Embedding punctured n-manifolds in Euclidean (2n-1)-space

拟Fuchsian穿孔环面空间的坐标

Coordinates for quasi-Fuchsian punctured torus space

我们考虑复Fenchel-Nielsen坐标的准Fuchsian空间穿孔环面。这些坐标产生于克拉的管道建设的概括，并与Teichmueller空间的地震有关。它们还允许我们在Teichmueller空间上的两个坐标系之间插值，即具有Fenchel-Nielsen坐标和Maskit嵌入的经典Fuchsian空间。我们还展示了它们如何与Keen和Series的褶皱坐标相关。

Geometric embeddings of braid groups do not merge conjugacy classes

An embedding of the m-times punctured disc into the n-times punctured disc, for n>m, yields an embedding of the braid group on m strands B_m into the braid group on n strands B_n, called a geometric embedding. The main example consists of adding n-m trivial strands to the right of each braid on m strands. We show that geometric embeddings do not merge conjugacy classes, meaning that if the images of two elements in B_m by a geometric embedding are conjugate in B_n, the original elements are conjugate in B_m. We also show that the result does not hold, in general, for geometric embeddings of ma…

Embeddings of k-connected n-manifolds into R^{2n-k-1}

We obtain estimations for isotopy classes of embeddings of closed k-connected n-manifolds into R^{2n-k-1} for n>2k+5 and k\ge0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of H_{k+1}(N;Z_2) on the set of embeddings. (For k\ne1 classification results were obtained by algebraic methods without direct construction of embeddings or homology invariants.) The proof involves reduction to classification of embeddings of punctured manifold and uses parametric connected sum of embeddings.

Linear slices of the quasifuchsian space of punctured tori

After fixing a marking (V, W) of a quasifuchsian punctured torus group G, the complex length l_V and the complex twist tau_V,W parameters define a holomorphic embedding of the quasifuchsian space QF of punctured tori into C^2. It is called the complex Fenchel-Nielsen coordinates of QF. For a complex number c, let Q_gamma,c be the affine subspace of C^2 defined by the linear equation l_V=c. Then we can consider the linear slice L of QF by QF \cap Q_gamma,c which is a holomorphic slice of QF. For any positive real value c, L always contains the so called Bers-Maskit slice BM_gamma,c. In this pap…

欧几里得空间中流形的嵌入与打结

Embedding and knotting of manifolds in Euclidean spaces

对高维对象拓扑结构的清晰理解在纯数学和应用数学的许多分支中都很重要。在这个调查中，我们试图提出一些结果的高维拓扑结构的方式，使清晰的视觉和直观的部分的建设和论点。特别是，我们展示了如何抽象的代数结构自然出现在几何问题的研究。在给出一个一般的结构之前，我们用简单但重要的特殊情况来说明主要思想，在这些情况下，本质并不被技术细节所掩盖。更具体地说，我们提出了几个经典和现代的结果嵌入和打结的流形在欧几里得空间。我们陈述了许多具体的结果（特别是，最近明确的分类打结环面）。他们的声明（但不是证据!）对非专业人士来说是简单易行的。我们通过经典的van Kampen-Shapiro-Wu-Haefliger-Weber“删除产品”阻塞概述了嵌入的一般方法。这种方法将嵌入的合痕分类归结为等变映射的同伦分类，从而得到了上述的具体结果。我们描述了复兴的兴趣在这个美丽的分支拓扑结构，提出新的结果在这方面（弗里德曼，Krushkal，Teichner，西格尔，Spiez和作者）：一个推广的Haefliger-韦伯嵌入定理低于亚稳维范围和例子表明，其他类似物这个定理是假的亚稳维范围之外。

On the local structure and the homology of CAT$(κ)$ spaces and euclidean buildings

We prove that every open subset of a euclidean building is a finite dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a proof for the rigidity of homeomorphisms of euclidean buildings. A key step in our approach to this result is the following: the space of directions $Σ_oX$ of a CAT$(κ)$ space $X$ is homotopy quivalent to a small punctured disk $B_\eps(X,o)\setminus o$. The second ingredient is the local homology sheaf of $X$. Along the way, we prove some results about the loc…

