All Questions
Tagged with homeomorphism gt.geometric-topology
12 questions
2
votes
0
answers
161
views
Stable homeomorphism theorem for bi-Lipschitz mappings
The stable homeomorphism theorem says that:
Every orientation preserving and surjective homeomorphism $f:\mathbb{R}^n\to\mathbb{R}^n$ can be written as $f=f_1\circ\ldots\circ f_k$, where $f_i:\mathbb{...
- 28k
2
votes
0
answers
193
views
Stable homeomorphism theorem and the annulus theorem
Brown and Gluck [BG] proved in 1964 that the stable homeomorphism conjecture implies the annulus conjecture.
Is the proof of this implication difficult?
Is there any other place with the proof of ...
- 28k
7
votes
1
answer
379
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 ...
- 28k
9
votes
1
answer
519
views
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 ...
- 1,095
1
vote
1
answer
110
views
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
1
answer
421
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 ...
- 5,407
1
vote
1
answer
208
views
Is every homeomorphism approximately a product of homeomorphisms?
Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
- 5,407
1
vote
0
answers
96
views
Constructing homeomorphisms from continuous functions and matrix exponentials
Fix a $d\times d$ matrix $A$, let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous function, and define the induced map $F_{f,A}:\mathbb{R}^d\rightarrow \mathbb{R}^d$ by
$$
x \mapsto \exp(f(x)A)...
- 5,407
3
votes
0
answers
75
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 $S^...
- 396
6
votes
2
answers
522
views
generators for the handlebody group of genus two
Is the handlebody group of genus two surface generated by Dehn twists along properly embedded disks and annuli?
Are there alternative ways to describe a set of generators that are conceptually simple ...
- 858
7
votes
1
answer
682
views
Proof of the stable homeomorphism conjecture
I would like to get a demonstration of the stable homeomorphism conjecture (SHC$_n$), stating that any orientation-preserving homeomorphism $\mathbb{R}^n \rightarrow \mathbb{R}^n$ is stable in ...
- 73
13
votes
3
answers
619
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 ...
- 131