On Haken's theorem and its extension (Q789066)

The paper is motivated by the following theorem of \textit{W. Haken} [Stud. mod. Topology, MAA Stud. Math. 5, 39-98 (1968; Zbl 0194.249)]: Let M be a closed connected 3-manifold with a fixed Heegaard splitting (M,F). If M contains an incompressible 2-sphere, then there exists an incompressible 2-sphere \(S^ 2\) in M such that \(F\cap S^ 2\) is a single circle which is not contractible in F. In the paper, this theorem is re-proved and the corresponding result on 2-sided projective planes is proved. It is the following theorem: Under the same assumption as above, if there exists a 2-sided projective plane in M, then there exists a 2-sided projective plane \(P^ 2\) in M such that \(F\cap P^ 2\) is a single circle. Also, an \({\mathbb{Z}}_ 2\)-equivariant version of Haken's theorem is proved. The key point of the proofs is the following lemma, which reveals a common property of spheres and projective planes with holes. Lemma. Let Q be a sphere or a projective plane with at least two holes. Let \(\alpha_ 1,...,\alpha_ n\) be disjoint essential arcs in Q, such that each component of \(Q\backslash(\alpha_ 1\cup...\cup \alpha_ n)\) is a 2- disk. Then, for some component C of \(\partial Q\), each intersection \(\alpha_ i\cap C\) (1\(\leq i\leq n)\) contains at most one point.