Questions tagged [embeddings]

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Prove that Takens' embedding is a smooth one-to-one map with a smooth inverse

Let $f: \mathcal{M} \rightarrow \mathcal{M}$ be a smooth diffeomorphism and $\phi: \mathcal{M} \rightarrow \mathbb{R}$ be a smooth function, where $\mathcal{M}$ is a $d$-dimensional manifold (which we ...
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8 votes
0 answers
91 views

Is every simplicial $d$-sphere linearly embeddable in $\Bbb R^{d+1}$?

A simplicial $d$-sphere is a simplicial complex homeomorphic to the $d$-sphere. It is known that not every such complex can be embedded into $\Bbb R^{d+1}$ as the boundary complex of a convex ...
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  • 9,887
8 votes
0 answers
229 views

A conjecture about homotopy $S^1\times B^3$'s

$\textbf{Conjecture}:$ Let $X^4$ be a homotopy $S^1\times B^3$ with the following properties: Attaching a four dimensional 2-handle gives a standard $B^4$. The $k$-fold cyclic cover is diffeomorphic ...
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0 answers
111 views

Can a non-reflexive space embed into a reflexive space?

My question is inspired from the concept of super-reflexivity which was defined by James here: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/superreflexive-banach-...
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10 votes
1 answer
159 views

The knot $K\subset \Bbb S^3$ is smoothly slice, but the disc $D\subset \Bbb D^4$ is only locally flat. Can $D$ be smoothed?

Suppose I am given a smoothly slice knot $K\subset\Bbb S^3$. But I am only given a locally flat disc $D\subset \Bbb D^4$ with boundary $K$. Question: Is there a smooth disc $D'\subset\Bbb D^4$ with ...
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  • 9,887
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0 answers
71 views

Compact embedding of anisotropic Sobolev space

I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
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6 votes
0 answers
157 views

Sobolev embedding theorems on manifolds

I had asked the following question on math.stackexchange but did not get any response: I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
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3 votes
0 answers
60 views

Finite approximations to the Kuratowski/Fréchet embedding

Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with $$ \left\{B\left(x_k,\frac1{n}\right)...
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1 vote
0 answers
64 views

Best estimate on doubling constant of a finite metric space

Let $(X,d)$ be a finite metric space. Clearly, $(X,d)$ is a doubling metric space but is there a 'best' estimate of $(X,d)$'s doubling constant? Probability based on its cardinality, diameter, and ...
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2 votes
1 answer
56 views

Completing a tree to a 2-connected outerplanar graph

Let $T$ be a given (finite) tree. Question 1: Is it always possible to add edges to $T$ to obtain a $2$-connected outerplanar supergraph $G$? Question 2: If the answer to Question #1 is negative, can ...
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9 votes
2 answers
353 views

Embedding of a bundle with connection into a bundle with flat connection?

I'm looking for a generalization of Nash's embedding theorem (for Riemannian manifolds) to vector bundles with a connection. Given a smooth manifold $M$ together with a vector bundle $V$ on $M$ ...
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0 answers
20 views

Separability of graph component embeddings

I have an undirected graph. I also have an embedding of the graph in $\mathbb{R}^3$. Assume that the graph has 2 connected components. I want to know whether there exists a plane (or better - any ...
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1 vote
0 answers
103 views

The best lower bound for isometric immersions

I just read Azov's article in the considered two classes of Riemannian metrics, \begin{align*} ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\ ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2 ,&g>0\...
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2 votes
0 answers
99 views

Approximating maps by embedding

A continuous map $f$ between two metric spaces is said to be a $r$-map if preimage of each point under $f$ has diameter atmost $r$. Suppose $D^n=\{x\in \mathbb{R}^n\mid ||x||\leq 1\}\subset \mathbb{R}^...
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11 votes
2 answers
400 views

Given a 2-disc embedded in $\Bbb R^4$, can I fit another 2-disc with the same boundary?

I am given a 2-disc $D^2$ embedded into $\Bbb R^4$, that is, I have an injective continuous map $\phi:D^2\to\Bbb R^4$. I want to "double" this disc in the sense that I am looking for a ...
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11 votes
1 answer
352 views

Haefliger trefoil $S^3\hookrightarrow S^6$

It is known that the Haefliger trefoil $S^3\hookrightarrow S^6$ is PL trivial but non-trivial smoothly. I wonder, where exactly does the problem come? Consider its tubular neighborhood $T\cong S^3\...
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  • 1,676
3 votes
0 answers
165 views

Reference Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173

I have been searching without success for the reference: Aubin T.- Espaces de Sobolev sur les varietes Riemanniennes. Bull. Sc. Math. 100, (1976) 149-173 It is cited in many related works. In ...
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4 votes
0 answers
134 views

Specify the embedding of special unitary group in a Spin group via their representation map

How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations? By ...
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1 vote
1 answer
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Does the rational normal curve embedding extend as a mapping from the "bulk" to some bigger ambient space?

