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Results tagged with homotopy-theory
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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.
13
votes
2
answers
2k
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Quasifibrations and homotopy pullbacks
I'm wondering about the theoretical placement of quasifibrations.
One nice thing about "weak fibrations" (maps homotopy equivalent in the category of maps to Hurewicz fibrations) is that a pullback s …
- 301
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} …
- 63
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 …
- 63.9k
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
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
$$
u: [A, M] \to \langle A …
- 63.9k
2
votes
0
answers
209
views
Products of cones and cones of joins
The join of $A$ and $B$ is the pushout of the diagram
$$
CA \times B \gets A\times B \to A\times CB,
$$
which can be formulated in either the pointed or unpointed topological
category. This pushout is …
- 12.5k
9
votes
0
answers
379
views
When is an increasing union a colimit?
Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$
$$
X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots
$$
of pointed spaces,
indexed by some ordinal $\lambda$, in which each $X_\xi$ i …
- 12.5k
12
votes
1
answer
734
views
Open subspaces of CW complexes
I am looking at the paper
Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West
and a claim is made in the proof of their first main theorem t …
- 9,263
7
votes
0
answers
107
views
Stable splitting of products
This question concerns the well-known homotopy equivalence
$$
\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
19
votes
4
answers
2k
views
Difference between represented and singular cohomology?
Ordinary cohomology on CW complexes is determined by the coefficients. There are (more than) two nice ways to define cohomology for non-CW-complexes: either by singular cohomology or
by defining $\ …
- 1,243
38
votes
2
answers
2k
views
Finite complexes whose homotopy groups are not "finitely generated"
I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$.
It seems likely that t …
- 3,451
4
votes
0
answers
193
views
Some notation from a paper by H. Toda
I'm trying to read Toda's paper "Complex of the standard paths and n-ad homotopy groups" and I'm running into trouble with a definition---here's the text (slightly rephrased)
Let $K$ be a CW-compl …
- 18.9k
9
votes
1
answer
294
views
Retracting a wedge of spheres off a homotopy fiber
There is a general principle that, for finite simply-connected CW complexes, things that are true rationally are usually true once you localize away from a finite list of primes.
I'm interested in …
- 5,596