A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture.
First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3-manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem.
Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3-manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the figure-8 knot is virtually Haken.
Suppose that M is a fibered three-manifold whose fiber is a surface of positive genus with one boundary component. Assume that M is not a semi-bundle. We show that infinitely many fillings of M along dM are virtually Haken. It follows that infinitely many Dehn-surgeries of any non-trivial knot in the three-sphere are virtually Haken.
In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [CG] and extends [MMZ].
We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, non-fibered knot manifolds in the cusped Snappea census. For each such manifold M, we compute a number c(M), such that, for any n>c(M), the n-fold cyclic cover of M is large.
We address a conjecture that $π_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $π_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds, which is used in constructing the first known example of minimum Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds.
We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question. An appendix to this paper by Agol, Groves, and Manning proves a generalization of the main result of "Residual finiteness, QCERF and fillings of hyperbolic groups".
We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurston's equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures on the existence of solutions to Thurston's equation and Haken equation are made. Resolutions of these conjecture will lead to a new proof of the Poincaré conjecture without using the Ricci flow. We approach these conjectures by a finite dimensional variational principle so that its critical points are related to solutions to Thurston's gluing equation and Haken's normal surface equation. The action…
Haken n-manifolds have been defined and studied by B. Foozwell and H. Rubinstein in analogy with the classical Haken manifolds of dimension 3, based upon the the theory of boundary patterns developed by K. Johannson. The Euler characteristic of a Haken manifold is analyzed and shown to be equal to the sum of the Charney-Davis invariants of the duals of the boundary complexes of the n-cells at the end of a hierarchy. These dual complexes are shown to be flag complexes. It follows that the Charney-Davis Conjecture is equivalent to the Euler Characteristic Sign Conjecture for Haken manifolds. Sin…
A famous example of Casson and Gordon shows that a Haken 3-manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3-manifold has only finitely many irreducible Heegaard splittings, up to isotopy. This is much stronger than the Waldhausen conjecture. Another immediate corollary is that for any irreducible non-Haken 3-manifold M, there is a number N, such that any two Heegaard splittings of M are equivalent after at most N stabilizations.
We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings. We first give a necessary and sufficient condition, in terms of spectral geometry, for a finitely presented group to have a finite index subgroup with infinite abelianisation. For negatively curved 3-manifolds, we show that this is equivalent to a statement about generalised Heegaard splitt…
We prove a volume inequality for 3-manifolds having C^0 metrics "bent" along a hypersurface, and satisfying certain curvature pinching conditions. The result makes use of Perelman's work on Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume orientable hyperbolic 3-manifold. An appendix by Dunfield compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed emb…
We apply representation theory to study the homology of equivariant Dehn-fillings of a given finite, regular cover of a compact 3-manifold with boundary a torus. This yields a polynomial which gives the rank of the part of the homology carried by the solid tori used for Dehn-filling. The polynomial is a symmetrized form of the group determinant studied by Frobenius and Dedekind. As a corollary every such hyperbolic 3-manifold has infinitely many virtually Haken Dehn-fillings.
Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings of…
In 1999, Kauffman-Harary conjectured that every non-trivial Fox $p$-coloring of a reduced, alternating knot diagram with prime determinant $p$ is heterogeneous. Ten years later this conjecture was proved by W. Mattman and P. Solis. Mathew Williamson generalized this conjecture to alternating virtual knots and proved it for certain families of virtual knots. In the present note, we use the methods of W. Mattman and P. Solis to give an affirmative answer to the Kauffman-Harary conjecture for alternating virtual knots.
We propose a finite dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston's gluing equation and Haken's normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary. Combining the result in this paper and the work of Futer-Guéritaud, Segerman-Tillmann and Luo-Tillmann, we obtain a new finite dimensional variational formulation of the Poncare-conjecture. This provides a step toward a new proof the Poincaré conjec…
Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.
Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye's paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing ba…
Let K be a knot in S^3, and M and M' be distinct Dehn surgeries along K. We investigate when M covers M'. When K is a torus knot, we provide a complete classification of such covers. When K is a hyperbolic knot, we provide partial results in the direction of the conjecture that M never covers M'.
We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken n-manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions $n \le 4$. Our main application is to Coxeter groups whose Davis complexes are manifolds; we show that the natural action of these groups on the Davis complex has a hierarchy. Our second result is that the Singer conjecture is equivalent to the cocompact action dimension conjecture, which is a statement about all groups, not just fundamental groups of closed aspherical manifold…
The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle. Our main results are 1…
We study virtual isotopy sequences with classical initial and final diagrams, asking when such a sequence can be changed into a classical isotopy sequence by replacing virtual crossings with classical crossings. An example of a sequence for which no such virtual crossing realization exists is given. A conjecture on conditions for realizability of virtual isotopy sequences is proposed, and a sufficient condition for realizability is found. The conjecture is reformulated in terms of 2-knots and knots in thickened surfaces.
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism $f$ of a closed manifold is isotopic to a map realizing the Nielsen number of $f$, which is a lower bound for the number of fixed points among all maps homotopic to $f$. The main theorem of this paper proves this conjecture for all orientation preserving maps on geometric or Haken 3-manifolds. It will also be shown that on many manifolds all maps are isotopic to fixed point free maps. The proof is based on the understanding of homeomorphisms on 2-orbifolds and 3-manifolds. Thurston's classification of surface homeomorphis…
The article has been withdrawn by the author. Wolfgang Lueck and Peter Linnell pointed out that the proof of Lemma 3.8 does not apply to the unrestricted case of wreath product. It is not clear at this stage how to complete the proof of Theorem 3.1 using the present version of Lemma 3.8. The valid results originating from this article will be added in a later paper.
The virtual Betti number conjecture states that any hyperbolic three-manifold has a finite cover with positive first Betti number. We show that this would follow if it were known that the derived series of the fundamental group $G$ of a hyperbolic three-manifold satisfies a certain stability property. The stability property is the statement that if all the quotients $G^i/G^{i+1}$ of the derived series $G^i$ of $G$ are finite, then the derived series stabilises. The proof involves basic facts regarding finite group actions on homology spheres.
This article has two purposes. In \cite{R3} (math.KT/0405211) we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by \cal C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of \cal C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M in {\cal C} there is a closed 3-manifold P su…
We prove the inequality $$ \dim_{mc}\Wi M\le n-2$$ for the macroscopic dimension of the universal covers $\Wi M$ of almost spin $n$-manifolds $M$ with positive scalar curvature whose fundamental group $π_1(M)$ is a virtual duality group that satisfies the coarse Baum-Connes conjecture.
This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that π_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. This result should be viewed as an analogue of Dehn's Lemma for π_1-injective singular surfaces. The geometric application states that if M is a closed orientable hyperbolic 3-manifold with volume less than 3.08 then the rank of H_1(M;Z/2Z) is at most 6. The proof of the geometric t…
We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. In fact, this follows from a more general result we prove here, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to pr…
Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether $H_n(T^n)^+= 0$ for the $n$-torus $T^n$ where $H_n(T^n)^+\subset H_n(T^n)$ is the subgroup of homology classes which can…
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.
We prove the Farrell-Jones Conjecture for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture by Bartels-Lück-Reich, as well as the structure theory of mapping tori by Dahmani-Krishna.
Given a reducible $3$-manifold $M$ with an aspherical summand in its prime decomposition and a homeomorphism $f\colon M\to M$, we construct a map of degree one from a finite cover of $M\rtimes_f S^1$ to a mapping torus of a certain aspherical $3$-manifold. We deduce that $M\rtimes_f S^1$ has virtually infinite first Betti number, except when all aspherical summands of $M$ are virtual $T^2$-bundles. This verifies all cases of a conjecture of T.-J. Li and Y. Ni, that any mapping torus of a reducible $3$-manifold $M$ not covered by $S^2\times S^1$ has virtually infinite first Betti number, except…
