On Brittenham's theorem (Q5944146)

Let \(M\) and \(N\) be closed orientable 3-manifolds, \(f: M\to N\) a homotopy equivalence, and \(\lambda\), \(\mu\) essential laminations on \(N\), \(M\), respectively. The main result of the paper says that if \(f\) is transverse to \(\lambda\) and pulls back to \(\mu\), then \(f\) is homotopic to a homeomorphism. This fact was first proved by \textit{M. Brittenham} [Topology Appl. 60, No. 3, 249-265 (1994; Zbl 0848.57003)] under the additional hypothesis that \(\lambda\) is transversely oriented. In the paper under review the author shows how to remove the transverse orientability hypothesis. Results of this type may be useful in the resolution of the topological rigidity conjecture for laminar manifolds, that is, if \(f: M\to N\) is a homotopy equivalence between closed orientable irreducible 3-manifolds and \(N\) is laminar, then \(f\) is homotopic to a homeomorphism [see for example \textit{D. Gabai}, AMS/IP Stud. Adv. Math. 2 (pt. 2), 1-33 (1997; Zbl 0888.57025)].