The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at ...
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1 vote
1 answer
109 views

Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space [closed]

Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so $$ G \supset J. $$ If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$. If $G$ has a universal cover $\widetilde{G}$, ...
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2 votes
0 answers
83 views

Almost Lipschitz embedding of compact metric measure spaces into Euclidean spaces

Let $(X,d)$ be a compact metric space, $m$ be a metric outer-measure on $X$. Are there 'mild conditions' on $X$ ensuring the existence of a positive integer $N\geq 3$ such that there exist $x_1,\dots,...
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6 votes
1 answer
250 views

$H_0^1(\Omega, D) \hookrightarrow L^2(D)$ is compact, for $\Omega$ quasi-open in $D$ - Proof verification

In their paper The spectral drop problem, Buttazzo and Velichkov state that the embedding $H_0^1(\Omega, D) \hookrightarrow L^2(D)$ is compact, where $\Omega \subset D$ is quasi-open, $D \subset \...
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1 vote
0 answers
56 views

Injectivity of post-composition operator

Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
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1 vote
1 answer
71 views

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 ...
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3 votes
1 answer
169 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 ...
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5 votes
1 answer
292 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 $(X,d)$. ...
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4 votes
3 answers
345 views

How to show that random graphs cannot be embedded with short edges

For each (not necessarily planar) embedding of a graph in $\mathbb{R}^k$ one can calculate the ratio $$\gamma = \frac{\textsf{mean Euclidean length of edges}}{\textsf{mean Euclidean distance between ...
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2 votes
0 answers
208 views

Interpolation inequality involving negative Sobolev space

$\newcommand\norm[1]{\left\|#1\right\|}\newcommand\inner[2]{\langle #1,#2\rangle}$ Let $u\in \dot{H}^1(\mathbb{R}^n)$ for $n\geq 3$ where $\dot{H}^{1}$ denotes the homogeneous Sobolev space that is ...
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3 votes
1 answer
174 views

Is this sequence of embeddings possible?

Working in a suitable extension of $\sf Z$ like $\sf ZfC + wholeness \ axiom$, or $\sf ZFj + Reinhardt \ axiom$. Can we have a sequence $(j_n)_{n \in \mathbb N} $ of nontrivial elementary embeddings ...
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7 votes
1 answer
363 views

Do holomorphic symplectic manifolds admit (high codimension) embeddings in some standard space?

Per the Whitney embedding theorem, any manifold $M$ can be embedded into a sufficiently high dimensional Euclidean space. According to Gromov's h-principle for contact embeddings, any contact manifold ...
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0 votes
0 answers
86 views

Reflective embeddings outside separation?

One of the known ways to axiomatize ZF is via these two axiom schemata: Separation: $\forall \vec{w} \forall A \exists! x: x=\{y \in A: \phi\}$ Reflection: $\forall \vec{w} \exists \alpha: \phi \to \...
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6 votes
1 answer
582 views

Can there exist such a sequence of elementary embeddings of the universe to itself?

Working in ZfC + Wholeness: Can we have a countable sequence of non-trivial elementary embeddings of the universe to itself, such that the range of each embedding is a subclass of the range of its ...
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24 votes
1 answer
1k views

On a curious map from the complex projective plane into $S^5$

I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a ...
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  • 3,656
2 votes
1 answer
190 views

Does the Hawaiian Earring Group embed into the permutation group of $\mathbb N$?

Recall that the Hawaiian earring group, $\mathbb G$, is the fundamental group of the Hawaiian Earing using the point at the origin. It can be understood more combinatorially as a subgroup of the ...
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2 votes
0 answers
401 views

Compact embedding of the Sobolev space $H^m(\Omega)$ and $L^2(\Omega)$ from Rellich-Kondrachov theorem

From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...
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  • 627
4 votes
1 answer
118 views

Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation

We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...
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2 votes
1 answer
209 views

Fast algorithm for computing $\sum_m (n \mod m)/m!$

I'm interested in quickly computing an embedding of the profinite integers $\widehat{\mathbb{Z}}$ into the unit interval $\left[0,1\right]$. $\widehat{\mathbb{Z}}$ can be represented as compatible ...
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1 vote
0 answers
324 views

Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$

Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
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  • 1,235
5 votes
0 answers
125 views

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 ...
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0 votes
2 answers
259 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, ...
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0 votes
0 answers
157 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 ...
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3 votes
1 answer
122 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 $r>s>2$? ...
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2 votes
0 answers
73 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 ...
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  • 5,008
6 votes
0 answers
227 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 ...
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5 votes
1 answer
208 views

Is identification of double points of an immersion smooth?

Let $f:M^m\to N^n$ be a generic map between smooth manifolds $n>m$. Depending on the pair $(m,n)$ generic maps will have a singular set of double points $\Sigma_2\subset M$. Let $\phi:\Sigma_2\to \...
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1 vote
0 answers
75 views

What is known about this generalization of planar dual?

So it is well known that given a planar graph, $G$, embedded in the plane (without edge crossing, so a planar embedding). One can construct the planar dual, $G^*$. What is perhaps slightly less well-...
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3 votes
0 answers
105 views

Which metric spaces embed isometrically in $\ell_p$?

It is known that each metric space $X$ embeds isometrically in the Banach space $\ell_\infty(X)$ of bounded (not necessarily continuous) functions $X \to \mathbb R$. Since $\ell_\infty(X)$ does not ...
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  • 1,606
2 votes
1 answer
308 views

Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension?

Background: I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded ...
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  • 627
2 votes
1 answer
179 views

complemented subspace of the direct sum of two Banach spaces

When I was reading a paper, I saw something like: If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented ...
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  • 383
5 votes
0 answers
67 views

subanalytic realization of smooth abstract stratification

Consider an $C^\infty$ abstract stratification $A$ (in the Thom-Mather sense, see Mather's note). Can we embed $A$ in some $\mathbb{R}^n$ (or in an analytic manifold) as a subanalytic set? If not, ...
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