Homotopy and isotopy in dimension three. (English) Zbl 0805.57008

Hass, Joel; Scott, Peter

Comment. Math. Helv. 68, No. 3, 341-364 (1993).

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.

Reviewer: L.P.Neuwirth (Princeton)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

homotopic homeomorphisms; isotopic; 3-plane and 1-line-intersection properties; homotopy equivalent to an irreducible 3-manifold

Citations:

Zbl 0771.57007