All Questions
Tagged with gn.general-topology gt.geometric-topology
329 questions
34
votes
1
answer
2k
views
Square roots of $\mathbb R^{2n}$
Recently, Richard Dore asked us if $\mathbb R^3$ is the cartesian square of some space, and Tyler Lawson answered beautifully in the negative.
The even powers of $\mathbb R$ were left out in that ...
- 47.3k
0
votes
0
answers
365
views
Finding paths in a path connected space
I'm looking for such literature as exists relevant to the following problem.
Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path ...
- 627
23
votes
13
answers
7k
views
What should be taught in a 1st course on smooth manifolds?
I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic ...
1
vote
1
answer
606
views
About deformation retract
Let $X\subset Y$ be CW-complexes. Denote $i\colon X\to Y$ be an inclusion map.
Is it true that $i$ is deformation retract if and only if $i$ is homotopy equivalence?
When I saw some papers about h-...
- 13
7
votes
2
answers
594
views
Computational cost of converting between 3-manifold presentations
Given a 3-manifold presented as a triangulation, a Heegaard splitting, or a Dehn surgery, what is the computational cost of converting to the other two presentations? I would like Heegaard splittings ...
- 135
18
votes
2
answers
2k
views
Which platonic solids can form a topological torus?
8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...
- 453
3
votes
1
answer
958
views
When does an antipodal map on a manifold extend to the antipodal map on a spheres
So I have been mulling the following question over in my head for awhile now, and want to see if anyone else might have any ideas.
Begin with $M$ a manifold and suppose that $M$ has an antipodal map $\...
- 5,191
2
votes
2
answers
1k
views
When is the group of homeomorphisms of a compact space locally compact?
When is the group of homeomorphisms of
a compact space locally compact?
I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate topology ...
- 1,771
8
votes
6
answers
2k
views
Uncountable preimage of every point
Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable?
Added: Must it have a point $x$ that $dim_H(f^{-1}(x))=0$ ? ($dim_H$ means the ...
- 5,053
18
votes
5
answers
2k
views
Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
I'm curious about the following:
Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Thanks.
EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...
- 761
8
votes
1
answer
223
views
local structure of free $\mathbb{R}$ actions
Assume the topological group $\mathbb{R}$ acts properly on a space $X$. Does then the projection map $p:X\rightarrow \mathbb{R}\backslash X$ have local sections ?
(for every $\mathbb{R}x\in \mathbb{R}...
- 11.1k
2
votes
1
answer
727
views
pseudo-Anosov maps on surfaces with boundary
In "Automorphisms of Surfaces after Nielsen & Thurston" by Casson & Bleiler (on pages 75 - 80) they discuss classifying automorphisms of a surface. They show that, if $S$ is a closed ...
- 3,165
20
votes
2
answers
1k
views
Rugged manifold
It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary.
Is it still true for topological manifold?
Comments:
To proof the smooth ...
- 45k
27
votes
1
answer
4k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 6,214
11
votes
3
answers
2k
views
Permute Wada Lakes keeping the coastline intact? (still open in dim >2)
Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical ...
- 4,188
3
votes
3
answers
384
views
Collapsing contractible subsets of the two-disk.
This question is quite specific, but it may admit answers in more general contexts.
Consider a subset $\Lambda \subset D^2$ where $D^2$ is the two dimensional disk.
We consider in $\Lambda$ an ...
- 3,928
5
votes
1
answer
296
views
Solenoid of a continuous map of a ball, is it contractible?
Let $B$ be the closed unit ball in $\mathbb R^n$ and $f\colon B\to B$ a continuous map.
Consider the infinite product $B^{\mathbb Z}$ equipped with the product topology. Consider the solenoid
$$
S_f=\...
- 4,188
5
votes
1
answer
1k
views
Do continuous maps give continuity in the 'topology' of Hausdorff distance?
I was reading this question:
limiting behaviour of converging loops on a torus
And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their ...
- 3,230
7
votes
1
answer
789
views
Counting submanifolds of the plane
After thinking about this question and reading this one I am led to ask for an uncountable collection of homeomorphism types of boundaryless connected path-connected submanifolds of the plane.
My ...
- 28.2k
7
votes
2
answers
419
views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
30
votes
5
answers
2k
views
Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?
Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)?
If not true in general, is there any condition ...
10
votes
2
answers
367
views
existence of a connected set with given connected projections.
Suppose A and B are compact connected sets in the XY plane and XZ plane respectively in R^3. Suppose further that the the range of x-values taken by A and B are the same (i.e, projections of A and B ...
- 515
4
votes
1
answer
2k
views
Fiber bundle = principal bundle + fiber?
This question is heavily related to this question.
Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\...
- 1,085
4
votes
3
answers
1k
views
Morse theory and Euler characteristics
Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of ...
- 1,129
5
votes
2
answers
1k
views
Relating Euler characteristic, intersection product, Morse theory (plus SU(2) and 3-manifolds)
Suppose we have a (closed, oriented) 3-manifold M with a Heegard surface F of genus g. Let F* denote F with a puncture. Then the space H of representations of pi_1(F*) on SU(2) is just SU(2)^2g, and ...
- 1,129
-4
votes
4
answers
677
views
What is the max number of points in R^3, interconnected by generic curves?
The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of it....
- 555
5
votes
1
answer
320
views
Ramified covers of S^n
This question has been inspired by covering 3-torus post.
Is it true that any good (smooth, compact, oriented) $n$-manifold can be mapped to $S^n$ in such a way that the map is true covering away ...
- 15.1k
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 44.4k
4
votes
2
answers
439
views
Legendrian homotopy of curves in a contact structure?
I'm aware of the great body of work on Legendrian knot theory in contact geometry, but suppose I'm curious just about homotopy and not isotopy. How does one understand the space of Legendrian loops ...
- 13.6k