A Proof of a Homeomorphism Theorem of Waldhausen

We give a short proof of Waldhausen’s homeomorphism theorem for orientable Haken manifolds. We give here a short proof of Waldhausen’s homeomorphism theorem [4] for orientable Haken manifolds. Recall that a compact irreducible 3-manifold is Haken (i.e., it has a hierarchy) if and only if it is sufficiently large (i.e., it admits an incompressible surface). This follows from the Haken-Kneser argument which shows that there is a bound on the number closed disjoint non-parallel incompressible surfaces in a compact irreducible 3-manifold. A readable proof of this is given in Hempel’s book [2]. We now state Waldhausen’s homeomorphism theorem. The proof that follows is a variant of the standard arguments. In the standard proofs, some of the special cases of Waldhausen’s theorem are proved during the course of the proof. We avoid this by appealing to a result of Hopf and Kneser that algebraic and geometric degrees are equal (see [1] and [3]). We will always assume that we are in the orientable case and that all our submanifolds are two sided. The argument also works in the case of non-orientable Haken manifolds. Theorem 1. Suppose (M,∂M) and (N, ∂N) are compact irreducible 3-manifolds with (possibly empty) boundary and N is sufficiently large. Then any homotopy equivalence f : (M,∂M) → (N, ∂N), whose restriction to ∂M is a homeomophism, is homotopic to a homeomorphism. We first consider the case when ∂N is non-empty. Then there exists an incompressible, ∂-incompressible surface S ⊂ N such that ∂S 6= φ. Let f(S) = ∪ i=1 Fi, where each Fi is connected. The following lemma is by now standard and stems from the work of Whitehead and Stallings. Lemma 2. After a homotopy of f , we may assume that each Fi is essential. Henceforth assume that each Fi is essential. Further, we assume that f is a homeomorphism in a neighbourhood of ∂M . Proposition 3. f(S) is connected and f : f(S) → S is degree-one. Proof. Let fk : Fk → S, 1 ≤ k ≤ m be the restrictions of f . We shall use the fact that the degree of a map can be computed locally. Thus, let ∂S = γ1∪· · ·∪γm. Then, as f is a homeomorphism in a neighbourhood of the boundary, for p in a neighbourhood of any γi, the inverse image f (p) consists of a single point. It follows that