All Questions
Tagged with gn.general-topology manifolds
72 questions
2
votes
0
answers
252
views
Special orthogonal groups over spheres
In Norman Steenrod's book "The Topology of Fibre Bundles", on page 37, one can find the following conjecture: if $n$ is a power of two then the fibre bundle with the projection $SO(n)\to SO(n)/SO(n-1)=...
1
vote
0
answers
80
views
Extending maps to disc homeomorphisms isotopic to the identity
Consider the closed unit disc $\mathbb D^n$ in $\mathbb R^n$ and its closed subdisc $D$ centered at the origin with radius $1/2$. Denote by $V$ the interior of $\mathbb D^n$. I wonder whether the ...
4
votes
0
answers
112
views
Bundle structures on spheres
Given a positive integer $n$, there is a well known free action of $\mathbb T^1$ on $\mathbb S^{2n-1}$ due to Hopf, which makes $\mathbb S^{2n-1}$ a fibre bundle with the fibre $\mathbb T^1$. Moreover,...
2
votes
0
answers
305
views
Are homotopy equivalent manifolds with homeomorpic boundaries themselves homeomorphic?
Let $f:M \to M′$ be a homotopy equivalence of topological manifolds with boundary such that $dim(M)=dim(M′)$ and $f:\partial M \to \partial M′$ is a homeomorphism. Does this imply the existence of a ...
- 79
1
vote
1
answer
932
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Every topological manifold is a ENR? (Reference)
It seems to be widely known that every topological manifold can be embedded as a neighbourhood retract in euclidean space, I can not find a reference, though.
The reason, why I'm asking this, is that ...
- 1,082
3
votes
1
answer
292
views
Is the complement of the ends of a manifold bounded?
Let $M$ be a connected manifold with precisely $k$ ends $\epsilon_1,...,\epsilon_k$. Choose a collection $(U_i)_{i=1}^k$ of pairwise disjoint open $\epsilon_i$-neighborhoods. Then I wonder how to ...
- 1,255
5
votes
1
answer
214
views
Which combinations of normality, separability, and paracompactness do complex manifolds possess?
I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have.
Is there a non-separable complex manifold? Can a non-separable complex ...
3
votes
1
answer
117
views
Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces?
For any $n$ dimensional closed manifold $M^n$, can we find an open covering $\{U_i\}_{i\in[2^n]}$ such that $M=\cup U_i$ and each $U_i\cong \mathbb R^n$? How about complex manifolds (replacing $\...
- 301
2
votes
1
answer
274
views
Digital topology, animal problem, 2-sphere and torus
I have the following question relating digital topology, surfaces, particularly $S^2$ and torus.
Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to ...
40
votes
1
answer
2k
views
Are there only countably many compact topological manifolds?
Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...
- 3,017
12
votes
2
answers
520
views
Homeo-Fixed point property
Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...
- 356
1
vote
0
answers
91
views
Tubular neighbourhood which is nowhere piecewise linear
I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
- 11
1
vote
0
answers
178
views
Proving that two given functionally structured spaces are isomorphic
The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
- 111
0
votes
1
answer
341
views
Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
- 500
11
votes
2
answers
2k
views
When is the connected sum of manifolds orientation-independent?
Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed?
If $N$ ...
- 931
19
votes
4
answers
4k
views
When is a finite cw-complex a compact topological manifold?
I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
- 732
4
votes
3
answers
1k
views
cayley transform for non-square matrices
Hi,
I am optimizing a function over a matrix $U$, where $U \in \mathbb{R}^{m \times n}$ and $U^TU = I$. I do not want to run a constrained maximization program, since employing the constraint $U^TU = ...
- 263
0
votes
1
answer
271
views
Numbers associated with boundaries of manifolds
I don't know what name if any is attached to the numbers I'm about to describe.
For a line segment, [a,b]
the number is 1 if for any k in (a,b)
and 2 if k=a or k=b.
For a square, [a,b] ...
- 379
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
5
votes
0
answers
196
views
Is there a Whitney-type theorem Cauchy manifolds?
Let $M$ be a Cauchy space whose induced topological space is a second-countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^m$.
Does it follow that there exists a subspace $N$ of $\...
user5810
58
votes
8
answers
9k
views
Is there a Whitney Embedding Theorem for non-smooth manifolds?
For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can ...
- 825
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