# Questions tagged [embeddings]

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### Finite graph not embeddable in $H(n,2)$

Let $n\in\mathbb{N}$ and consider $x,y \in\{0,1\}^n$. The Hamming distance of $x,y$ is defined by $$d_H(x,y) = |\{i\in \{0,\ldots, n-1\}:x_i\neq y_i\}|.$$ For $n\geq 2$ let $H(n,2)$ be the graph given ...
1answer
125 views

### Embedding CW-complexes into infinite-dimensional topological vector spaces

Sometimes it is desirable to embed CW-complexes into real vector spaces, to use a simple linear algebra to work with them. Result on embedding into Euclidean spaces are well known, check Hatcher‘s ...
0answers
35 views

### The linear embedding complexity of subsets of $0/1$ cube

We say $\pi$ is a subset of the $0/1$ integer points in $t$ dimensions represented by coordinates $(x_1,\dots,x_t)$ of complexity $\log^ct$ if there is an $A$ of $2^{\log^ct}\times2^{\log^ct}$ ...
1answer
190 views

### When is a metric space a snowflake?

Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$. ...
3answers
297 views

0answers
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### Correspondence between Riemannian metrics and Euclidean embeddings

Given a sufficiently smooth manifold M, a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely) an embedding of M into ...
2answers
214 views

### Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?

Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$. I would like to show that, ...
0answers
102 views

### Examples of (infinite) graphs which cannot be embedded into 3d space?

I was thinking about the concept of embedding graphs into Euclidean spaces. Specifically, i was looking for examples of infinite graphs which cannot be embedded in $\mathbb{R}^3$ but can be embedded ...
1answer
108 views

### Banach embedding of finite dimensional spaces

Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]). I am wondering whether are any result for the case when $r>s>2$? ...
0answers
70 views

### Dense embeddings into Euclidean space

The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
0answers
221 views

### Can a knotted sphere isometrically embed into $\mathbb R^3$?

All smooth simple closed curves in $\mathbb R^3$ (knotted or not) can be isometrically embedded into $\mathbb R^2$ as a circle of equal arclength. The situation for knotted spheres seems more ...
1answer
155 views

2answers
356 views

### Is Cohen immersion conjecture (theorem) known for vector bundles?

R. Cohen proved the immersion conjecture in a 1985 Annals paper: Cohen, Ralph L., The immersion conjecture for differentiable manifolds, Ann. Math. (2) 122, 237-328 (1985). ZBL0592.57022. Any smooth ...
1answer
394 views

### Local diffeomorphism on a neighborhood of an embedding

In my reading of the (excellent!) paper of Grabowski and Rotkiewicz on higher vector bundles (https://arxiv.org/abs/math/0702772), I have encountered the following argument which I do not understand. ...
2answers
504 views

### Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is ...
1answer
266 views

### Hyperbolic embedding of a directed acyclic graph defined over strings

For integer $n$ and alphabet $\Sigma$ we construct a DAG (directed acyclic graph) $G=(V,E)$ over strings $s\in\Sigma^\star$ as follows: $$V = \{s\in\Sigma^\star\colon |s|\le n\}$$ E = \{(s_1,s_2)\...
2answers
568 views

### Embeddings of flag manifolds

Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...
2answers
315 views

### About the commutativity of the $1^{st}$ homotopy group of the space of knots

I would like to know if the fundamental group of the connected component of a knot space could be non commutative. I am specially interested in the case of $\mathbb{R}^3$, $\mathbb{S^3}$ or some other ...
0answers
112 views

### Intrinsic Reach for a Riemannian manifold

The reach of a set $X\subseteq \mathbb{R}^d$ is the supremum of all $r \geq 0$ such that for all $y\in X^c$ with $dist(y,X)<r$ there is a unique $x\in X$ with $dist(y,x)= dist(y,X)$. My question: ...
1answer
312 views

### Can there be elementary embedding between a universe and a universe inside it?

[EDIT] the prior question (see the second section below) was trivially false, however the intention is to arrange a possible world of such universes, in other words the question is about if it is ...
1answer
515 views