Homotopy implies isotopy for some Seifert fibre spaces
''Let M be a closed orientable irreducible 3-manifold. It is a longstanding problem to decide if homotopic homeomorphisms of M must be isotopic. No counterexamples are known. Waldhausen showed that this is true if M is Haken (sufficiently large).'' The Seifert fibre spaces - i.e. closed 3-manifolds with a foliation by circles - not covered by Waldhausen's theorem have the 2-sphere as base space (space of leaves) and at most 3 exceptional fibers. In the present paper it is shown that homotopic homeomorphisms are isotopic for Seifert fibre spaces with 3 exceptional fibers of orders p,q and r with p,q\(\geq 4\). This takes care of most Seifert fibre spaces with infinite fundamental group. The proof uses immersed surfaces instead of embedded ones in the Haken case and carefully analyses the intersection of an immersed torus with its image under a homeomorphism homotopic to the identity. For various Seifert fibre spaces with finite fundamental group, especially lens spaces, the result is also known. The remaining cases are the Seifert fibre spaces (finite or infinite fundamental group) of type \((p,q,r)=(2,3,p)\) and (3,3,q). A solution for these cases has recently been announced in a preprint by M. Boileau and J. P. Otal.
scientific article; zbMATH DE number 3921538