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3-Manifold

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J. Hyam Rubinstein – One of the best experts on this subject based on the ideXlab platform.

  • Four-dimensional Haken cobordism theory
    Illinois Journal of Mathematics, 2016
    Co-Authors: Bell Foozwell, J. Hyam Rubinstein
    Abstract:

    = ∅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

  • Triangulations of 3-Manifolds with essential edges
    arXiv: Geometric Topology, 2014
    Co-Authors: Craig D. Hodgson, J. Hyam Rubinstein, Henry Segerman, Stephan Tillmann
    Abstract:

    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.

  • Four-dimensional Haken cobordism theory
    arXiv: Geometric Topology, 2012
    Co-Authors: Bell Foozwell, J. Hyam Rubinstein
    Abstract:

    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.

Bell Foozwell – One of the best experts on this subject based on the ideXlab platform.

  • Four-dimensional Haken cobordism theory
    Illinois Journal of Mathematics, 2016
    Co-Authors: Bell Foozwell, J. Hyam Rubinstein
    Abstract:

    = ∅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

  • Four-dimensional Haken cobordism theory
    arXiv: Geometric Topology, 2012
    Co-Authors: Bell Foozwell, J. Hyam Rubinstein
    Abstract:

    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.

  • Homotopy hyperbolic 3-Manifolds are hyperbolic
    Annals of Mathematics, 2003
    Co-Authors: David Gabai, G. Robert Meyerhoff, Nathaniel Thurston
    Abstract:

    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.

  • Homotopy Hyperbolic 3-Manifolds are Hyperbolic
    arXiv: Geometric Topology, 1996
    Co-Authors: David Gabai, G. Robert Meyerhoff, Nathaniel Thurston
    Abstract:

    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}