Singular surfaces, mod 2 homology, and hyperbolic volume, I

Ian Agol, Marc Culler, Peter B. Shalen
2010 Transactions of the American Mathematical Society
If M is a simple, closed, orientable 3-manifold such that π 1 (M ) contains a genus-g surface group, and if H 1 (M ; Z 2 ) has rank at least 4g − 1, we show that M contains an embedded closed incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the rank of H 1 (M ; Z 2 ) is at most 6. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
more » ... AR SURFACES, MOD 2 HOMOLOGY, HYPERBOLIC VOLUME, I 3465 ofφ * : H 1 (K; Z 2 ) → H 1 (N n ; Z 2 ) implies that the rank of H 1 (N n ; Z 2 ) is at most 2g. It follows that in this case N n is a proper submanifold of M n , and hence ∂N n = ∅. We in fact show, using elementary arguments based on Poincaré-Lefschetz duality, that if the mapφ * : H 2 (K; Z) → H 2 (N n ; Z) is trivial, then ∂N n has a component F of genus at most g. In the case whereφ * : H 2 (K; Z) → H 2 (N n ; Z) is nontrivial, we use Gabai's results from [7] to show that N n contains a non-separating incompressible closed surface F of genus at most g. The rest of the proof consists of showing that if a given M j , with 0 < j ≤ n, contains a closed incompressible surface of genus at most g, then N j−1 also contains such a surface. The surface in N j−1 will be incompressible in M j−1 , as well as in N j−1 , because ∂N j−1 is incompressible in M j−1 . It is at this step that we need to know that closed manifolds in the tower have first homology with Z 2 -coefficients of rank at least 4g−2. Indeed, it is a consequence of Proposition 4.4 that if a 2-sheeted covering of a simple compact 3-manifold N contains a closed incompressible surface of genus at most g, then N itself must contain such a surface, unless N is closed and H 1 (N ; Z 2 ) has rank at most 4g − 3. Proposition 4.4 involves the notion of a "book of I-bundles" which we define formally in Definitions 2.2. Books of I-bundles in PL 3-manifolds arise naturally as neighborhoods of "books of surfaces" (Definition 2.6). We may think of a book of surfaces as being constructed from a 2-manifold with boundaryΠ, whose components have Euler characteristic ≤ 0, and a closed 1-manifold Ψ, by attaching ∂Π to Ψ by a covering map. The components of Ψ and Π = intΠ are, respectively, "bindings" and "pages." A book of I-bundles comes equipped with a corresponding decomposition into "pages" which are I-bundles over surfaces, and "bindings" which are solid tori. (In the informal discussion that we give in this Introduction, the extra structure defined by the decomposition will be suppressed from the notation.) With these notions as background we shall now sketch the proof of Proposition 4.4. An incompressible surface F in a two-sheeted covering space of N , if it is in general position, projects to N via a map which has only double-curve singularities. After routine modifications one obtains a map from F to N with the additional property that its double curves are homotopically non-trivial. In particular, the image of such a map is a book of surfaces X. A regular neighborhood W of X in N is then a book of I-bundles, which has Euler characteristic ≥ 2 − 2g if F has genus at most g. Using the simplicity of N one can then produce a book of I-bundles V with W ⊂ V ⊂ N and χ(W ) ≥ 2 − 2g, such that each page of W has strictly negative Euler characteristic. (This step is handled by Lemma 2.5.) We now distinguish two cases. In the case where some page P 0 of V has the property that P 0 ∩ ∂V is contained in a single component of ∂V , we show that by splitting bindings of the book of surfaces X, one can produce an embedded (possibly disconnected) closed, orientable surface S which is homologically non-trivial in N . Ambient surgery on S in N then produces a non-empty incompressible surface whose components have genus at most g. In the case where no such page P 0 exists, a Euler characteristic calculation shows that the boundary components of V have genus at most g. In this case, ambient surgery on ∂V produces a non-empty incompressible surface whose components have genus at most g. We show that this surface is non-empty unless V carries π 1 (N ). But for a book of I-bundles V whose Euler characteristic is at least 2 − 2g and whose pages are all of negative License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use IAN AGOL, MARC CULLER, AND PETER B. SHALEN Euler characteristic, one can show that H 1 (V ; Z 2 ) has rank at most 4g − 3 (this is included in Lemma 2.11); so in the case where V carries π 1 (N ), the rank of H 1 (N ; Z 2 ) is at most 4g − 3. The details and background needed for the proof of Proposition 4.4 occupy Sections 2-4. Section 5 provides the combinatorial background needed to construct the tower, while Sections 6 and 7 provide the homological background. The application of Gabai's results mentioned above appears in Section 7. The material on towers proper, and the proof of the main topological theorem and its corollary, are given in Section 8, and the geometric applications are given in Section 9. We are grateful to Michael Siler for pointing out a number of errors in an earlier version of the manuscript. The rest of this Introduction will be devoted to indicating some conventions that will be used in the rest of the paper. 1.1. In general, if X and Y are subsets of a set, we denote by X \ Y the set of elements of X that do not belong to Y . In the case where we know that Y ⊂ X and wish to emphasize this, we will write X − Y for X \ Y . 1.2. A manifold may have a boundary. If M is a manifold, we shall denote the boundary of M by ∂M and its interior M − ∂M by int M . In many of our results about manifolds of dimension ≤ 3 we do not specify a category. These results may be interpreted in the category in which manifolds are topological, PL or smooth, and submanifolds are, respectively, locally flat, PL or smooth. These three categories are equivalent in these low dimensions as far as classification is concerned. In much of the paper the proofs are done in the PL category, but the applications to hyperbolic manifolds in Section 9 are carried out in the smooth category. 1.3. A (possibly disconnected) codimension-1 submanifold S of a manifold M is said to be separating if M can be written as the union of two 3-dimensional submanifolds M 1 and M 2 such that M 1 ∩ M 2 = S.