裤子图之间的单纯嵌入

Simplicial embeddings between multicurve graphs

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.

Geometric simplicial embeddings of arc-type graphs

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the "obvious way". We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex …

作为不可定向曲面泰希米勒空间组合模型的裤子图

The pants graph as a combinatorial model for the Teichmüller space of a non-orientable surface

研究不可定向曲面裤子图与其可定向双覆盖裤子图的关系，通过提升裤子分解构造映射，证明其为拟等距嵌入，进而得出不可定向曲面裤子图与配备诱导度量的泰希米勒空间拟等距。

On Codimension one Embedding of Simplicial Complexes

We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.

少顶点单纯复形的嵌入维数

Embedding dimensions of simplicial complexes on few vertices

该研究给出少顶点单纯复形可嵌入d-球面的充要条件，由此重获经典定理并推广相关结论，还证明这类复形的连续与线性嵌入等价。

Hardness of almost embedding simplicial complexes in $\mathbb R^d$

A map $f\colon K\to \mathbb R^d$ of a simplicial complex is an almost embedding if $f(σ)\cap f(τ)=\emptyset$ whenever $σ,τ$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. (a) Assume that $P\ne NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb R^d$ but for which there exists an equivariant map $\tilde K\to S^{d-1}$. (b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb R^d$ is NP hard. The proof is based on the technique from the Mat…

Kolmogorov-Barzdin嵌入估计的推广

Generalizations of the Kolmogorov-Barzdin embedding estimates

我们考虑几种方法来衡量“几何复杂性”的嵌入从一个单纯复形到欧几里得空间。其中之一是基于Kolmogorov和Barzdin的一篇论文的“厚度”版本。我们证明不等式的厚度和数量的简单的单纯复形，推广估计，Kolmogorov和Barzdin证明的图形。我们还考虑了扭曲的结。我们给另一个证明定理的赦免，有合痕类的结需要任意大的失真。这个证明是基于算术双曲流形的扩张式性质。

Hardness of almost embedding simplicial complexes in $\mathbb{R}^d$, II

A map $f: K \to \mathbb{R}^d$ of a simplicial complex is an almost embedding if $f(σ) \cap f(τ) = \varnothing$ whenever $σ, τ$ are disjoint simplices of $K$. Fix integers $d,k \geqslant 2$ such that $k+2 \leqslant d \leqslant\frac{3k}2+1$. Assuming that the "preimage of a cycle is a cycle" we prove $\mathbf{NP}$-hardness of the algorithmic problem of recognition of almost embeddability of finite $k$-dimensional complexes in $\mathbb{R}^d$. Assuming that $\mathbf{P} \ne \mathbf{NP}$ (and that the "preimage of a cycle is a cycle") we prove that the embedding obstruction is incomplete for $k$-dim…

Embedding simply connected 2-complexes in 3-space -- IV. Dual matroids

We introduce dual matroids of 2-dimensional simplicial complexes. Under certain necessary conditions, duals matroids are used to characterise embeddability in 3-space in a way analogous to Whitney's planarity criterion. We further use dual matroids to extend a 3-dimensional analogue of Kuratowski's theorem to the class of 2-dimensional simplicial complexes obtained from simply connected ones by identifying vertices or edges.

单纯复形嵌入$\R^d$的困难性

Hardness of embedding simplicial complexes in $\R^d$

设EMBED（k，d）是以下算法问题：给定一个维数至多为k的有限单纯复形K，是否存在K到R^d中的（分段线性）嵌入？已知的结果很容易暗示EMBED（k，2）（k= 1，2; k=1，d=2的情况是图的平面性）和EMBED（k，2k）对于所有k>2（即使k不被认为是固定的）的多项式性。我们观察到Novikov关于识别5-球面的算法不可解性的著名结果意味着EMBED（d，d）和EMBED（d-1，d）对于每个d>4都是不可判定的。我们的主要结果是EMBED（2，4）的NP-硬度，更一般地，对于所有k，d且d>3和d \geq k \geq（2d-2）/3，EMBED（k，d）的NP-硬度。这些尺寸落在Haefliger和Weber定理的所谓亚稳范围之外，该定理使用删除的产品阻碍来表征嵌入性。我们的减少是基于例子，由于西格尔，Spiestan，弗里德曼，Krushkal，Teichner，和Skopenkov，表明在亚稳范围之外的删除的产品阻塞是不足以表征嵌入性。

Parsimonious cones

We construct embeddings of simplicial complexes into a (surface of a) simplicial ball whose triangulation has bounded degrees and low volume. This construction can be used either to efficiently "simplify a complicated space" by realizing it as a part of a ball/sphere, or to "complexify" a sphere - to give it a specific metric that inherits desired properties from an embedded complex.

