No embeddings of solenoids into surfaces
Boju Jiang, Shicheng Wang, Hao Zheng
本文给出了Bing关于螺线管不能嵌入曲面定理的简短证明，并由此推出Gumerov关于螺线管覆盖度的结果。
本文给出了标题所指的Bing定理的一个简短证明。该证明同时也得出了Gumerov关于螺线管覆盖度的结果。
A quick proof of Bing's theorem indicated by the title is given. The proof also concludes Gumerov's result on covering degrees of solenoids.
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The Higson compactification of any simply connected proper geodesic metric space admits an embedding into a product of adelic solenoids that induces an isomorphism of 1-dimensional cohomology.
On tame embeddings of solenoids into 3-space
本文研究螺线管到三维空间的驯顺嵌入，将其作为圆周驯顺嵌入的逆向极限给出分类结果，并构造出严格非手性的驯顺嵌入螺线管，揭示其与紧多面体非手性嵌入必为平面这一已知事实的鲜明对比。
When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements…
In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H^1(L;R)= 0 and if the foliation is a product foliation in some saturated open neighbourhood U of L, then there exists a foliation F' on M which is C^1-close to F, and F' has an uncountable set of solenoidal minimal sets contained in U that are pair wise non-homeomorphic. If H^1(L;R) is not 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighbourhood. Thus, our result can be thought of as a…
Previous work of the author has developed coordinates on bundles over the classical Teichmueller spaces of punctured surfaces and on the space of cosets of the Moebius group in the group of orientation-preserving homeomorphisms of the circle, and this work is surveyed here. Joint work with Dragomir Saric is also sketched which extends these results to the setting of the Teichmueller space of the solenoid of a punctured surface, which is defined here in analogy to Dennis Sullivan's original definition for the case of closed surfaces. Because of relations with the classical modular group, the pu…
The punctured solenoid $§$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of $§$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $§$. Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of $§$ giving a combinatorial description of its decorated Teichmüller space itself. This is used to obtain a non-trivial set of generators of the modular group of $§$, which is presumably the main res…
On equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds
本文研究了闭双曲曲面到闭双曲3-流形的全测地嵌入及其等变版本，特别探讨了可定向曲面上的大反向作用及不可定向曲面上的大作用。
On geodesic embeddings of hyperbolic surfaces into hyperbolic 3-manifolds
本文研究闭可定向双曲曲面全测地嵌入闭可定向双曲3-流形的问题，证明所有有限不可约群作用下的准柏拉图曲面均存在等变测地嵌入，并讨论更一般有限群作用的情形。
Let F be a closed orientable surface. We give an explicit formula for the number mod 2 of quadruple points occurring in any generic regular homotopy between any two regularly homotopic embeddings e,e':F -> R^3. The formula is in terms of homological data extracted from the two embeddings.
An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the lowest genus of a surface on which such an embedding is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ admits such an embedding (…
Systolic embedding of graphs on translation surfaces
该研究提出平移曲面上图的收缩嵌入概念，证明任意图可实现本质-收缩嵌入并估计亏格，还研究了圆楔和及多顶点图族的胞腔-收缩嵌入与对应曲面亏格范围。
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.
Embedding calculus for surfaces
本文证明了Goodwillie-Weiss嵌入微积分在至多二维流形（含曲面微分同胚）中的收敛性，并揭示了曲面映射类群的Johnson滤过与嵌入微积分滤过之间的联系。
Embedding surfaces in 4-manifolds
本文证明了具有良好基本群的4-流形在存在对偶球面且法丛无限制时的曲面嵌入定理，引入Kervaire-Milnor不变量作为新障碍，并借助带特征曲面给出其组合计算公式。
The relative chromatic number $c\_0(S)$ of a compact surface $S$ with boundary is defined as the supremum of the chromatic numbers of graphs embedded in $S$ with all vertices on $\partial S$. This topological invariant was introduced for the study of the multiplicity of the first Steklov eigenvalue of $S$. In this article, we show that $c\_0(S)$ is also relevant for the study of tight polyhedral embeddings of $S$ byproving two results. The first one is that if there is a tight polyhedral embedding of $S$ in $\R^n$ which is not contained in a hyperplane, then $n\leq c\_0(S)-1$. The second resul…
In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.
We prove by an algebraic method that the embedding of the Teichmuller space in the space of geodesic currents is totally linearly independent. We prove a similar result for all negatively curved surfaces using an ergodic argument.
On the isotopy classes of embeddings of surfaces in 5-manifolds
该研究证明了在特定条件下，两个同伦的曲面嵌入在5维流形中必须是同胚的，并构造了一个基于同伦群的不变量来分类这些嵌入的同胚类。
The universal order 1 invariant f^U of immersions of a closed orientable surface into R^3, whose existence has been established in [N3], takes values in the group G_U = K \oplus Z/2 \oplus Z/2 where K is a countably generated free Abelian group. The projections of f^U to K and to the first and second Z/2 factors are denoted f^K, M, Q respectively. An explicit formula for the value of Q on any embedding has been given in [N2]. In the present work we give an explicit formula for the value of f^K on any immersion, and for the value of M on any embedding.
We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.
An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates
本文利用辫代数在柄体辫模上的作用，构造了曲面 Kauffman 括号辫代数到局部化量子环面的嵌入，并借此研究了单位根处的表示，给出了 Bonahon-Wong 唯一性猜想的新证明及唯一表示的显式重构方法。
We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages.
