Homotopy and isotopy in dimension three (Q1313256)

The authors enlarge the class of 3-manifolds for which homotopic homeomorphisms are always isotopic. The class was known to include Haken manifolds and Seifert fiber spaces, and is conjectured to include any closed \(\mathbb{P}^ 2\)-irreducible 3-manifold. The authors prove Theorem 1.1: Let \(M\) be a closed orientable irreducible 3-manifold which is neither Haken nor a Seifert fiber space. If there is a closed orientable surface \(F\), not \(S^ 2\), and an immersion \(f: F\to M\) which injects \(\pi_ 1(F)\) and has the 3-plane and 1-line-intersection properties, then homotopic homeomorphisms of \(M\) are isotopic. The authors had previously shown in [Topology 31, No. 3, 493-517 (1992; Zbl 0771.57007)] that under the hypotheses of the theorem if \(M\) is homotopy equivalent to an irreducible 3-manifold \(N\), then \(M\) is homeomorphic to \(N\). Definitions of the 3-plane and 1-line-intersection properties are given in the paper.