Incompressible surfaces and the topology of 3-dimensional manifolds. (English) Zbl 0813.57017

This paper is a survey of the role of incompressible surfaces in the topology of (orientable) 3-manifolds. In the first, shorter part the authors present the classical theory of embedded incompressible surfaces, developed mainly in the sixties; they tell about existence of incompressible surfaces, Haken manifolds, hierarchies, and classification of Haken manifolds, by homotopy data. The second part is devoted to the problem of existence of incompressible immersions of closed surfaces in (non-Haken) irreducible 3-manifolds, the relevance of such immersions for the existence of finite Haken coverings of non-Haken manifolds, and related questions. Much of this latter work has been done quite recently, in the nineties and late eighties. The first part of the paper can easily be read by anyone acquainted with the fundamentals of manifold theory and algebraic topology. In the second part the reader is assumed to be better informed about the specifics of 3-manifold theory. Proofs are not included, except for sketches of the proofs of two recent results.


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M35 Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57Q35 Embeddings and immersions in PL-topology