All Questions
Tagged with gn.general-topology homotopy-theory
132
questions
1
vote
0
answers
201
views
Examples of when $X$ is homotopy equivalent to $X\times X$
I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
2
votes
0
answers
86
views
Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
- 3,697
2
votes
0
answers
109
views
Homotopy type of a 3-manifold produced via Dehn surgery?
My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology.
I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...
- 285
3
votes
2
answers
264
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Cut a homotopy in two via a "frontier"
Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$.
(...
3
votes
0
answers
189
views
Contractibility of the pseudo-boundary of the Hilbert cube
Let the separable Hilbert cube $Q=\prod_{i=1}^{+\infty}[0,1]$ embed into the real Hilbert space $H=l^2(\mathbb{Z}^+)$, whose coordinate unit vectors are $\{ e_i \}_{i=1}^{+\infty}$, as the subset $\...
- 1,069
4
votes
0
answers
420
views
Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$
Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3).
How to show the composition
$$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$
is non-trivial ...
4
votes
0
answers
192
views
path category and classifying space
Let $\mathbf{Top}$ be the category of topological spaces and continuous maps, and $\mathbf{Cat}$ be the category of small categories and functors.
There is a path functor $\mathcal{P}:\mathbf{Top}\to \...
- 123
5
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0
answers
115
views
Under what assumption on a proper map does the preimage of sufficiently small neighborhood is homotopy equivalent to the fiber?
Let $\pi\colon X\rightarrow Y$ be a proper map of topological spaces. Let's assume that both $X$ and $Y$ are paracompact, Hausdorff and locally weakly contractible. Then is it enough to conclude that ...
- 790
1
vote
0
answers
67
views
Powersets of simplicial sets vs. powersets of topological spaces
Motivation. Recently I've been trying to understand how well- or ill-behaved are the many different powerset topologies one can put on the powerset of a topological space, and in particular I'm trying ...
- 9,727
3
votes
2
answers
217
views
Uniformly continuous homotopy equivalence
Suppose $M$ and $N$ are complete metric spaces and $f, g: M \to N$ are uniformly continuous maps between them with common modulus of continuity $m$. Further suppose $f$ and $g$ are homotopy equivalent....
- 2,209
11
votes
1
answer
449
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A topological tree is weakly contractible
Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
- 559
-1
votes
2
answers
229
views
Function space and contractibility
$\DeclareMathOperator\map{map}$I have the following question:
Let $X$ and $Y$ be topological spaces. Let $\map(X,Y)$ denote the space of non-constant continuous functions from $X$ to $Y$. Suppose ...
- 315
30
votes
1
answer
2k
views
What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
- 25.6k
2
votes
1
answer
278
views
(Homotopy) colimit and manifold
Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by $D$, over the CW ...
- 37
8
votes
1
answer
373
views
Finite domination and compact ENRs
Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only ...
- 18.5k
1
vote
1
answer
83
views
Vanishing of $H^*(f^{-1}[0,c], f^{-1}(0))$ for small $c$, and $f\in C^0(X, [0,+\infty))$
Let $X$ be a topological space and consider a continuous function $f:X\to [0,+\infty)$. For $c\geq 0$ set $X_c := f^{-1} ([0,c])$.
Furthermore, suppose that $X_0 \neq \emptyset$ and $f$ is proper.
...
- 2,503
8
votes
1
answer
806
views
What's the point of a point-free locale?
In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...
- 4,760
4
votes
0
answers
179
views
are trivial fibrations of finite CW-complexes soft for normal maps?
Are trivial Hurewicz fibrations of finite CW-complexes soft for normal maps,
i.e. is it true that for any trivial Hurewicz fibration $f:Y_1\to Y_2$
and a closed subset $A$ of a hereditary normal space ...
- 317
2
votes
0
answers
196
views
are acyclic fibrations of nice spaces absolute extensors for perfectly normal spaces?
A space $Y$ is called an absolute extensor for normal spaces (also sometimes solid) if, for any normal space $X$, closed subset $A$ of $X$, and map $f:A\to Y$, there exists a map $f′:X\to Y$ such that ...
- 317
3
votes
2
answers
441
views
Can the loops in the definition of the fundamental group be considered injective?
Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all ...
- 6,164
1
vote
0
answers
143
views
Lifting theorem for finite spaces: replacing perfect normality by normality
In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below),
can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to
"$A\to X$ has the right ...
- 317
1
vote
1
answer
142
views
Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space [closed]
Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
$$
G \supset J.
$$
If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, ...
- 317
7
votes
1
answer
199
views
Quasifibrations and transfinite filtrations
This question takes place in the category $\mathrm{CGWH}$
of compactly generated weak Hausdorff spaces.
Let $\lambda$ be a limit ordinal, and suppose we have
a diagram $\Phi: \lambda \to \mathrm{CGWH}$...
- 12.4k
3
votes
1
answer
648
views
Motives and topological data analysis
Here is some meta mathematics question.
During the last decade there has been some progress in the field of applied maths, called topological data analysis.
The setup starts with some set of points in ...
- 1,397
5
votes
1
answer
363
views
$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?
Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?
My understanding so far —
An $\...
- 10.3k
1
vote
0
answers
237
views
Spaces homotopy equivalent over the topologist's sine curve
Consider $$T=\left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in [-1, 0)\cup(0,1] \right\} \cup \{(0,0)\}\subset \mathbb{R}^2$$
with the subspace topology.
Denote $p=(-1, \sin -1), q=(1, \sin 1)\...
- 51
1
vote
1
answer
144
views
Do Locally Contractible, Path-Connected Groups have Accessible Bases?
Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\...
- 409
3
votes
0
answers
129
views
Null-homotopic cellular loops are elementary null-homotopic?
I've got a 2-dimensional cell complex $X$ and a cellular closed loop $l \subset X$ that I happen to know is null-homotopic in $X$.
There are some very simple sorts of homotopies of cellular loops (or ...
- 5,309
6
votes
2
answers
372
views
Is an open subset of a cofibration a cofibration?
Suppose $A \to X$ is a cofibration in topological spaces, and $U \subseteq X$ is an open subset. Is $U \cap A \to U$ a cofibration?
Sorry if this is rather simple, but I don't have much experience ...
- 1,317
13
votes
2
answers
1k
views
Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\mathbb{R}^3$
I was wondering if there were a proof of the fact that $$\mathbb{R}^3 \setminus \{p_1,\dots,p_n\} \: \text{is not homeomorphic to} \: \mathbb{R}^3$$
for every $n \geq 1$
that does not use cohomology ...
- 1,313
4
votes
1
answer
234
views
Density of compactly-supported homeomorphisms
**Disclaimer:**I posted the following question on MSE, but since there were no answers. I'm migrating it here.
Let $Homeo_0(\mathbb{R}^n)$ ($Homeo_c(\mathbb{R}^n)$) be the space of all (compactly-...
- 5,001
2
votes
1
answer
1k
views
How do we know that a surface minus finite number of points is homotopy equivalent to a bouquet of circles? [closed]
In this post (Homotopy Equivalence of Punctured Tori), the author of the first answer states that a surface minus finite number of points is homotopy equivalent to a bouquet of circles. However, it ...
- 21
2
votes
0
answers
193
views
Products of cones and cones of joins
The join of $A$ and $B$ is the pushout of the diagram
$$
CA \times B \gets A\times B \to A\times CB,
$$
which can be formulated in either the pointed or unpointed topological
category. This pushout is ...
- 12.4k
9
votes
0
answers
327
views
Homotopical characterization of CW complexes
Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
- 117
9
votes
0
answers
198
views
Homotopical characterization of manifolds
Let $X$ be a compact metrizable topological space of covering dimension $4$.
Assume that for any point $x\in X$ any neighbourhood of $x$ contains a contractible open neighbourhood $U$ such that $U\...
- 117
1
vote
0
answers
134
views
Complement of contractible locally Euclidean subspace
Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
- 11
1
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0
answers
152
views
Homotopy groups of ball complement
Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
- 19
5
votes
1
answer
437
views
Any continuous map is homotopic to one assuming fixed values at finitely many points
Let $X$ and $Y$ be topological spaces. Assume $X$ is locally contractible and has no dense finite subset. Assume $Y$ is path-connected.
Given $n$ pairs of points $(x_i, y_i)$ where $x_i\in X$ and $y_i\...
user145520
4
votes
0
answers
335
views
When every closed and connected subset is path connected
Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?
6
votes
0
answers
202
views
Intereresting classes of topological spaces locally modelled on some fixed spaces
A substantial part of mathematics studies manifolds which are defined as second countable Hausdorff locally Euclidean topological spaces. That always seemed kind of random to me since what is so ...
user158636
3
votes
0
answers
186
views
Does the suspension spectrum functor preserve weak equivalences?
Let $\Sigma^{\infty}$ denote the suspension spectrum functor from pointed topological spaces (=CGWH spaces) to orthogonal spectra. As usual, a weak equivalence of spaces is a continuous map inducing a ...
- 858
4
votes
1
answer
389
views
Contractible chain complex from non-contractible space
Recall that a chain complex $(C_*,d)$ of abelian groups is contractible if it is homotopic to the zero map. Or equivalently: there exists a degree 1 map $F: C_* \to C_*$ such that $\operatorname{Id}= ...
- 177
6
votes
1
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433
views
Map which is null-homotopic on compacts
This is the missing ingredient towards answering my previous question.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...
- 5,275
4
votes
1
answer
687
views
Classifying space BG and contractable space EG
This question is probably not research level that's why I asked it previously on MSE a week ago. Unfortunately it doesn't get much attention there and I thought I would try it here.
Choose a ...
- 5,716
3
votes
0
answers
323
views
About the Moore composition of paths
1) QUESTION (EDIT: 04/28/2020 to remove a possible counterexample)
I work with weak Hausdorff $k$-spaces (so all spaces are $T_1$). The internal hom is denoted by $\mathbf{TOP}(-,-)$. Let $\mathcal{G}...
- 3,697
1
vote
0
answers
131
views
Identifying the two points of a subspace homeomorphic to a Sierpinski space
Let $X$ be a $\Delta$-generated space having a subset $A=\{a,b\}$ such that the relative topology is the Sierpinski topology with for example $\{a\}$ closed and $\{b\}$ open (the Sierpinsky space is a ...
- 3,697
13
votes
1
answer
677
views
Explicit isomorphism $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1})$
From covering space theory we know that $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_{n+1}(\mathbb{S}^n)$.
From wikipedia I can notice that $\pi_{n+1}(\mathbb{S}^n) \cong \pi_1(\mathbb{RP}^{n-1})$.*
My ...
- 1,253
11
votes
1
answer
700
views
Space with semi-locally simply connected open subsets
A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
- 283
12
votes
1
answer
688
views
Open subspaces of CW complexes
I am looking at the paper
Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West
and a claim is made in the proof of their first main theorem ...
- 12.4k
10
votes
1
answer
849
views
In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?
I previously asked In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also?
Given the broad scope of this question I ...
- 4,590