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2-Manifolds

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William P. Thurston – One of the best experts on this subject based on the ideXlab platform.

  • Finite covers of random 3-manifolds

    Inventiones mathematicae, 2006
    Co-Authors: Nathan M. Dunfield, William P. Thurston

    Abstract:

    A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π_1(Σ)→ Q . If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.

  • Finite covers of random 3-manifolds
    Inventiones Mathematicae, 2006
    Co-Authors: Nathan M. Dunfield, William P. Thurston

    Abstract:

    A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.

G. Elber – One of the best experts on this subject based on the ideXlab platform.

  • Contouring 1- and 2-Manifolds in arbitrary dimensions
    International Conference on Shape Modeling and Applications 2005 (SMI' 05), 2005
    Co-Authors: Joon-kyung Seong, G. Elber

    Abstract:

    We propose an algorithm for contouring k-manifolds (k = 1,2) embedded in an arbitrary n-dimensional space. We assume (n -k) geometric constraints are represented as polynomial equations in n variables. The common zero-set of these (n-k) equations is computed as an 1-or 2-manifold, respectively, for k = 1 or k = 2. In the case of 1-manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitly-defined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2-Manifolds, the algorithm is similar to techniques for contouring iso-surfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 -2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zero-set as a piecewise linear 1or 2-manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.

  • SMI – Contouring 1- and 2-Manifolds in arbitrary dimensions
    International Conference on Shape Modeling and Applications 2005 (SMI' 05), 2005
    Co-Authors: Joon-kyung Seong, G. Elber

    Abstract:

    We propose an algorithm for contouring k-manifolds (k = 1,2) embedded in an arbitrary n-dimensional space. We assume (n -k) geometric constraints are represented as polynomial equations in n variables. The common zero-set of these (n-k) equations is computed as an 1-or 2-manifold, respectively, for k = 1 or k = 2. In the case of 1-manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitly-defined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2-Manifolds, the algorithm is similar to techniques for contouring iso-surfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 -2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zero-set as a piecewise linear 1or 2-manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.

Nathan M. Dunfield – One of the best experts on this subject based on the ideXlab platform.

  • Finite covers of random 3-manifolds

    Inventiones mathematicae, 2006
    Co-Authors: Nathan M. Dunfield, William P. Thurston

    Abstract:

    A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π_1(Σ)→ Q . If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.

  • Finite covers of random 3-manifolds
    Inventiones Mathematicae, 2006
    Co-Authors: Nathan M. Dunfield, William P. Thurston

    Abstract:

    A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.