Embedding simply connected 2-complexes in 3-space -- II. Rotation systems

We prove that 2-dimensional simplicial complexes whose first homology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they are simply connected.

单连通2-复形在3维空间中的嵌入

Embedding simply connected 2-complexes in 3-space

本文用九个禁用子式刻画局部3-连通单连通2维单纯复形在3维空间中的可嵌入性，并将结果推广到所有单连通2维单纯复形。

Embedding simply connected 2-complexes in 3-space -- I. A Kuratowski-type characterisation

We characterise the embeddability of simply connected locally 3-connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski's characterisation of graph planarity, by excluded minors. This answers questions of Lovász, Pardon and Wagner.

Topological embeddings into random 2-complexes

We consider 2-dimensional random simplicial complexes $Y$ in the multi-parameter model. We establish the multi-parameter threshold for the property that every 2-dimensional simplicial complex $S$ admits a topological embedding into $Y$ asymptotically almost surely. Namely, if in the procedure of the multi-parameter model, each $i$-dimensional simplex is taken independently with probability $p_i=p_i(n)$, from a set of $n$ vertices, then the threshold is $p_0 p_1^3 p_2^2 = \frac{1}{n}$. This threshold happens to coincide with the previously established thresholds for uniform hyperbolicity and tr…

Embedding simply connected 2-complexes in 3-space -- V. A refined Kuratowski-type characterisation

This paper is the last paper in a series of five papers. Building on earlier papers in this series, we prove an analogue of Kuratowski's characterisation of graph planarity for three dimensions. More precisely, a simply connected 2-dimensional simplicial complex embeds in 3-space if and only if it has no obstruction from an explicit list of obstructions. This list of obstructions is finite except for one infinite family.

有界上同调的等距嵌入

Isometric Embeddings in Bounded Cohomology

本文主要研究有界上同调群之间保模映射的构造。对于具有顺从棱道群的群图，我们构造了顶点群的有界上同调的直和在群图的基本群的有界上同调中的等距嵌入.用类似的技巧证明了：如果（X，Y）是一对CW-复形，且Y的每个连通分支的基本群是顺从的，则（X，Y）的相对有界上同调与X的至少2次有界上同调之间的同构是等距的.作为应用，我们给出了Gromov等价定理和单形体积关于沿π_1-内射边界分量的可加性的简单完备的证明，其中边界分量具有顺从的基本群.

Path lifting properties and embedding between RAAGs

For a finite simplicial graph $Γ$, let $G(Γ)$ denote the right-angled Artin group on the complement graph of $Γ$. In this article, we introduce the notions of "induced path lifting property" and "semi-induced path lifting property" for immersions between graphs, and obtain graph theoretical criteria for the embedability between right-angled Artin groups. We recover the result of S.-h.{} Kim and T.{} Koberda that an arbitrary $G(Γ)$ admits a quasi-isometric group embedding into $G(T)$ for some finite tree $T$. The upper bound on the number of vertices of $T$ is improved from $2^{2^{(m-1)^2}}$ t…

${\mathbb R}^d$中嵌入的几何复杂度

Geometric complexity of embeddings in ${\mathbb R}^d$

给定一个单纯复$K$，我们考虑几个概念的几何复杂性的嵌入的$K$在欧几里德空间${\mathbb R}^d$：厚度，失真和细化复杂性（PL嵌入所需的最小数量的单纯形）。我们证明了任何$n$ -复形的$N$单形拓扑嵌入${\mathbb R}^{2n}$，$n>2$，可以PL嵌入${\mathbb R}^{2n}$与细化复杂性$O(e^{N^{4+ε}})$。构造单纯$n$ -复形$K$的族，使得$K$到${\mathbb R}^{2n}$中的任何嵌入都具有作为$K$的单纯形数目的函数的厚度和精化复杂度的指数下界.这对比了稳定范围$K\subset {\mathbb R}^{2n+k}$、$k>0$中的嵌入，其中几何复杂度函数的所有已知界限都是多项式。此外，我们给出了一个几何参数的界限膨胀图的失真在欧氏空间。讨论了几个相关的公开问题，包括嵌入的复杂性函数的增长率问题，以及经典环的交叉数和绳长问题。

