2\) (resp. \(W_1\) and \(W_2\)) with \(F \cap V_m = F \cap W_m\). \(V_i\) (resp. \(W_j\)) contains an essential non-separating disk \(\Delta\) (resp. \(\Sigma\)) disjoint from \(D\) (resp. \(E\)). When \(i \neq j\), a circle \(C''\) in \(F - (\partial \Delta \cup \partial \Sigma)\) can be extended to a Haken sphere \(S''\) if \(C''\) separates the circles \(\partial \Delta\) and \(\partial \Sigma\). Then we say that \(S\) and \(S''\) are \((i,j)\)-related. When \(i = j\), and \(V_i \cup W_i \cong S^2 \times S^1-\)(an open ball), we can take \(\Delta\) and \(\Sigma\) so that \(\partial \Delta = \partial \Sigma\). A circle \(C^*\) in \(F - \partial \Delta\) can be extended to a Haken sphere \(S^*\) if \(C^*\) separates \(F\) into two once-punctured tori. Then we say that \(S\) and \(S^*\) are \((i,i)\)-related. In the sequence, no pair of \(S_k\) and \(S_{k+1}\) are \((i,i)\)-related if \(M\) is a sum of two lens spaces. This case was studied by the first author in [Haken spheres in the connected sum of two lens spaces, preprint]. \textit{M. Scharleman} and \textit{A. Thompson} studied the case of \(M = S^3\) in [Proc. Lond. Math. Soc. (3) 87, No. 2, 523--544 (2003; Zbl 1047.57008)].