# Questions tagged [homeomorphism]

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### Is it known whether a homeomorphism close to the identity of a compact manifold with nonzero Euler characteristic necessarily has a fixed point?

I recently saw in Kirby's list of open problems that it isn't known if two commuting homeomorphisms of a compact manifold close to the identity necessarily share a common fixed point, when the ...
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### Galois connection for homeomorphisms

let $M = \mathbb{R^2}$ and $X = \{0\}$ and $G = Aut_X(M)$ the group of homeorphisms fixing $X$ (pointwise). Then we have, in analogy to classical Galois theory for field extensions, a Galois ...
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### Decidability of whether two polynomial bijections generate a free group

I am wondering about the decidability of the following question: Given two polynomial bijections $f, g$ from the real numbers to the real numbers (with say rational coefficient just to simplify what &...
1 vote
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### Banach tori: classification up to Fréchet homeomorphisms

Consider the set $T$ in $l_p$ defined as closure of \begin{equation} T = \{ (x_1,\dotsc,x_n,\dotsc): x_j = \frac{1}{2^{(j/p)}} e^{it_j}, t_j \in \mathbb{R}/\mathbb{Z} \}. \end{equation} This seems to ...
359 views

### Exotic homeomorphisms of a cube

If $\varphi:\mathbb{R}\to\mathbb{R}$ is continuous, non-constant, non-decreasing, and differentiable a.e. with $\varphi'=0$ a.e., then the mapping $$\Phi(x,y)=(x+\varphi(x),y+\varphi(y))$$ is a ...
1 vote
60 views

### When a support of an isotopy is disjoint from a subset

Let $M$ be a compact connected manifold, $X\subset M$ a closed subset, and $f:M \times [0;1] \to M$ an isotopy such that each $f_t:M \to M$ is fixed on some open neighborhood $N_t$ of $X$, but there ...
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### Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?

Suppose we are given embeddings $f_1,f_2:[0,1]\to\mathbb R^3$. Does there exist a homeomorphism $g:\mathbb R^3\to\mathbb R^3$ such that $g\circ f_1=f_2$? This question seems to be classical eighty ...
110 views

### When is a composition of homeomorphisms topologically transitive provided one of the two is?

Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...
1 vote
225 views

### Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?

I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below). It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be written as an ...
1 vote
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### What is t-equivalence in function spaces?

In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...
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### Induced homeomorphism from a quasi-isometry between hyperbolic spaces

Theorem. Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$. If $X$ and $Y$ are proper, then ϕ induces a homeomorphism between their boundaries. The proof ...
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### What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?

If $M$ is a smooth connected closed $n$-dimensional manifold, its universal covering space is homeomorphic to Euclidean space $R^{n}$, and its fundamental group is $Z^{n}$, then is it homeomorphic ...
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### Existence of a Hölder homeomorphism satisfying prescribed norm constraints

Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k_1> k_2>0$. Does there ...
1 vote
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### Disjoint union of clopen sets such that the fibers has constant cardinality [closed]

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that : $X=\sqcup_{i=1}^{n}U_i$ ...
1 vote
93 views

### Existence of global section, étale map and totally disconnected space

I am trying to show the following result : Let $Y$ be a totally disconnected space and compact space, $X$ a locally compact space and $p:Y\to X$ a surjective local homeomorphism. Then, there exist ...
1k views

### Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\mathbb{R}^3$

I was wondering if there were a proof of the fact that $$\mathbb{R}^3 \setminus \{p_1,\dots,p_n\} \: \text{is not homeomorphic to} \: \mathbb{R}^3$$ for every $n \geq 1$ that does not use cohomology ...
1 vote
351 views

### Extension of homeomorphisms

Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical ...
1 vote
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### 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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### 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 ...
1 vote
83 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 ...
406 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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### 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 $\mathrm{Borel}(M)$ ...
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 ...
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 ...