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group. Similarly, we show that any connected semisimple Lie group contains a dense copy of any fully residually free group.
In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman and Chillingworth and of Gonçalves and Guaschi to show that the mapping class group MCG(N ; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. We then compute the cohomological dimension of the braid groups of all…
Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space
该研究探讨尖点双曲曲面到闭黎曼曲面模空间的收缩映射及其Teichmüller空间提升是否为拟等距嵌入，证明温和条件下其性质由单值化完全刻画，且适用于全纯映射。
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is an embedding under which the formal adjacency relation on half-edges corresponds to the adjacency relation induced by the embedding. *-graphs are a natural generalization of four-valent framed graphs, which are four-valent graphs with an opposite half-edge structure. In [5], the…
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admits a Z2-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a …
We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.
Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3-manifolds homotopy equivalent to S. This 3-dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has bizarre local topology, possessing many points that are not closed. There is, however, a natural embedding of M(S) into AI(S) and a compactification of AI(S) such that this embedding extends to an embedding of the Deligne-Mumford compactification of M(S) into the compactification of AI(S).
Let $M$ be a sphere with handles and holes, $f:M\to\mathbb R^3$ an embedding, and $H_1=H_1(M;\mathbb Z)$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to\mathbb Z$. Let $\cap:H_1\times H_1\to\mathbb Z$ be the intersection form of $M$. Then the Seifert form is $\cap$-symmetric, i.e., $L(f)(β,γ)-L(f)(γ,β)=β\capγ$ for any $β,γ\in H_1$. If $M$ has non-empty boundary, then any $\cap$-symmetric bilinear form $H_1\times H_1\to\mathbb Z$ is realizable as $L(f)$ for some embedding $f$. We present a characterization of realizable forms for the torus $M$. The …
Embedding periodic maps of surfaces into those of spheres with minimal dimensions
本文研究了亏格为 $g$ 的闭定向曲面上阶为 $n$ 的周期映射等变嵌入球面 $S^m$ 的最小维数问题，在可定向光滑范畴下确定了 $n\geq 3g$ 时的最小 $m$，并证明对任意整数 $k>1$ 存在无穷多个映射使最小 $m$ 等于 $k$。
We study embedding of closed totally geodesic hyperbolic 2-orbifolds in the Bianchi orbifolds $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$. Our main result shows that there is a constant $c$ such that for $d$ large enough there are at least $cd$ closed embedded totally geodesic hyperbolic 2-orbifolds. Moreover we provide a list which conjecturally consists of those $d$ for which $\mathbb{H}^3/PSL(2,\mathcal{O}_d)$ does not contain a closed embedded totally geodesic hyperbolic 2-orbifold.
On spaces of embeddings of circles in surfaces
该研究探讨了在非正曲率曲面中嵌入圆的空间结构，通过有限根树索引连通分支，并展示了其与编织自同构群的关系。
The Deformation Space of Geodesic Triangulations and Generalized Tutte's Embedding Theorem
本文证明了负曲率闭曲面上测地三角剖分形变空间的可缩性，通过推广Tutte嵌入定理解决了Connelly等人1983年提出的双曲曲面公开问题。
We restrict our discussion to the orientable category. For $g > 1$, let $OE_g$ be the maximum order of a finite group $G$ acting on the closed surface $Σ_g$ of genus $g$ which extends over $(S^3, Σ_g)$, where the maximum is taken over all possible embeddings $Σ_g\hookrightarrow S^3$. We will determine $OE_g$ for each $g$, indeed the action realizing $OE_g$. In particular, with 23 exceptions, $OE_g$ is $4(g+1)$ if $g\ne k^2$ or $4(\sqrt{g}+1)^2$ if $g=k^2$, and moreover $OE_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$.
Isometric embeddings of surfaces for scl
本文证明自由群几何单射诱导稳定交换子长度的等距嵌入，并推广到子曲面相对Gromov半范数的同调等距嵌入，通过容许曲面的标准形式与同调消失条件实现。
We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by $π_1$-injective embeddings of the corresponding surfaces. We also prove the rigidity of the multicurve graphs.
Call a periodic map $h$ on the closed orientable surface $Σ_g$ extendable if $h$ extends to a periodic map over the pair $(S^3, Σ_g)$ for possible embeddings $e: Σ_g\to S^3$. We determine the extendabilities for all periodical maps on $Σ_2$. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair $(S^3, Σ_g)$. To do this we first list all periodic maps on $Σ_2$, and indeed we exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be an interesting piece. A by-…
Constructing embedded surfaces for cellular embeddings of leveled spatial graphs
针对空间图胞腔嵌入曲面的存在性问题，定义分层性质，通过分解为球面与圆柱手柄上的子图构造曲面，并提出算法推广至任意层数，猜想连通分层嵌入均可实现。
A simple characterization is given of open subsets of a complex surface that smoothly perturb to Stein open subsets. As applications, complex 2-space C^2 contains domains of holomorphy (Stein open subsets) that are exotic R^4's, and others homotopy equivalent to the 2-sphere but cut out by smooth, compact 3-manifolds. Pseudoconvex embeddings of Brieskorn spheres and other 3-manifolds into complex surfaces are constructed, as are pseudoconcave holomorphic fillings (with disagreeing contact and boundary orientations). Pseudoconcave complex structures on Milnor fibers are found. A byproduct of th…
The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest pat…
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