All Questions
Tagged with gn.general-topology homotopy-theory
141 questions
9
votes
1
answer
425
views
Delta-generated spaces vs CW complexes
$\newcommand\Top{\mathrm{Top}}\newcommand\CW{\mathrm{CW}}\newcommand\Deltagenerated{\text{$\Delta$-generated}}\newcommand\Spaces{\mathrm{Spaces}}\newcommand\DeltaSpaces{\text{$\Delta$-Spaces}}$I am ...
- 521
6
votes
1
answer
360
views
On connected sum of compact manifolds along a submanifold
Let $M_1$ and $M_2$ be two compact manifolds of dimension $n\ge 3$. Let us have embeddings $i_1: K \to M_1$ and $i_2: K \to M_2$ for a closed manifold $K$ of dimension at most $n-1$ such that the ...
- 506
2
votes
0
answers
156
views
Testing for weak homotopy equivalences with compact Hausdorff spaces
Let $f \colon X \to Y$ be a weak homotopy equivalence between topological spaces. If I am not mistaken, then one can rephrase this by stating that the induced map $[K,X] \to [K,Y]$ between homotopy ...
- 2,998
1
vote
0
answers
177
views
If $X$ is a strong deformation retract of $\mathbb{R}^n$, then is $X$ simply connected at infinity?
Let $X \subseteq \mathbb{R}^n$, and assume there is a strong deformation retract from $\mathbb{R}^n$ to $X$. Is $X$ necessarily simply connected at infinity?
(Edit) Follow up question: if there is a ...
- 637
2
votes
0
answers
414
views
$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]
If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
- 705
4
votes
0
answers
249
views
Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation
The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
- 2,858
6
votes
1
answer
231
views
Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
- 331
2
votes
0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
1
vote
0
answers
237
views
Examples of when $X$ is homotopy equivalent to $X\times X$
I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
2
votes
0
answers
92
views
Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
- 3,901
2
votes
0
answers
122
views
Homotopy type of a 3-manifold produced via Dehn surgery?
My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology.
I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...
- 295
3
votes
2
answers
285
views
Cut a homotopy in two via a "frontier"
Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$.
(...
3
votes
0
answers
200
views
Contractibility of the pseudo-boundary of the Hilbert cube
Let the separable Hilbert cube $Q=\prod_{i=1}^{+\infty}[0,1]$ embed into the real Hilbert space $H=l^2(\mathbb{Z}^+)$, whose coordinate unit vectors are $\{ e_i \}_{i=1}^{+\infty}$, as the subset $\...
- 1,543
4
votes
0
answers
425
views
Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$
Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3).
How to show the composition
$$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$
is non-trivial ...
4
votes
0
answers
231
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 \...
- 153
5
votes
0
answers
141
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
83
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 ...
- 11.8k
3
votes
2
answers
235
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....
- 2,221
11
votes
1
answer
493
views
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 ...
- 609
-1
votes
2
answers
259
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
33
votes
2
answers
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, ...
- 30.3k
2
votes
1
answer
313
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
468
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 ...
- 18.8k
1
vote
1
answer
85
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.
...
- 2,533
4
votes
0
answers
181
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
201
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
509
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 ...
- 6,962
1
vote
0
answers
150
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
156
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.
$$
If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, ...
- 317
7
votes
1
answer
200
views
Quasifibrations and transfinite filtrations
This question takes place in the category $\mathrm{CGWH}$
of compactly generated weak Hausdorff spaces.
Let $\lambda$ be a limit ordinal, and suppose we have
a diagram $\Phi: \lambda \to \mathrm{CGWH}$...
- 12.5k
3
votes
1
answer
757
views
Motives and topological data analysis
Here is some meta mathematics question.
During the last decade there has been some progress in the field of applied maths, called topological data analysis.
The setup starts with some set of points in ...
- 1,479
5
votes
1
answer
372
views
$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?
Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?
My understanding so far —
An $\...
- 10.5k
1
vote
0
answers
259
views
Spaces homotopy equivalent over the topologist's sine curve
Consider $$T=\left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in [-1, 0)\cup(0,1] \right\} \cup \{(0,0)\}\subset \mathbb{R}^2$$
with the subspace topology.
Denote $p=(-1, \sin -1), q=(1, \sin 1)\...
- 51
1
vote
1
answer
155
views
Do Locally Contractible, Path-Connected Groups have Accessible Bases?
Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\...
- 439
3
votes
0
answers
158
views
Null-homotopic cellular loops are elementary null-homotopic?
I've got a 2-dimensional cell complex $X$ and a cellular closed loop $l \subset X$ that I happen to know is null-homotopic in $X$.
There are some very simple sorts of homotopies of cellular loops (or ...
- 5,349
6
votes
2
answers
406
views
Is an open subset of a cofibration a cofibration?
Suppose $A \to X$ is a cofibration in topological spaces, and $U \subseteq X$ is an open subset. Is $U \cap A \to U$ a cofibration?
Sorry if this is rather simple, but I don't have much experience ...
- 1,347
13
votes
2
answers
1k
views
Elementary proof that $\mathbb{R}^3 \setminus \{p_1,\dots,p_n\}$ is not homeomorphic to $\mathbb{R}^3$
I was wondering if there were a proof of the fact that $$\mathbb{R}^3 \setminus \{p_1,\dots,p_n\} \: \text{is not homeomorphic to} \: \mathbb{R}^3$$
for every $n \geq 1$
that does not use cohomology ...
- 1,343
4
votes
1
answer
258
views
Density of compactly-supported homeomorphisms
**Disclaimer:**I posted the following question on MSE, but since there were no answers. I'm migrating it here.
Let $Homeo_0(\mathbb{R}^n)$ ($Homeo_c(\mathbb{R}^n)$) be the space of all (compactly-...
- 5,405
2
votes
1
answer
2k
views
How do we know that a surface minus finite number of points is homotopy equivalent to a bouquet of circles? [closed]
In this post (Homotopy Equivalence of Punctured Tori), the author of the first answer states that a surface minus finite number of points is homotopy equivalent to a bouquet of circles. However, it ...
- 21
2
votes
0
answers
214
views
Products of cones and cones of joins
The join of $A$ and $B$ is the pushout of the diagram
$$
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 ...
- 12.5k
9
votes
0
answers
333
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
200
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
152
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
vote
0
answers
154
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
519
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
365
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
232
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
423
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
answer
504
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). ...
- 5,529