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Results tagged with gt.geometric-topology
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user 61536
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
7
votes
2
answers
644
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A generic metric on $X\cup\mathbb Z$
$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$.
Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that
$d(x,y)=d_X(x, …
- 4,776
47
votes
3
answers
3k
views
A metric characterization of the real line
Is the following metric characterization of the real line true (and known)?
A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real …
- 41.8k
2
votes
A generic metric on $X\cup\mathbb Z$
The affirmative answer to this problem follows from a general result on extension of graph metrics. In the following definitions, the unordered pair $\{x,y\}$ of two elements $x,y$ is denoted by $xy$. …
- 41.8k
6
votes
1
answer
491
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is isom …
- 1,446
8
votes
2
answers
354
views
Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean...
Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $ …
- 658
4
votes
0
answers
182
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d) …
- 41.8k
5
votes
0
answers
104
views
Is each Lipschitz action of a finite group on the 3-sphere equivalent to a linear action?
It is known that each action of a compact group on the 2-dimensional $S^2$ sphere is equivalent (=conjugated) to the linear action of a subgroup of $O(3)$ on $S^2$.
On the other hand, there exists a t …
- 41.8k
5
votes
Retracting off a compact set
Let us show how to find such a retraction for $n=2$ (I do not know if this method generalizes to higher dimensions).
Given a compact set $C\subset\mathbb R^2$ and an open neighborhood $U\subseteq\mat …
- 41.8k
2
votes
Accepted
Two paths to the boundary with no holes in between
Yes, this is true because every topological copy of $[0,1]$ in the plane is unknotted and can be transformed by a homeomorphism of the plane into the straight arc $[0,1]\times\{0\}$. In the latter cas …
- 41.8k
8
votes
2
answers
2k
views
Classification of closed 3-manifolds with finite first homology group?
I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$.
Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ea …
- 10.5k
9
votes
Accepted
Lowest Dimension for Counterexample in Topological Manifold Factorization
As asked Dusan Repovs (who is an expert in the theory of topological manifolds), and he sent me the following answer:
This is indeed best possible result, since whenever a product of two spaces is a …
- 41.8k
6
votes
1
answer
440
views
Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
Questio …
- 4,713
46
votes
Accepted
If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \time...
This question is answered by two classical theorems of infinite-dimensional topology, which can be found in the books of Bessaga and Pelczynski, Chigogidze or Sakai.
Factor Theorem. For any Polish abs …
CommunityBot
- 1
1
vote
Square lying on moving chord of a simple closed curve
This is yet another idea to to resolve this resistant problem.
Let us parametrize everything. Identify the plane containing the moving square with the complex plane $\mathbb C$ and let $\mathbb T=\{z\ …
- 41.8k
8
votes
Symmetry of a distance metric for a generating set of Topology
Non-symmetric distances are called quasi-metrics and they often are used in topology and topological algebra, (e.g., for studying paratopological groups, see http://arxiv.org/abs/1412.2239). For gener …
- 41.8k