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Results tagged with gt.geometric-topology
Search options questions only
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user 121665
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).
6
votes
1
answer
459
views
Rational homology spheres and geometric properties of the Wu manifold
I am interested in simply connected rational homology spheres. The first such example is in dimension 5 and it is the Wu manifold $SU(3)/SO(3)$. You can find a discussion about simply connected ration …
- 28k
9
votes
1
answer
658
views
A wild embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$
Can one construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$
so that every orthogonal projection onto a two dimensional plane
is a unit disc?
It is easy to construct an embedding of $ …
- 28k
9
votes
4
answers
1k
views
Homology sphere with $\mathbb{R}^3$ as the universal cover
Question. Is there a $3$-dimensional integer homology sphere whose universal cover is $\mathbb{R}^3$?
I believe the answer is in the positive and I am looking for (precise) references. If not in …
- 28k
2
votes
0
answers
147
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
31
votes
3
answers
1k
views
Non embedding of $Y\times Y$ into $\mathbb{R}^3$
I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type.
Theorem. Let $Y$ be the union of two segments wi …
- 28k
10
votes
0
answers
783
views
Topological dimension, Hausdorff dimension, and Lipschitz mappings
I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure.
Theorem. Suppose that $f:\mathbb{R}^n\supset\Omeg …
- 28k
8
votes
1
answer
339
views
Characterizations of metric trees
Let $X$ be a geodesic space. Then the following conditions are equivalent:
For any $x,y\in X, x \neq y$, there is a unique arc (homeomorphic to the interval $[0,1]$) with endpoints $x$ and $y$.
No su …
- 28k
10
votes
0
answers
241
views
Bi-Lipschitz mappings
Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a neig …
- 28k
12
votes
0
answers
406
views
Is the Lipschitz structure on $\mathbb{S}^4$ unique?
Sullivan [S] proved that on any $n$-dimensional, $n\neq 4$, topological manifold there is a unique Lipschitz structure. Although it is generally accepted result, it seems that the proof lacks some det …
- 28k
10
votes
2
answers
673
views
Bi-Lipschitz extension
Given a bi-Lipschitz homeomorphism
$\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ s …
- 28k
7
votes
1
answer
375
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 home …
- 28k
2
votes
0
answers
185
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 thi …
- 28k
12
votes
1
answer
730
views
Isotopic diffeomorphisms of the sphere
Assume that $f:\mathbb{S}^n\to\mathbb{S}^n$ is a diffeomorphism and assume that there is an orientation preserving diffeomorphism $F:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that $F|_{\mathbb{S}^n}=f …
- 28k
15
votes
1
answer
920
views
Extending diffeomorphisms
Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
Question. Is it possible to ex …
- 28k
15
votes
3
answers
1k
views
Linking topological spheres
Is there a simple proof of the fact that:
If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$
embedded into $S^3\setminus A$ that such that the circles $A$ and $B$
are link …
- 28k