Questions tagged [homeomorphism]

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What can we say about the complement of two homotopic simple closed curves on a compact orientable surface?

Do two homotopic simple closed curves separate a compact orientable surface? If they are disjoint they do and bound a cylinder. But if they intersect? What can we say about the components of their ...
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35 views

Continuous surjection from $X(D_n)$ onto $\operatorname{Homeo}_0(D_n)$

Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of ...
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69 views

Approximative extension of the autohomeomorphism of the complement of the trivial knot?

Let $S^1\subset \mathbb{R}^3$ be the unit circle and suppose $h\colon \mathbb{R}^3\setminus S^1\to \mathbb{R}^3\setminus S^1$ is a homeomorphism. Clearly it might be that $h$ cannot be extended to $S^...
10
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1answer
663 views

Why are homeomorphism groups important?

For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
6
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2answers
291 views

Do mixing homeomorphisms on continua have positive entropy?

I am trying to understand relations between various measures of topological complexity. I have read that expansive homeomorphisms on continua, for example, have positive entropy. But I do not know ...
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62 views

Contractable and Simply Connected Doubling Spaces Homeomorphic to Euclidean Space

Is there a characterization of all simply connected, contractable doubling metric spaces which are homeomorphic to a simply connected subset of Euclidean space?
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2answers
183 views

Extending homeomorphisms between compact metric subsets

Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism. Is it possible to extend $h$ to a ...
10
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1answer
188 views

homeomorphisms induced by composant rotations in the solenoid

Let $S$ be the dyadic solenoid. Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$. $X$ is called a composant of $S$. It is well-known ...
8
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2answers
248 views

Are there invariants of cell complexes similar to the Euler characteristic?

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely \begin{equation} \chi=\...
6
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1answer
340 views

Automorphisms of Erdős spaces

A surjective homeomorphism $h:X\to X$ is minimal if $\overline{\{h^n(x):n\in \mathbb N\}}=X$ for every $x\in X$. In other words, the orbit of each point is dense. Does either of the Erdös spaces $\...
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2answers
315 views

generators for the handlebody group of genus two

Is the handlebody group of genus two surface generated by Dehn twists along properly embedded disks and annuli? Are there alternative ways to describe a set of generators that are conceptually simple ...
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163 views

Similarities between isomorphism classes of homeomorphic directed graphs

To clarify, I'm speaking of homeomorphisms in a graph theoretic context, defined by subdivisions of arcs in a directed graph. A subdivision of an arc $(x,z)$ in a directed graph is obtained by ...
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0answers
79 views

Homeomorphism type of pair of faces in a regular CW complex

Let $X$ be a regular CW complex, $\sigma$ an $n$-dimensional cell of $X$ and $\tau$ an $(n-1)$-dimensional face of $\sigma$. Is it true that the pair $(\bar\sigma, \bar\tau)$ is homeomorphic to the ...
10
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3answers
362 views

Is an open subset of a rigid space rigid?

Let $X$ be a locally compact Hausdorff space. Call $X$ rigid if its only autohomeomorphism is the identity, $\operatorname{Homeo}(X)=\{1\}$. Questions: Let $X$ be rigid. Is it true that every open ...
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2answers
415 views

Homeomorphisms vs Borel automorphisms

Let $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ be the groups of homeomorphic and Borel automorphisms of a space $M$, respectively. Question: Are $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ ...
5
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1answer
406 views

Proof of the stable homeomorphism conjecture

I would like to get a demonstration of the stable homeomorphism conjecture (SHC$_n$), stating that any orientation-preserving homeomorphism $\mathbb{R}^n \rightarrow \mathbb{R}^n$ is stable in ...
9
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2answers
425 views

Extending homeomorphisms from closed countable sets to S^2

Let $A, B \subset S^2$ be closed, countable sets and $\phi \colon A \rightarrow B$ be a homeomorphism. Can we extend $\phi$ to a homeomorphism from $S^2$ to itself? It is well-known that the answer ...