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3Manifold
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J. Hyam Rubinstein – One of the best experts on this subject based on the ideXlab platform.

Fourdimensional Haken cobordism theory
Illinois Journal of Mathematics, 2016CoAuthors: 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 torusbundle 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 surfacebundles 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 wellknown 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 3Manifolds with essential edges
arXiv: Geometric Topology, 2014CoAuthors: Craig D. Hodgson, J. Hyam Rubinstein, Henry Segerman, Stephan TillmannAbstract:We define essential and strongly essential triangulations of 3Manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3Manifolds, EpsteinPenner 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 semiangle structure is a sufficient condition for a triangulation of a 3Manifold 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 3Manifold is essential or strongly essential are given.

Fourdimensional Haken cobordism theory
arXiv: Geometric Topology, 2012CoAuthors: 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 4dimensional Haken cobordisms whose boundary consists of any two closed Haken 3Manifolds. In particular, each closed Haken 3Manifold is the $\pi_1$injective boundary of some Haken 4manifold.
Bell Foozwell – One of the best experts on this subject based on the ideXlab platform.

Fourdimensional Haken cobordism theory
Illinois Journal of Mathematics, 2016CoAuthors: 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 torusbundle 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 surfacebundles 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 wellknown 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

Fourdimensional Haken cobordism theory
arXiv: Geometric Topology, 2012CoAuthors: 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 4dimensional Haken cobordisms whose boundary consists of any two closed Haken 3Manifolds. In particular, each closed Haken 3Manifold is the $\pi_1$injective boundary of some Haken 4manifold.
Nathaniel Thurston – One of the best experts on this subject based on the ideXlab platform.

Homotopy hyperbolic 3Manifolds are hyperbolic
Annals of Mathematics, 2003CoAuthors: David Gabai, G. Robert Meyerhoff, Nathaniel ThurstonAbstract:This paper introduces a rigorous computerassisted procedure for analyzing hyperbolic 3Manifolds. This procedure is used to complete the proof of several longstanding rigidity conjectures in 3Manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3Manifold.

Homotopy Hyperbolic 3Manifolds are Hyperbolic
arXiv: Geometric Topology, 1996CoAuthors: David Gabai, G. Robert Meyerhoff, Nathaniel ThurstonAbstract:This paper introduces a rigorous computerassisted procedure for analyzing hyperbolic 3Manifolds. This technique is used to complete the proof of several longstanding rigidity conjectures in 3Manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3Manifold. We prove the following result: \it\noindent Let $N$ be a closed hyperbolic 3Manifold. Then \begin{enumerate} \item[(1)] If $f\colon M \to N$ is a homotopy equivalence where $M$ is a closed irreducible 3Manifold, 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}