穿孔环面群的褶不变量

Pleating invariants for punctured torus groups

本文用{ \em折叠不变量}对拟Fuchsian穿孔环面群的空间$ \QF $进行了完整的描述。这些都是自然的不变量的边界$\bch$的凸核心的相关双曲3-流形$M$和非富克斯群$\QF - \F$的坐标。$\bch$的一个分量的褶裥不变量由它的弯曲测度的投影类以及在这个类中固定选择的横向测度的层积长度组成。我们的描述补充了明斯基在\cite{MinskyPT}，在其中他描述了空间的所有穿孔环面群的{ \em结束不变量}的特点的渐近几何的结束$M$。褶裥不变量给出了一个准Fuchsian模拟的Kerckhoff-Thurston描述Fuchsian空间的临界线和地震horocycles。临界线延伸到{ \em褶皱平面}，在该平面上$\bch$的褶皱轨迹是恒定的，并且时环延伸到{ \em BM-片}，在该片上$\bch$的一个分量的褶皱不变量是固定的。我们证明了对应于有理叠层的褶皱平面是稠密的，并且它们的边界可以{ \em explicitly}找到。这意味着，回答了Bers在20世纪60年代后期提出的问题，可以计算出任意精确的$\QF$嵌入$\CC^2$的形状。

Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, Σ)$ among all possible embedded closed/bordered surfaces with given geometric/algebraic genus $>1$ in $\mathbb{R}^3$. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces.

Equivariant embeddings of manifolds into Euclidean spaces

Suppose a finite group $G$ acts on a manifold $M$. By a theorem of Mostow, also Palais, there is a $G$-equivariant embedding of $M$ into the $m$-dimensional Euclidean space $\RR^{m}$ for some $m$. We are interested in some explicit bounds of such $m$. First we provide an upper bound: there exists a $G$-equivariant embedding of $M$ into $\RR^{d|G|+1}$, where $|G|$ is the order of $G$ and $M$ embeds into $\RR^d$. Next we provide a lower bound for finite cyclic group action $G$: If there are $l$ points having pairwise co-prime lengths of $G$-orbits greater than $1$ and there is a $G$-equivariant …

An equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean space and CR regular embeddings into complex space

We show that a pseudo-holomorphic embedding of an almost-complex $2n$-manifold into almost-complex $(2n + 2)$-Euclidean space exists if and only if there is a CR regular embedding of the $2n$-manifold into complex $(n + 1)$-space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when $n$ is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into $\R^8$ equipped with an almost-complex structure that need not be…

Influence of Fractal Embedding in Three-Dimensional Euclidean Space on Wave Propagation in Electro- Chromodynamics

In this paper two zero-dimensional compact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obtained, that the embedding of compact influences on characteristics of wave in final state. Thus, the embedding of Cantor set in Euclidean space is additional property of a fractal which can be important both for applications of fractal electrodynamics and for physics of strong in…

On embeddings of almost complex manifolds in almost complex Euclidean spaces

We prove that any compact almost complex manifold $(M, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in a Euclidean space of dimension $4m + 2$, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class of $(M, J)$, for the existence of an embedding or an immersion in an almost complex Euclidean $4m$-space. We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in $R^6$.

Coarse embeddings of symmetric spaces and Euclidean buildings

Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. In this paper, we will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results for symmetric spaces and buildings of higher rank by Anderson-Schroeder, Kleiner, Kleiner-Leeb, Eskin-Farb and Fisher-Whyte. In particular, it is well known that the rank of these spaces is monotonous under quasi-isometric embeddings. This i…

将图的积嵌入欧几里得空间

Embedding products of graphs into Euclidean spaces

本文研究图积嵌入欧几里得空间的最小维度，证明了Kuratowsky图的n次幂不可嵌入2n维欧几里得空间，解决了Menger 1929年的问题。

Topological manifolds are locally-flat Euclidean neighbourhood retracts

We show that every topological n-manifold M admits a locally flat closed embedding $ι\colon M \hookrightarrow \mathbb{R}^{2n+1}$ and is a retract of some neighbourhood $U \subseteq \mathbb{R}^{2n+1}$

Riemann曲面在欧几里得空间中的等变嵌入及其最小维度

Equivariant embeddings of Riemann surfaces in Euclidean spaces with minimal dimensions

本文研究了Riemann曲面在欧几里得空间中的等变嵌入问题，证明了特定自同构群下的最小嵌入维度与素数p的关系，并给出了相关上界。

Isometric embeddings of pretangent spaces in E^n

We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.

环面上的圆模式空间及其辛形式

Space of circle patterns on tori and its symplectic form

本文研究闭环面上圆模式空间的辛几何与拓扑结构，证明其通过嵌入获得的 Weil-Petersson 辛形式具有非退化性，并证明该空间同胚于闭环面的 Teichmüller 空间。