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Shicheng Wang

  1. Chern Simons Theory and the volume of 3-manifolds.

    Tue, 11/29/2011 - 15:01 — SomNambul
    Authors: Shicheng Wang, Pierre Derbez
    Subjects: Geometric Topology
    Abstract

    We give some applications of the Chern Simons gauge theory to the study of
    the set ${\rm vol}(N,G)$ of volumes of all representations $\rho\co\pi_1N\to
    G$, where $N$ is a closed oriented three-manifold and $G$ is either ${\rm
    Iso}_e\t{\rm SL_2(\R)}$, the isometry group of the Seifert geometry, or ${\rm
    Iso}_+{\Hi}^3$, the orientation preserving isometry group of the hyperbolic
    3-space. We focus on three natural questions:

    (1) How to find non-zero values in ${\rm vol}(N, G)$? or weakly how to find
    non-zero elements in ${\rm vol}(\t N, G)$ for some finite cover $\t N$ of $N$?

  2. On fibered commensurability.

    Tue, 03/02/2010 - 16:01 — SomNambul
    Authors: Shicheng Wang, Danny Calegari, Hongbin Sun
    Subjects: Geometric Topology
    Abstract

    This paper initiates a systematic study of the relation of commensurability
    of surface automorphisms, or equivalently, fibered commensurability of
    3-manifolds fibering over the circle. We show that every hyperbolic fibered
    commensurability class contains a unique minimal element, whereas the class of
    Seifert manifolds fibering over the circle consists of a single
    commensurability class with infinitely many minimal elements. The situation for
    non-geometric manifolds is more complicated, and we illustrate a range of
    phenomena that can occur in this context.

  3. A spin obstruction for codimension-two diffeomorphism and homeomorphism extension.

    Tue, 10/27/2009 - 16:01 — SomNambul
    Authors: Fan Ding, Shicheng Wang, Jiangang Yao, Yi Liu
    Subjects: Geometric Topology
    Abstract

    Let $\imath: M^p\to \RR^{p+2}$ be a codimension-2 smooth embedding from
    closed orientable manifold to the Euclidean space, then there is a spin
    structure $\imath^#(\varsigma^{p+2})$ on $M$ canonically induced from the
    embedding. If an orientation-preserving self-diffeomorphism $\tau$ of $M$
    extends over $\imath$ as an orientation-preserving self-homeomorphism of
    $\RR^{p+2}$, then $\tau$ preserves the induced spin structure.

  4. Only rational homology spheres admit $\Omega(f)$ to be union of DE attractors.

    Tue, 09/08/2009 - 12:16 — SomNambul
    Authors: Fan Ding, Jianzhong Pan, Shicheng Wang, Jiangang Yao
    Subjects: Geometric Topology
    Abstract

    If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$
    such that the non-wandering set $\Omega(f)$ consists of finitely many
    orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a
    rational homology sphere; moreover all those attractors are of topological
    dimension $n-2$.

    Expanding maps are expanding on (co)homologies.

  5. Extending $T^p$ automorphisms over $\RR^{p+2}$ and realizing DE attractors.

    Tue, 09/08/2009 - 12:16 — SomNambul
    Authors: Fan Ding, Shicheng Wang, Jiangang Yao, Yi Liu
    Subjects: Geometric Topology
    Abstract

    We show that for any expanding map $\phi: T^p\to T^p$, there is an
    orientation-preserving self-diffeomorphism of $\RR^{p+2}$ realizing a
    hyperbolic attractor derived from $\phi$. The construction is based on a result
    in differential topology that for the standard unknotted embedding
    $\imath_p:T^p\to\RR^{p+2}$, the subgroup $E_{\imath_p}$ of
    $\Aut(T^p)\cong\SL(p,\ZZ)$ which consists of automorphisms that extend over
    $\RR^{p+2}$ as orientation-preserving diffeomorphisms, has index at most
    $2^p-1$.

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