Mom technology and volumes of hyperbolic 3-manifolds

David Gabai, Robert Meyerhoff, Peter Milley
2011 Commentarii Mathematici Helvetici  
This paper introduces Mom technology to understand low volume hyperbolic 3manifolds; it is used in [GMM3] and [M1] to show that the Weeks manifold is the unique closed orientable hyperbolic 3-manifold of least volume. Here we enumerate the hyperbolic Mom-n manifolds for n Ä 3, offer a conjectural enumeration when n D 4, and establish important technical results about embedding hyperbolic Mom-n manifolds into hyperbolic 3-manifolds. Mathematics Subject Classification (2010). Primary 57M50;
more » ... rimary 57M50; Secondary 51M10, 51M25. goal of this paper is to begin to explain, in an organized fashion, this phenomenon as summarized by the following: Hyperbolic Complexity Conjecture 0.1 (Thurston, Weeks, Matveev-Fomenko). The complete low-volume hyperbolic 3-manifolds can be obtained by filling cusped hyperbolic 3-manifolds of small topological complexity. Remark 0.2. Part of the challenge of this conjecture is to clarify the undefined adjectives low and small. We propose that the manifolds obtained by Dehn filling a Mom-n, n Ä 3 (or n Ä 4) include all the reasonably low volume complete hyperbolic 3-manifolds. Indeed, in [GMM3] we show that the 10 lowest volume 1-cusped hyperbolic 3-manifolds (or equivalently the manifolds of volume Ä 2:848) are obtained by filling a Mom-n manifold with n Ä 3. Furthermore, based on computer calculation all the cusped orientable manifolds in the SnapPea census of volume Ä 3:177, Ä 4:059, and Ä 5:468 are obtained by respectively filling Mom-2, Mom-3, and Mom-4 manifolds. (This includes 2948 manifolds.) In the late 1970s, Troels Jorgensen proved that for any positive constant C there is a finite collection of cusped hyperbolic 3-manifolds from which all complete hyperbolic 3-manifolds of volume less than or equal to C can be obtained by Dehn filling. If it could be proved that all cusped manifolds of volume Ä 5:0 can be obtained by filling a Mom-n manifold with n Ä 4, and if a corresponding result (with a somewhat lower volume bound) could be proved for closed manifolds, then we would have a concrete and satisfying realization of Jorgensen's Theorem for "low" values of C . A special case of the Hyperbolic Complexity Conjecture is the following: ). The Weeks manifold M W , obtained by .5; 1/, .5; 2/ surgery on the two components of the Whitehead link, is the unique orientable hyperbolic 3manifold of minimum volume. Note that the volume of M W is 0:942 : : : : The proof required understanding all the very low volume manifolds obtained by filling a Mom-n manifold for n Ä 3, an analysis carried out in [M1]. All manifolds in this paper will be orientable and all hyperbolic structures are complete. We call a compact manifold hyperbolic if its interior supports a complete hyperbolic structure of finite volume. Unless said otherwise, all compact hyperbolic 3-manifolds in this paper are compactified cusped hyperbolic 3-manifolds. Definition 0.4. A Mom-n structure .M; T; / consists of a compact 3-manifold M whose boundary is a union of tori, a preferred boundary component T , and a handle decomposition of the following type. Starting from T I , n 1-handles and n Vol. 86 (2011) Mom technology and volumes of hyperbolic 3-manifolds Definition 1.2. Let M be a compact connected 3-manifold M with B @M a compact surface which may be either disconnected or empty. A handle structure Vol. 86 (2011) Mom technology and volumes of hyperbolic 3-manifolds 149 on .M; B/ is the structure obtained by starting with B I , adding a finite union of 0-handles, then attaching finitely many 1 and 2-handles to B f1g and the 0-handles. We call B I (resp. B I [ 0-handles) the base (resp. extended base) and say that the handle structure is based on B. Most of the time B will be a component of @M . We strongly recommend that the reader think only about this case until absolutely necessary. Sometimes, B will be an 1 -injective annulus in @M . In that case, M will be a manifold with corners, the corners being @B f0; 1g; see the manifold M 1 in Example 7.1 for a typical case of this. In the most general case (in this paper), B will have several components, each being either an annulus or torus as above. In this paper all k-handles will attach to lower dimensional handles in a standard way. E.g. if a 1-handle is parametrized by D 2 I , then D 2 @I is the attaching zone and a 2-handle will attach to the 1-handle in regions of the form˛ I , wherę is an embedded arc in @D 2 . For a precise statement see Definition 2.1. The valence of a 1-handle is the number of times, counted with multiplicity, the various 2-handles run over it and the valence of a 2-handle is the number of 1-handles, counted with multiplicity, it runs over. Following the terminology of Schubert [Sch] and Matveev [Mv1] we call the 0-handles, 1-handles and 2-handles balls, beams and plates respectively. We call islands (resp. bridges) the intersection of the extended base with beams (resp. plates) and the components of the closure of the complement of the islands and bridges in B f1g [ @.0-handles/ are the lakes. We say that is full if each lake is a disc. If B D ;, then we say that is a classical handle structure. Let M be a compact 3-manifold with @M a union of tori and let T be a component of @M . We say that .M; T; / is a weak Mom-n if is a handle structure based on T without 0-handles or 3-handles, such that each 1-handle is of valence 2 and each 2-handle is of valence 2 or 3. Furthermore, there are exactly n 2-handles of valence 3. A weak Mom-n with no valence-2 2-handles is a Mom-n. A weak Mom-n is strictly weak if there exists a valence-2 2-handle. Remark 1.3. For Euler characteristic reasons, if .M; T; / is a weak Mom-n, then has the same number of 1 and 2-handles. The following is a well-known existence result stated in our language. Proposition 1.4. A compact 3-manifold M has a weak Mom-structure if and only if @M is a union of at least two tori. Proof. If M has a weak Mom-n structure, then by definition all of its boundary components are tori and there is at least one such boundary component. Further, because there are no 3-handles in , there must be another (torus) boundary component. D. Gabai, R. Meyerhoff, and P. Milley CMH Lemma 1.11. If M is compact, irreducible, and non-elementary, and if @M is a union of tori, then @M is incompressible. Proof. If M 0 is obtained by compressing @M , then @M 0 has a 2-sphere component S that bounds a 3-ball B in M and hence in M 0 . It follows that either the compression is inessential or M is a solid torus. The latter implies that M is elementary. Definition 1.12. Let N be a compact hyperbolic 3-manifold. An internal Mom-n structure on N consists of a non-elementary embedding f W M ! N where .M; T; / is a Mom-n and each component of @M is either boundary parallel in N or bounds a solid torus in N . We will sometimes suppress mention of the embedding and simply say that .M; T; / is an internal Mom-n structure on N . In the natural way we define the notion of weak internal Mom-n structure on N . Lemma 1.13. A non-elementary embedding of the Mom-n manifold M into the compact hyperbolic 3-manifold N will fail to give an internal Mom-n structure on N if and only if some component of @M maps to a convolutube. In that case, a reimbedding of M , supported in a neighborhood of the convolutubes gives rise to an internal Mom-n structure on N . Proof. As in the proof of Lemma 1.7, there exists a finite set of pairwise disjoint embedded 3-balls such that each convolutube is contained in exactly one such ball. Reimbed M in N by unknotting each convolutube. The boundary components of the resulting reimbedded M will be either tubes or essential annuli. Further the reimbedded M will have the same 1 -image as M .
doi:10.4171/cmh/221 fatcat:hngzba3l5zch7osgcpinoxdmwi