Some 3-manifolds with 2-generated fundamental group
For an orientable closed \(3\)-manifold \(M\), the Heegaard genus gives an upper bound for the rank of the fundamental group \(\pi_1(M)\). \textit{M. Boileau} and \textit{H. Zieschang} [Invent. Math. 76, 455--468 (1984; Zbl 0538.57004)] computed the rank of the fundamental groups for some Seifert fibered manifolds and showed that there exists a family of Seifert fibered manifolds with Heegaard genus \(3\) and rank two fundamental group. In this paper, the author constructs a family of graph manifolds consisting of two Seifert fibered manifolds whose orbifolds are the Möbius band with at most one exceptional fiber and the disk with two exceptional fibers. It is shown that these manifolds have Heegaard genus \(3\) and rank two fundamental group. In fact, the examples of Boileau and Zieschang can be considered as a special case of the family here as the union of two Seifert fibered manifolds.
scientific article; zbMATH DE number 2051641