1
vote
0answers
62 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 ...
1
vote
0answers
85 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 ...
0
votes
1answer
173 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, ...
8
votes
2answers
714 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 $N$ ...
6
votes
3answers
911 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 ...
2
votes
2answers
466 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 = ...
0
votes
1answer
253 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] ...
15
votes
5answers
1k 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 ...
5
votes
0answers
179 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 ...
5
votes
5answers
772 views

topologically homogeneous space?

I want to know the example which satisfies the following. X is topological space. for every point x,y in X, there exist open nbhd Ux,Uy of x,y which are homeomorphic. X has some kind of good ...
20
votes
7answers
3k 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 ...
7
votes
1answer
529 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 ...