All Questions
38 questions
4
votes
1
answer
1k
views
Original proof of the Borsuk-Ulam theorem
I am looking for the original proof by Borsuk of the Borsuk-Ulam theorem. I would appreciate very much if someone could outline the proof.
- 356
5
votes
1
answer
246
views
Does a "good" homotopy equivalence between pairs imply homotopy equivalence between quotient spaces?
If $(X,A)$ and $(Y,B)$ are (good) pairs of topological spaces, and $f:X\rightarrow Y$ is a homotopy equivalence such that the restriction $f\restriction_A$ is a homotopy equivalence between $A$ and $B$...
- 11.1k
4
votes
0
answers
107
views
Reference request for a theorem of Jaworowski
Jan Jaworowski, in 2000, proved the following theorem (I came to know about it from here)
Jaworowski (2000) : Let $Y$ be a finite simplicial complex of dimension $k$ and let $n\ge 2k$. If $f:S^n\to Y$...
- 141
0
votes
0
answers
150
views
Connectedness of deleted symmetric product
Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
- 10.9k
2
votes
0
answers
185
views
Properties of universal fibration
I am trying to read the following paper [1] (Becker, James C.; Gottlieb, Daniel Henry
Coverings of fibrations.
Compositio Math.26(1973)) where the authors mentioned that for any fiber $F$,
there ...
- 179
2
votes
0
answers
101
views
Concrete topological objects and notions in the category of locales
I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...
- 121
9
votes
0
answers
211
views
Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?
Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...
- 15.9k
1
vote
0
answers
110
views
Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
1
vote
1
answer
388
views
About isotopy of simple close curve
In the Primer mapping class group by farb Margalit. We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...
- 28.2k
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 ...
- 35.9k
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:\...
- 10.9k
9
votes
0
answers
308
views
Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
- 4,519
0
votes
1
answer
554
views
Is the meaning of "irreducible manifold", "not reducible to other manifold"?
This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or ...
CommunityBot
- 1
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 ...
- 188k
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. ...
- 1,869
4
votes
1
answer
314
views
Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request
I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
- 30.3k
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
2
votes
2
answers
349
views
Fibration of principal bundles
Let $G$ be a topological group, let $f:X\rightarrow Z$ be a $G$-equivariant map of (left) $G$-spaces such that
$X\rightarrow X/G$ and $Z\rightarrow Z/G$ are principal $G$-bundles.
$f$ is a ...
- 11.1k
6
votes
0
answers
169
views
Whitney stratification for proper morphisms
Let $f: X \to \Delta$ be a flat, projective morphism, smooth over the punctured disc $\Delta^*:=\Delta \backslash \{0\}$ and central fiber $f^{-1}(0)$ is a reduced, simple normal crossings divisor. ...
- 1,593
4
votes
1
answer
220
views
Cofibrations and mapping spaces in compactly generated weak Hausdorff spaces
Assume that $X$ and $Y$ are compactly generated weak Hausdorff spaces (CGWH spaces for short). Assume that they are also well-pointed (so the inclusions of the base points are Hurewicz cofibrations).
...
- 56.9k
15
votes
0
answers
716
views
Is this "Homology" useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
- 356
3
votes
1
answer
155
views
A pair of spaces equivalent to a pair of CW-complexes
Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that
$Z-A$ is homeomorhic to $X-Y$ and
$Z/A$ homeomorphic to $X/Y$ and
The closure of $Z-A$ ...
- 55.9k
24
votes
2
answers
4k
views
complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
- 4,034
7
votes
0
answers
119
views
The automorphism group of the fibered cylinder
My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...
- 41.9k
0
votes
1
answer
377
views
How to prove that there does not exist any plane isotopy from the logarithmic spiral onto the real line? [closed]
Questions.
EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a strong a-forteriori-reason, kindly ...
- 91.8k
4
votes
1
answer
153
views
The homological negligibility of certain subsets in compact manifolds
Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).
I need a reference to the following facts (which I believe are true at least in dimension $n=3$):
Fact 1. For every ...
- 17.5k
1
vote
1
answer
179
views
Is there any upper bound on the LS-category of open $n$-dimensional submanifolds of $\mathbb{R}^n$?
Suppose $X\subseteq\mathbb{R}^n$ is a connected open bounded $n$-dimensional submanifold.
1) I wonder if for the class of such spaces there is any upper bound on the LS-category of $X$?
2) Is it ...
- 3,173
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 $K$ ...
- 8,803
17
votes
2
answers
1k
views
Who first defined _simply connected_, reference?
The following definition is due to Donald J. Newman:
A connected open subset $D$ of the plane $\mathbb C$
is simply connected
if and only if its complement $\widetilde D = \mathbb C \setminus D$
...
- 1,375
12
votes
0
answers
313
views
For a Banach space $X$, when is $X$ homeomorphic to $X \setminus A$?
$\mathbb{R}^n\not\cong\mathbb{R}^n\setminus\{0\}$ are not homeomorphic is a triviality from Algebraic Topology. On the other hand, if $X$ is an infinite dimensional Banach space, then $X \cong X\...
- 43.2k
1
vote
0
answers
251
views
Copylefted introduction to topology
Is there a textbook in topology with a copyleft license?
$$ $$
- 45k
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
12
votes
2
answers
1k
views
Concrete examples of covering from the 3-torus to the 3-sphere
There is a two-fold branched covering from 2-torus to the 2-sphere, $T^2 \rightarrow S^2$, whose covering transformation group is generated by the map $x \mapsto -x$ (Note that $T^2$ is an abelian ...
- 19.4k
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+...
- 18.7k
8
votes
1
answer
480
views
Face poset of a subcomplex complement
Let $P$ denote the face poset of a simplicial complex, $\Delta$ the order complex of a poset, and $\sim$ homotopy equivalence. It's known that for any finite simplicial complex $\mathcal{K}$ that $\...
- 191
22
votes
0
answers
676
views
Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?
Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \...
- 15.5k
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
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 34.7k