All Questions
22 questions
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.6k
21
votes
7
answers
1k
views
Reference for topological graph theory (research / problem-oriented)
I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
16
votes
2
answers
820
views
Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
- 45k
16
votes
1
answer
521
views
Extending a map from $S^n\to M^n$ to a nice map from $B^{n+1}\to M^n$
Let $S^n$ and $B^{n+1}$ be the unit sphere and unit ball in $\mathbb{R}^{n+1}$, and let $M^n$ be a contractible space of dimension $n$.
If necessary, assume that $M^n$ is a contractible simplicial $n$-...
- 368
15
votes
3
answers
1k
views
What do absolute neighborhood retracts look like?
In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. ...
- 64k
14
votes
4
answers
1k
views
Obtain any 3-manifold from repeating surgeries on knots in $S^3$
In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...
- 755
14
votes
1
answer
938
views
Classification of 3-dimensional manifolds with boundary
It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as
$$\mathcal{M}=P_{1}\#\dots\# P_{n}$$
where $P_{i}$ are prime manifolds, i.e. ...
- 1,429
13
votes
3
answers
2k
views
A quotient space of complex projective space
Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...
- 331
12
votes
4
answers
1k
views
Elementary proof that knot complements are path-connected
The complement of any (topological) knot is path-connected. More precisely, if $K$ is a subset of $\mathbb{R}^3$ (or $S^3$) homeomorphic to $S^1$, then $\mathbb{R}^3\setminus K$ (or $S^3\setminus K$) ...
- 35.9k
7
votes
1
answer
183
views
Stability Question for Isotopies Between Compact Sets
Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$.
Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
- 439
6
votes
1
answer
500
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 41.9k
6
votes
1
answer
300
views
Proof of Denjoy-Riesz Theorem and Moore's Generalization?
The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
- 439
5
votes
1
answer
216
views
Continuity of taking collapse maps
Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their ...
- 666
5
votes
0
answers
265
views
Quotienting disk inside sphere result in sphere
Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where $S^...
- 2,023
4
votes
0
answers
154
views
Is there a notion of "locally flat" for CW complexes?
A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\...
- 13.6k
4
votes
0
answers
182
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 41.9k
4
votes
0
answers
183
views
In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?
I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...
- 13.6k
3
votes
1
answer
319
views
Fixed point property for intersection of spaces which are homeomorphic to a disk
The following question is question 9.8 from Miller's paper ``Some interesting problems
'':
Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every
$n\in \omega, D_{n+...
- 32.2k
3
votes
0
answers
88
views
Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?
Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
- 13.6k
3
votes
0
answers
102
views
Find a certain triangulation subordinate to a given covering of a manifold
Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...
- 2,789
3
votes
0
answers
359
views
Cubical approximation theorem for cubical complexes
A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.
I have found a claim ...
- 985
2
votes
0
answers
208
views
Retracting to a bigger compact
Consider the topological spaces $X$ with the following property:
For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a retract of $X$.
Let ...
- 128k