Related papers: Hardness of almost embedding simplicial complexes …

A map $f: K \to \mathbb{R}^d$ of a simplicial complex is an almost embedding if $f(\sigma) \cap f(\tau) = \varnothing$ whenever $\sigma, \tau$ are disjoint simplices of $K$. Fix integers $d,k \geqslant 2$ such that $k+2 \leqslant d…

We study conditions under which a finite simplicial complex $K$ can be mapped to $\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\to \mathbb R^d$ such that the images of any $r$ pairwise…

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;…

We show that the decision problem of determining whether a given (abstract simplicial) $k$-complex has a geometric embedding in $\mathbb R^d$ is complete for the Existential Theory of the Reals for all $d\geq 3$ and $k\in\{d-1,d\}$. This…

The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is…

Denote by $\Delta_M$ the $M$-dimensional simplex. A map $f\colon \Delta_M\to\mathbb R^d$ is an almost $r$-embedding if $f\sigma_1\cap\ldots\cap f\sigma_r=\emptyset$ whenever $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint faces. A…

We provide a simple characterization of simplicial complexes on few vertices that embed into the $d$-sphere. Namely, a simplicial complex on $d+3$ vertices embeds into the $d$-sphere if and only if its non-faces do not form an intersecting…

Given a simplicial complex $K$, we consider several notions of geometric complexity of embeddings of $K$ in a Euclidean space ${\mathbb R}^d$: thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL…

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$…

This thesis addresses the question of the maximal number of $d$-simplices for a simplicial complex which is embeddable into $\mathbb{R}^r$ for some $d \leq r \leq 2d$. A lower bound of $f_d(C_{r + 1}(n)) =…

A stable smooth map $f:N\to M$ is called "$k$-realizable" if its composition with the inclusion $M\subset M\times\Bbb R^k$ is $C^0$-approximable by smooth embeddings; and a "$k$-prem" if the same composition is $C^\infty$-approximable by…

A general position map $f:K\to M$ of a $k$-dimensional simplicial complex to a $2k$-dimensional manifold (for $k=1$, of a graph to a surface) is a $\mathbb Z_2$-embedding if $|f\sigma \cap f\tau|$ is even for any non-adjacent $k$-faces…

A map $\varphi:K\to R^2$ of a graph $K$ is approximable by embeddings, if for each $\varepsilon>0$ there is an $\varepsilon$-close to $\varphi$ embedding $f:K\to R^2$. Analogous notions were studied in computer science under the names of…

A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric…

For any $\varepsilon > 0$, we prove that $k$-Dimensional Matching is hard to approximate within a factor of $k/(12 + \varepsilon)$ for large $k$ unless $\textsf{NP} \subseteq \textsf{BPP}$. Listed in Karp's 21 $\textsf{NP}$-complete…

We relate the embeddability of the simplicial complex $[3]*K$ into $\mathbb{R}^{n+2}$ to that of $K$ into $\mathbb{R}^n$. In brief, the embeddability of $K$ into $\mathbb{R}^n$, in the metastable range $2n\geq 3(d+1)$, is equivalent to the…

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…

Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are $\bullet$ a…

In this survey-research paper, we first introduce the theory of Smith classes of complexes with fixed-point free, periodic maps on them. These classes, when defined for the deleted product of a simplicial complex $K$, are the same as the…

We show that for every $k\in\mathbb{N}$ and $\varepsilon>0$, for large enough alphabet $R$, given a $k$-CSP with alphabet size $R$, it is NP-hard to distinguish between the case that there is an assignment satisfying at least…