Homotopy and isotopy in dimension three

Abstract

Let M be a closed p2-irreducible 3-manifold. It is a long standing problem to decide if homotopic homeomorphisms of M must be isotopic. The answer is now known to be affirmative if M is Haken, [Wal], see also [L], or if M is a Seifert fiber space [Ho-R] [Bon] [B-R] [A] [R] [Scl] [B-O], and for a few other special manifolds [B-R]. Thus is now seems reasonable to conjecture that the answer is always affirmative. However, if one considers reducible manifolds, there is a counter example [F W]. In this paper, we further enlarge the class of 3-manifolds for which the above conjecture can be proved. If a closed P2-irreducible 3-manifold is non-orientable, it must be Haken, so we consider only orientable 3-manifolds in the rest of this paper. Let M be an orientable 3-manifold, let F be a closed orientable surface not S 2 and let f: F ~ M be an immersion which injects n~ (F). Let Mr denote the cover of M such that nj (Mr) equals f,(nl(F)) and let M denote the universal cover of M. We will suppose that the lift of f into MF is an embedding. (Note that this is automatic iff is least area in the smooth or PL sense.) Thus the pre-image in M of f(F) consists of an embedded plane /7 which covers F in M,, and the translates of // by hi(M). We will say that f has the k-plane property if, given k distinct translates of H, some pair is disjoint. In this paper we will consider the case when k equals 3. A map with the 3-plane property has no transverse triple points. We will say that f has the 1-line-intersection property if two distinct translates of /7 are disjoint or intersect transversely in a single line. The main result of this paper is 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 z, and an immersion f:F~M which injects nl(F) and has the 3-plane and l-line-intersection properties, then homotopic homeomorphisms of M are isotopic. In [H-S], we show that if M satisfies the hypotheses of this theorem, and M is homotopy equivalent to an irreducible 3-manifold N, then M and N are homeomor