A proof of Przytycki's conjecture on \(n\)-relator 3-manifolds
Let \(H\) be a 3-dimensional handlebody (possibly nonorientable) and \(C\) be a twosided closed 1-manifold in \(\partial H\) such that \(\partial H - C\) is incompressible in \(H\). Denote by \(H_C\) the 3-manifold obtained by attaching 2-handles to \(H\) along the components of \(C\). \textit{J. H. Przytycki} [Lond. Math. Soc. Lect. Note Ser. 112, 273-285 (1986; Zbl 0621.57005)] proposed a set of conditions on \(C\) which he conjectured to imply that \(H_C\) has incompressible boundary (or is a 3-ball); the conditions are purely algebraic and concern the homotopy classes of the components of \(C\) in \(\pi_1 (H)\). Przytycki himself verified his conjecture for \(C\) with no more than three components. In the present paper the author proves the conjecture in general.
scientific article; zbMATH DE number 751258