One-sided incompressible surfaces in Seifert fibered spaces
In his thesis [Invent. Math. 3, 308-333, and 4, 87-117 (1967; Zbl 0168.445)], \textit{F. Waldhausen} classified the two-sided incompressible surfaces in Seifert 3-manifolds, by showing that each of them was isotopic either to a ''vertical'' surface (union of fibers of the Seifert fibration) or to a ''horizontal'' surface (transverse to the fibration). In the present paper, the author carries out a similar analysis for one- sided incompressible surfaces in Seifert manifolds. More precisely, he shows that each one-sided incompressible surface is isotopic to either a ''pseudovertical'' or a ''pseudohorizontal'' surface, namely one which is vertical or horizontal outside of a tubular neighborhood of some fibers. Conversely, he also proves that pseudovertical surfaces are incompressible, and gives an isotopy classification of these pseudovertical surfaces. The general problem of classifying pseudohorizontal surfaces remains open.
one-sided incompressible surfaces in Seifert manifolds
scientific article; zbMATH DE number 3980031