The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
J. Hyam Rubinstein - One of the best experts on this subject based on the ideXlab platform.
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Four-dimensional Haken cobordism theory
Illinois Journal of Mathematics, 2016Co-Authors: Bell Foozwell, J. Hyam RubinsteinAbstract:= ∅is of particular interest.Our proof of this result will be obtained in a number of steps. The firststep is to show that if M is a torus-bundle over a circle, then there is aHaken 4–manifold Wwith boundary ∂W= M. We do this in section 4. Wethen show a similar result for general surface-bundles in section 5. To showthat Haken manifolds satisfy our main result, we use a result of Gabai [6]and Ni [12] in section 6.It is well-known that all closed 3–manifolds are null cobordant, i.e boundcompact 4–manifolds. Davis, Januszkiewicz and Weinberger [2] followingon from work in [1], show that if an aspherical closed n–manifold is nullcobordant, then it bounds an aspherical (n+1)–manifold, and furthermore,the inclusion map of the boundaryis π
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Triangulations of 3-Manifolds with essential edges
arXiv: Geometric Topology, 2014Co-Authors: Craig D. Hodgson, J. Hyam Rubinstein, Henry Segerman, Stephan TillmannAbstract:We define essential and strongly essential triangulations of 3-Manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-Manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold. We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3-Manifold to be essential, and a strict angle structure is a sufficient condition for a triangulation to be strongly essential. Moreover, algorithms to test whether a triangulation of a 3-Manifold is essential or strongly essential are given.
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Four-dimensional Haken cobordism theory
arXiv: Geometric Topology, 2012Co-Authors: Bell Foozwell, J. Hyam RubinsteinAbstract:Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $\pi_1(W)$ is injective. In this paper we prove that there are 4-dimensional Haken cobordisms whose boundary consists of any two closed Haken 3-Manifolds. In particular, each closed Haken 3-Manifold is the $\pi_1$-injective boundary of some Haken 4-manifold.
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Diffeomorphisms of Elliptic 3-Manifolds
Lecture Notes in Mathematics, 2012Co-Authors: Sungbok Hong, John Kalliongis, Darryl Mccullough, J. Hyam RubinsteinAbstract:This work concerns the diffeomorphism groups of 3-Manifolds, in particular of elliptic 3-Manifolds. These are the closed 3-Manifolds that admit a Riemannian metric of constant positive curvature, now known to be exactly the closed 3-Manifolds that have a finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-Manifold M, the inclusion from the isometry group of M to its diffeomorphism group 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. The main results establish the Smale Conjecture for all elliptic 3-Manifolds containing geometrically incompressible Klein bottles, and for all lens spaces L(m,q) with m at least 3. Additional results imply that for a Haken Seifert-fibered 3 manifold V, the space of Seifert fiberings has contractible components, and apart from a small list of known exceptions, is contractible. Considerable foundational and background material on diffeomorphism groups is included
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0-Efficient Triangulations of 3-Manifolds
Journal of Differential Geometry, 2003Co-Authors: William Jaco, J. Hyam RubinsteinAbstract:efficient triangulations of 3-Manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-Manifold M can be modified to a 0-efficient triangulation orM can be shown to be one of the manifolds S 3 , RP 3 or L(3,1). Similarly, any triangulation of a com- pact, orientable, irreducible, @-irreducible 3-Manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal tr is de- fined. It is shown if M is a compact, orientable, irreducible, @-irreducible 3-Manifold having no essential annuli and distinct from the 3-cell, then ◦ M admits an ideal triangulation; furthermore, it is shown that any ideal trian- gulation of such a 3-Manifold can be modified to a 0-efficient ideal triangula- tion. A 0-efficient triangulation of a closed manifold has only one vertex or the manifold is S 3 and the triangulation has precisely two vertices. 0-efficient tri- angulations of 3-Manifolds with boundary, and distinct from the 3-cell, have all their vertices in the boundary and then just one vertex in each bound- ary component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal sur- face. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-Manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-Manifold, an alternate algorithm for the 3-sphere recognition prob- lem, results on edges of low valence in minimal triangulations of 3-Manifolds, and a construction of irreducible knots in closed 3-Manifolds.
Bell Foozwell - One of the best experts on this subject based on the ideXlab platform.
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Four-dimensional Haken cobordism theory
Illinois Journal of Mathematics, 2016Co-Authors: Bell Foozwell, J. Hyam RubinsteinAbstract:= ∅is of particular interest.Our proof of this result will be obtained in a number of steps. The firststep is to show that if M is a torus-bundle over a circle, then there is aHaken 4–manifold Wwith boundary ∂W= M. We do this in section 4. Wethen show a similar result for general surface-bundles in section 5. To showthat Haken manifolds satisfy our main result, we use a result of Gabai [6]and Ni [12] in section 6.It is well-known that all closed 3–manifolds are null cobordant, i.e boundcompact 4–manifolds. Davis, Januszkiewicz and Weinberger [2] followingon from work in [1], show that if an aspherical closed n–manifold is nullcobordant, then it bounds an aspherical (n+1)–manifold, and furthermore,the inclusion map of the boundaryis π
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Four-dimensional Haken cobordism theory
arXiv: Geometric Topology, 2012Co-Authors: Bell Foozwell, J. Hyam RubinsteinAbstract:Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $\pi_1(W)$ is injective. In this paper we prove that there are 4-dimensional Haken cobordisms whose boundary consists of any two closed Haken 3-Manifolds. In particular, each closed Haken 3-Manifold is the $\pi_1$-injective boundary of some Haken 4-manifold.