随机复形的线性嵌入

Linear embeddings of random complexes

该研究针对多参数随机单纯复形模型，推导得出复形可线性嵌入$\mathbb{R}^{2d}$时参数$α_i$需满足的充要严格不等式，完善了随机复形线性嵌入的参数判据。

Fungible obstructions to embedding 2-complexes

We give examples of finite, simplicial $2$-complexes that do not PL embed in $\mathbb{R}^4$ and exhibit, for each such complex, a family of PL immersions into $\mathbb{R}^4$ that hide the obstruction to embedding in "higher and higher order Milnor invariants". In addition, we show that the embedding obstructions defined by Krushkal in [8] vanish for our examples. We also answer a question in a paper of Avramidi-Okun-Schreve.

Teichmüller空间的曲率和秩

Curvature and rank of Teichmüller space

设S是具有亏格g和n个边界分支的曲面，d（S）= 3g-3+n表示S的任意Pants分解中的曲线数.利用Pants分解图CP（S）的度量性质，证明了Teichmuller空间Teich（S）上的Weil-Petersson度量是Gromovv-双曲的当且仅当d（S）≤ 2.当d（S）>= 3时，Weil-Petersson度量在Gromov意义下具有较高的秩（它允许R^k的拟等距嵌入，k >= 2）;当d（S）<= 2时，我们结合曲线复形的双曲性和CP（S）的相对双曲性证明了Gromov-双曲性.当d（S）>= 3时，Teich（S）不存在有限余体积的测地线完备Gromov-双曲度量;当d（S）>= 2时，模空间M（S）上不存在紧致负曲率的完备黎曼度量.

Embedding mapping class groups into finite products of trees

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad-Nagata dimension, direct embeddings exhibiting $\ell^p$ compression exponent $1$ for all $p\geq 1$ and they quasi-isometrically embed into $\ell^1(\mathbb N)$. We deduce…

Reeb图的Book嵌入

Book embeddings of Reeb graphs

设$X$是具有分段线性函数$f:X\to\mathbb{R}$的单纯复形。Reeb图$Reeb(f,X)$是$X$的商，其中我们将$f^{-1}(t)$的每个连通分量折叠为单个点。设$Reeb(f,X)$的节点都是水平集$f^{-1}(t)$的相应分量的任何同调都改变的同调临界点。然后，我们可以用水平集的相应$d$维分量的Betti数$(β_1,β_2,\dots,β_d)$来标记$Reeb(f,X)$的每个弧。同调标记比经典的Reeb图给出了更多关于原始复合体$X$的信息。我们描述了一个Reeb图到一本多页书（一颗星穿过一条线）的一个典型嵌入，并给出了这本书嵌入的一个唯一线性码。

量化不连续性

Quantifying discontinuity

本文研究了无法嵌入到$\mathbb{R}^d$的紧空间的不连续性程度，并通过定义尺度不变的不连续性模量，获得了Haefliger--Weber类型的量化不可嵌入结果以及拓扑Tverberg定理的量化版本。

Combinatorial Lefschetz theorems beyond positivity

Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we introduce a version of the Kähler package beyond positivity, allowing us to prove the hard Lefschetz theorem for toric varieties (and beyond) even when the ample cone is empty. A particular focus lies on replacing the Hodge-Riemann relations by a non-degeneracy relation at torus-inva…

可嵌入性与部分矩阵的最小秩

Embeddability of joinpowers, and minimal rank of partial matrices

本文提出了高维复形到流形的Z₂嵌入性准则，将Kühnel问题转化为代数问题，结合了几何拓扑、组合学和线性代数。

拓扑扩张器、粗几何与复形的厚嵌入

Topological expanders, coarse geometry and thick embeddings of complexes

本文引入一族次线性函数量化有界度单纯复形的拓扑扩张，证明其单调性并计算欧氏空间等实例的上下界，进而证明高重 horocyclic 树乘积到若干粗几何空间不存在粗嵌入。

$k$-复形在$2k$-流形中的嵌入与部分对称矩阵的最小秩

Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices

本文证明了对于奇数$k\\\ge3$，一个$k$维单纯复形$K$嵌入到闭$(k-1)$-连通PL $2k$维流形$M$的充要条件，并将该结果推广到$\\mathbb Z_2$-和$\\mathbb Z$-可嵌入性，通过将相交函数扩展为低秩矩阵简化了已有条件。