All Questions
16 questions with no upvoted or accepted answers
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
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
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
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
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
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
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
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
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
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
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
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
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
0
answers
251
views
Copylefted introduction to topology
Is there a textbook in topology with a copyleft license?
$$ $$
- 45k
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 ...
- 674