Vyacheslav S. Krushkal
The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into $R^{2n}$ for $n\neq 2$, and it was recently shown to be incomplete for $n=2$. We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of 2-complexes in $R^4$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the ci…
The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T^4 of trace elements which define invariants of 4-thickenings under 2-deformations. In T^4 two subsets are identified : T^3, which produces invariants of 4-thickenings normalizable to in…
Vietoris thickenings and complexes of manifolds are homotopy equivalent
该研究证明了维特里斯加厚与流形复形在同伦上的等价性，并展示了其在度量空间中的应用。
We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the $2$-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equations does not have an integral solution, we show that some $2$-dimensional CW complexes cannot be embedded into the 3-sphere.
An important "stability" theorem in shape theory, due to D.A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this note we present straightforward and self-contained proof of that theorem.
Obstruction Complexes in Grid Homology
本文定义了完整网格（不移除方格）的障碍链复形并计算其同调，虽无法直接推广Manolescu-Sarkar构造，但证明了网格同调中符号赋值的存在性。
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.
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.
We say a $d$-dimensional simplicial complex embeds into double dimension if it embeds into the Euclidean space of dimension $2d$. For instance, a graph is planar iff it embeds into double dimension. We study the conditions under which the join of two simplicial complexes embeds into double dimension. Quite unexpectedly, we show that there exist complexes which do not embed into double dimension, however their join embeds into the respective double dimension. We further derive conditions, in terms of the van Kampen obstructions of the two complexes, under which the join will not be embeddable i…
We improve the bound on Kühnel's problem to determine the smallest $n$ such that the $k$-skeleton of an $n$-simplex $Δ_n^{(k)}$ does not embed into a compact PL $2k$-manifold $M$ by showing that if $Δ_n^{(k)}$ embeds into $M$, then $n\leq (2k+1)+(k+1)β_k(M;\mathbb Z_2)$. As a consequence we obtain improved Radon and Helly type results for set systems in such manifolds. Our main tool is a new description of an obstruction for embeddability of a $k$-complex $K$ into a compact PL $2k$-manifold $M$ via the intersection form on $M$. In our approach we need that for every map $f\colon K\to M$ the re…
We show that the Reshetikhin-Turaev-Walker invariant of 3-manifolds can be normalized to obtain an invariant of 4-dimensional thickenings of 2-complexes. Moreover when the underlying semisimple tortile category comes from the representations of a quantum group at a primitive prime root of unity, the 0-term in the Ohtsuki expansion of this invariant depends only on the spine and is the mod p invariant of 2-complexes defined previously from the second author. As a consequence it is shown that when the Euler characteristic is greater or equal to 1, the 2-complex invariant depends only on homology…
Algebraic realizability of knot Floer-like complexes
该研究聚焦类纽结复形局部等价类可实现性的代数障碍，完成了$\mathbb{F}[U,V]$上相关等价类的分类，解决了Dai等人提出的问题。
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.
Embedding obstructions in ${\mathbb R}^d$ from the Goodwillie-Weiss calculus and Whitney disks
本文利用 Goodwillie-Weiss 塔提出有限 CW 复形嵌入 $\mathbb{R}^d$ 的障碍理论，证明其几何类比与 van Kampen 经典障碍之后的首个障碍实质等价，并揭示其与构型空间 Arnold 类的关联。
On the Classification of Planar-Rips complexes and their corresponding unit disk graphs
本文对可实现为平面点集Vietoris-Rips复形的伪流形与弱伪流形进行单纯同构分类，并同伦分类二维纯闭平面Rips复形，进而刻画一类所有极大团基数相同的单位圆盘图。
Let EMBED(k,d) be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R^d? Known results easily imply polynomiality of EMBED(k,2) (k=1,2; the case k=1, d=2 is graph planarity) and of EMBED(k,2k) for all k>2 (even if k is not considered fixed). We observe that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBED(d,d) and EMBED(d-1,d) are undecidable for each d>4. Our main result is NP-hardness of EMBED(2,4) and, more generally, of EMBED…
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…
We consider $k$-dimensional random simplicial complexes that are generated from the binomial random $(k+1)$-uniform hypergraph by taking the downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and the zero-th cohomology group is isomorphic to $\mathbb{F}_2$. This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of the…
We introduce the notion of coupled embeddability, defined for maps on products of topological spaces. We use known results for nonsingular biskew and bilinear maps to generate simple examples and nonexamples of coupled embeddings. We study genericity properties for coupled embeddings of smooth manifolds, extend the Whitney embedding theorems to statements about coupled embeddability, and we discuss a Haefliger-type result for coupled embeddings. We relate the notion of coupled embeddability to the $\mathbb{Z}/2$-coindex of embedding spaces, recently introduced and studied by the authors. With …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。