Nathaniel Thurston - One of the best experts on this subject based on the ideXlab platform.
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Homotopy hyperbolic 3-Manifolds are hyperbolic
Annals of Mathematics, 2003Co-Authors: David Gabai, G. Robert Meyerhoff, Nathaniel ThurstonAbstract:This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-Manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-Manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-Manifold.
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Homotopy Hyperbolic 3-Manifolds are Hyperbolic
arXiv: Geometric Topology, 1996Co-Authors: David Gabai, G. Robert Meyerhoff, Nathaniel ThurstonAbstract:This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-Manifolds. This technique is used to complete the proof of several long-standing rigidity conjectures in 3-Manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-Manifold. We prove the following result: \it\noindent Let $N$ be a closed hyperbolic 3-Manifold. Then \begin{enumerate} \item[(1)] If $f\colon M \to N$ is a homotopy equivalence where $M$ is a closed irreducible 3-Manifold, then $f$ is homotopic to a homeomorphism. \item[(2)] If $f,g\colon M\to N$ are homotopic homeomorphisms, then $f$ is isotopic to $g$. \item[(3)] The space of hyperbolic metrics on $N$ is path connected. \end{enumerate}
David Gabai - One of the best experts on this subject based on the ideXlab platform.
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Homotopy hyperbolic 3-Manifolds are hyperbolic
Annals of Mathematics, 2003Co-Authors: David Gabai, G. Robert Meyerhoff, Nathaniel ThurstonAbstract:This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-Manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-Manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-Manifold.
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Homotopy Hyperbolic 3-Manifolds are Hyperbolic
arXiv: Geometric Topology, 1996Co-Authors: David Gabai, G. Robert Meyerhoff, Nathaniel ThurstonAbstract:This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-Manifolds. This technique is used to complete the proof of several long-standing rigidity conjectures in 3-Manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-Manifold. We prove the following result: \it\noindent Let $N$ be a closed hyperbolic 3-Manifold. Then \begin{enumerate} \item[(1)] If $f\colon M \to N$ is a homotopy equivalence where $M$ is a closed irreducible 3-Manifold, then $f$ is homotopic to a homeomorphism. \item[(2)] If $f,g\colon M\to N$ are homotopic homeomorphisms, then $f$ is isotopic to $g$. \item[(3)] The space of hyperbolic metrics on $N$ is path connected. \end{enumerate}
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On the geometric and topological rigidity of hyperbolic 3-Manifolds
arXiv: Geometric Topology, 1994Co-Authors: David GabaiAbstract:A homotopy equivalence between a hyperbolic 3-Manifold and a closed irreducible 3-Manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3-Manifolds which do not satisfy this condition.
Suhyoung Choi - One of the best experts on this subject based on the ideXlab platform.
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Convex and Concave Decompositions of Affine 3-Manifolds
Bulletin of the Brazilian Mathematical Society New Series, 2019Co-Authors: Suhyoung ChoiAbstract:A (flat) affine 3-Manifold is a 3-Manifold with an atlas of charts to an affine space \({{\mathbb {R}}}^3\) with transition maps in the affine transformation group \({\mathbf {Aff}}({{\mathbb {R}}}^3)\). We will show that a connected closed affine 3-Manifold is either an affine Hopf 3-Manifold or decomposes canonically to concave affine submanifolds with incompressible boundary, toral \(\pi \)-submanifolds and 2-convex affine manifolds, each of which is an irreducible 3-Manifold. It follows that if there is no toral \(\pi \)-submanifold, then M is prime. Finally, we prove that if a closed affine manifold is covered by a connected open set in \({{\mathbb {R}}}^{3}\), then M is irreducible or is an affine Hopf manifold.
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Convex and concave decompositions of affine $3$-manifolds
arXiv: Geometric Topology, 2014Co-Authors: Suhyoung ChoiAbstract:A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space $\mathbb{R}^3$ with transition maps in the affine transformation group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine $3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to concave affine submanifolds with incompressible boundary, toral $\pi$-submanifolds and $2$-convex affine manifolds, each of which is an irreducible $3$-manifold. It follows that if there is no toral $\pi$-submanifold, then $M$ is prime. Finally, we prove that if a closed affine manifold is covered by a domain in $\mathbb{R}^{n}$, then $M$ is irreducible or is an affine Hopf manifold.
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Closed flat affine 3-Manifolds are prime
arXiv: Geometric Topology, 2014Co-Authors: Suhyoung ChoiAbstract:An (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space ${\mathbf R}^3$ with transition maps in the affine transformation group $Aff({\mathbf R}^3)$. Equivalently an affine $3$-manifold is a $3$-manifold with a flat torsion-free affine connection. We show that a closed affine $3$-manifold is either irreducible or is finitely covered by an affine Hopf manifold. A real projective $3$-manifold is a manifold with an atlas of charts to a real projective space ${\mathbf R} P^3$ with transition maps in the projective transformation group $PGL(4, {\mathbf R})$. Using the convex concave decomposition of real projective manifolds, we will show that a closed real projective $3$-manifold decomposes into concave affine submanifolds, toral $\pi$-submanifolds and $2$-convex real projective manifolds.
