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Results tagged with gt.geometric-topology
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user 4354
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
Accepted
A question about how completely a rectifiable arc can fill a non-empty compact continuum tha...
No, the lengths must go to infinity.
If $s$ is a path of length $l$ in the plane and $\varepsilon>0$, then the area of the $\varepsilon$-neighborhood of $s$ is no greater than $20\varepsilon(l+\varep …
- 32.4k
7
votes
A question about axes of symmetry in the plane.
Here is a more general fact: if $J$ is central-symmetric w.r.t. a point $O$ (in our case, this is the intersection of the axes), then $O$ is inside.
Indeed, let $D$ be the domain bounded by $J$. We k …
- 32.4k
6
votes
Properties of the n-dimensional Stereographic Projection
I don't know what is the "tangent cone" argument you mention, but anyway here is my favorite proof of the fact.
The stereographic projection is the restriction of an inversion $I$ from $\mathbb R^{n …
- 32.4k
12
votes
Accepted
Compactness of the class of connected sets with perimetre smaller than 1?
No. Let $B$ be a closed ball of radius 1/2 and $I$ a diameter of $B$. Construct a Cantor-like set $K\subset I$ of lengths $\mathcal H^1(K)=0.9$ (where $\mathcal H^1$ denotes the 1-dimensional Hausdorf …
- 32.4k
11
votes
Accepted
Does the metric space of compact metric spaces satisfy the binary intersection property?
No, Let $B_n\in M$ be the $n$-dimensional Euclidean unit ball and $r=\frac12+\varepsilon$ where $\varepsilon=\frac1{100}$. Then the $r$-balls in $M$ centered at $B_n$ intersect pairwise. Indeed, for $ …
- 32.4k
9
votes
Accepted
Solenoid of a continuous map of a ball, is it contractible?
No, it is not even path-connected in general, already for $n=1$.
Consider the folding map $f:[0,1]\to[0,1]$, namely $f(t)=2t$ for $t\le 1/2$ and $f(t)=2(1-t)$ for $t\ge 1/2$. There is no path connect …
- 32.4k
5
votes
Accepted
Symmetric colorings of regular tessellations
The answer is yes. Moreover, for every two different faces $A$ and $B$ there is a symmetric coloring assigning different colors to $A$ and $B$.
The isometry group $G$ is residually finite, hence here …
- 32.4k
3
votes
What is the correct notion of boundary of a submanifold?
This is too long for a comment.
For your stated goal it suffices to define the boundary as manifold's boundary (i.e., it is empty for a submanifold as defined in your first paragraph). Then you'll ha …
- 32.4k
17
votes
Accepted
Are there unique geodesics in the NIL and SOL geometry?
The geodesics between points are not unique in both cases. Moreover the following is true: if $M$ is a universal cover of a compact Riemannian manifold whose fundamental group is virtually solvable bu …
- 32.4k
21
votes
Convex Hull of Path Connected sets
This is true in $\mathbb R^2$ but not in higher dimensions. For example, consider a path in $\mathbb R^3$ that lies in the half-space $z\ge 0$ and touches the $xy$-plane at three non-collinear points. …
- 32.4k
7
votes
Accepted
Minimal-length embeddings of braids into R^3 with fixed endpoints
UPDATE.
I revisited the question and realized that verifying the local CAT(0) property is not that easy. When I wrote the original answer, I was under impression that removing any collection of codim …
- 32.4k
5
votes
Fundamental polygons with infinite pairwise identifications
I am not sure what the question is, but will try to answer anyway. This is basic general topology stuff, sorry if you meant something deeper in your question.
For any topological space $X$ and any eq …
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3
votes
Uncountable preimage of every point
Here is a formalization of André Henriques' answer to the Hausdorff dimension variant of the question.
Let $K=\{0,1\}^\infty$ be the standard Cantor set. Define a map $f:K\to[0,1]$ as follows:
for a …
- 32.4k
7
votes
Accepted
Manifolds with rectifiable curves
This is just Lipschitz structure: only locally Lipschitz maps preserve rectifiability of all curves.
What is wrong with the $x\mapsto x^{1/3}$ map is explained in Tapio Rajala's answer. (A more expli …
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5
votes
Accepted
Generalization of Radon's theorem
In dimension 1, Radon's theorem says that for any 3 points on the real line, one of them belongs to the segment between the two others. This becomes false if one replaces the real line by the tripod ( …
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