On the stable equivalence of open books in three-manifolds

We show that two open books in a given closed, oriented three-manifold admit isotopic stabilizations, where the stabilization is made by successive plumbings with Hopf bands, if and only if their associated plane fields are homologous. Since this condition is automatically fulfilled in an integral homology sphere, the theorem implies a conjecture of J. Harer, namely, that any fibered link in the three-sphere can be obtained from the unknot by a sequence of plumbings and deplumbings of Hopf bands. The proof presented here involves contact geometry in an essential way. Let M be an oriented three-manifold. An open book in M (also called open book decomposition of M) is a pair (K, θ) consisting of: • a proper one-dimensional submanifold K in M ; • a fibration θ : M \K → S = R/2πZ which, in some neighborhood N = D ×K of K = {0} ×K, is the normal angular coordinate. The submanifold K is called the binding of the open book while the closures of the fibers of θ are named pages. The binding and the pages are cooriented by θ, and hence they are oriented since M is. On the other hand, any page F of an open book (K, θ) completely determines K = ∂F and also (though much less evidently [Ce, LB, Wa]) θ up to isotopy relative to F . Around 1920, as a corollary of his results on branched covers and the braiding of links, J. Alexander proved the existence of open books in any closed oriented three-manifoldM . On the other hand, given an open book in M , many others can be constructed by the following plumbing operation. Let F ⊂ M be a compact surface with boundary and C ⊂ F a proper simple arc. We say that a compact surface F ′ ⊂ M is obtained from F by H-plumbing along C — or, more explicitly, by plumbing a positive/negative Hopf band along C — if F ′ = F ∪ A where A is an annulus in M with the following properties: • the intersection A ∩ F is a tubular neighborhood of C in F ; • the core curve of A bounds a disk in M \F and the linking number of the boundary components of A is equal to ±1. According to results of J. Stallings [St] (see Section A), if F is a page of an open book (K, θ) in M then any surface F ′ obtained from F by H-plumbing is also a page of an open book (K , θ) in M . We will say that the open book (K , θ) itself is obtained from (K, θ) by Date: September 2005. 1991 Mathematics Subject Classification. 57M50, 57R17.