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Search: MSC category 57M50 ( Geometric structures on low-dimensional manifolds )

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1. CJM 2013 (vol 66 pp. 141)

Caillat-Gibert, Shanti; Matignon, Daniel
Existence of Taut Foliations on Seifert Fibered Homology $3$-spheres
This paper concerns the problem of existence of taut foliations among $3$-manifolds. Since the contribution of David Gabai, we know that closed $3$-manifolds with non-trivial second homology group admit a taut foliation. The essential part of this paper focuses on Seifert fibered homology $3$-spheres. The result is quite different if they are integral or rational but non-integral homology $3$-spheres. Concerning integral homology $3$-spheres, we can see that all but the $3$-sphere and the Poincaré $3$-sphere admit a taut foliation. Concerning non-integral homology $3$-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology $3$-spheres.

Keywords:homology 3-spheres, taut foliation, Seifert-fibered 3-manifolds
Categories:57M25, 57M50, 57N10, 57M15

2. CJM 2010 (vol 62 pp. 994)

Breslin, William
Curvature Bounds for Surfaces in Hyperbolic 3-Manifolds
A triangulation of a hyperbolic $3$-manifold is \emph{L-thick} if each tetrahedron having all vertices in the thick part of $M$ is $L$-bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant $L$ such that every complete hyperbolic $3$-manifold has an $L$-thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of $\pi_1$-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic $3$-manifolds.

Category:57M50

3. CJM 2008 (vol 60 pp. 164)

Lee, Sangyop; Teragaito, Masakazu
Boundary Structure of Hyperbolic $3$-Manifolds Admitting Annular and Toroidal Fillings at Large Distance
For a hyperbolic $3$-manifold $M$ with a torus boundary component, all but finitely many Dehn fillings yield hyperbolic $3$-manifolds. In this paper, we will focus on the situation where $M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling. For such a situation, Gordon gave an upper bound of $5$ for the distance between such slopes. Furthermore, the distance $4$ is realized only by two specific manifolds, and $5$ is realized by a single manifold. These manifolds all have a union of two tori as their boundaries. Also, there is a manifold with three tori as its boundary which realizes the distance $3$. We show that if the distance is $3$ then the boundary of the manifold consists of at most three tori.

Keywords:Dehn filling, annular filling, toroidal filling, knot
Categories:57M50, 57N10

4. CJM 2007 (vol 59 pp. 418)

Stoimenow, A.
On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials
It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree $9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d \le 10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

Categories:57M25, 57M27, 20F36, 57M50

5. CJM 2006 (vol 58 pp. 673)

Bart, Anneke; Scannell, Kevin P.
The Generalized Cuspidal Cohomology Problem
Let $\Gamma \subset \SO(3,1)$ be a lattice. The well known \emph{bending deformations}, introduced by \linebreak Thurston and Apanasov, can be used to construct non-trivial curves of representations of $\Gamma$ into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in $\H^1(\Gamma, \mink{4})$. Our main result generalizes this construction of cohomology to the context of ``branched'' totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in $S^3$ which is not infinitesimally rigid in $\SO(4,1)$. The first order deformations of this link complement are supported on a piecewise totally geodesic $2$-complex.

Categories:57M50, 22E40

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