Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with homotopy-theory
Search options not deleted
user 3634
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
3
votes
0
answers
106
views
Exponential law and cones reference
Given a map $\omega: A\to \Omega X$, one can set up the diagram
and construct the map $\sigma : \Sigma A\to X$.
It's pretty easy to check that the homotopy classes
of $\omega$ and $\sigma$ correspond …
- 12.5k
1
vote
0
answers
139
views
Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, …
- 12.5k
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
8
votes
1
answer
260
views
Pointed versus unpointed maps into a topological monoid
I've just stumbled on something that seems either too good to be true,
or else too good for me not to have heard of it before.
It has to do with the basepoint forgetting map
$$
u: [A, M] \to \langle A …
- 12.5k
2
votes
0
answers
209
views
Products of cones and cones of joins
The join of $A$ and $B$ is the pushout of the diagram
$$
CA \times B \gets A\times B \to A\times CB,
$$
which can be formulated in either the pointed or unpointed topological
category. This pushout is …
- 12.5k
24
votes
Accepted
Homotopy equivalent Postnikov sections but not homotopy equivalent
This is a pretty well-known phenomenon, linked with phantom maps.
One of the first existence results was Brayton Gray's paper
Spaces of the same $n$-type, for all $n$, Topology
5 (1966) 241--243
Cla …
- 12.5k
9
votes
0
answers
379
views
When is an increasing union a colimit?
Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$
$$
X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots
$$
of pointed spaces,
indexed by some ordinal $\lambda$, in which each $X_\xi$ i …
- 12.5k
20
votes
Accepted
Divisibility in the homotopy groups of spheres?
Yes. In fact $\bigcup_{n\geq 1} P_n $ is the set of all primes. Serre proved that, for each odd prime $p$, there is some very predictable $p$-torsion in $\pi_{k+(p-1)}(S^k)$, for example.
There is …
- 12.5k
5
votes
What is the smallest class of spaces closed under finite homotopy colimits, finite homotopy ...
(The discussion below is for pointed spaces.) I'll use
$\mathcal{F}_*$ for the pointed version of your $\mathcal{F}$.
As Nicholas Kuhn says, this is related to the closed classes studied by E. Dror …
- 12.5k
12
votes
1
answer
734
views
Open subspaces of CW complexes
I am looking at the paper
Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West
and a claim is made in the proof of their first main theorem t …
- 12.5k
40
votes
In a topological space if there exists a loop that cannot be contracted to a point does ther...
Every finite simplicial complex is weakly homotopy equivalent to a finite space. Therefore there are finite spaces with nontrivial loops; and these are obviously not embedded.
- 12.5k
7
votes
0
answers
107
views
Stable splitting of products
This question concerns the well-known homotopy equivalence
$$
\Sigma (X\times Y) \simeq \Sigma (X \vee \ Y) \vee \Sigma (X\wedge Y)
$$
(I'm happy to use only CW complexes). I can see that
there is …
- 12.5k
4
votes
0
answers
92
views
Homotopy colimits of long sequences
Let $\lambda$ be a limit ordinal, and let $F: \lambda\to \mathcal{T}_*$ be a diagram of pointed spaces with shape $\lambda$. Write $X = F(0)$ and $Y = \mathrm{hocolim} F$. I believe it to be true (I …
- 12.5k
8
votes
1
answer
219
views
Splitting low-dimensional $p$-local CW complexes for large $p$
Fix a prime $p$. I have a sketch of a proof that if $X$ is a finite simply-connected CW complex with $\mathrm{dim}(X) < p$ then for some $t\in \mathbb{N}$, the $p$-localization $\Sigma^t X_{(p)}$ is …
- 12.5k
3
votes
Accepted
Rationalization of topological groups and degree maps
Yes, since the group power map induces multiplication by $n$ on homotopy groups, turning them into rational vector spaces.
- 12.5k