The classification of 3-manifolds with 4-plane, quasi-finite immersed surfaces (Q1818711)

A famous conjecture in 3-manifold theory states that if two closed irreducible orientable 3-manifolds have isomorphic infinite fundamental groups, then they are homeomorphic. \textit{F. Waldhausen} obtained in [Ann. Math., II. Ser. 87, 56-88 (1968; Zbl 0157.30603)] an important partial result, showing that homotopy equivalence implies homeomorphism if one of the manifolds contains an embedded incompressible oriented surface. A natural generalization of an incompressible surface is an immersed incompressible surface. \textit{J. Hass} and \textit{G. P. Scott} [Topology 31, No. 3, 493-517 (1992; Zbl 0771.57007)] generalized Waldhausen's result to a class of 3-manifolds with immersed incompressible surfaces whose self-intersections are of a very simple form. The main result of this paper is the following Theorem. -- Let \(M\) be a closed orientable irreducible 3-manifold and let \(f:\Sigma\to M\) be an incompressible immersed surface, with \(\Sigma\) being orientable of genus at least one. If \(M'\) is a closed irreducible 3-manifold which is homotopy equivalent to \(M\), then \(M'\) is homeomorphic to \(M\). Another recent result related to the conjecture above is due to \textit{D. Gabai, R. Meyerhoff} and \textit{N. Thurston} who proved in [Homotopy hyperbolic 3-manifolds are hyperbolic, preprint] that if a closed, irreducible, orientable 3-manifold is homotopy equivalent to a hyperbolic manifold, then the two manifolds are homeomorphic.