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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.
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 …
- 12.5k
19
votes
1
answer
2k
views
Slick Proof of Kudo Transgression Theorem
The Kudo Trangression Theorem has to do with the transgression in the Leray-Serre spectral sequence for cohomology in $\mathbb{Z}/p$ ($p$ odd). It can be proved by the method of the universal example …
- 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 $\ …
- 12.5k
15
votes
2
answers
969
views
Pointed Hurewicz model structure
In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves
that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and homot …
- 12.5k
14
votes
5
answers
2k
views
Good reference for homology of $K(\mathbb{Z}, 2n)$?
The homology algebra $H_*( K(\mathbb{Z},2n); \mathbb{Z})$ contains a
divided polynomial algebra on a generator $x$ of dimension $2n$.
I suppose I could read through the Cartan seminar for a proof, bu …
- 12.5k
13
votes
2
answers
2k
views
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 …
- 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
12
votes
0
answers
655
views
Mapping cylinders of fibrations
If $p: E\to B$ is a fibration, is the map $q:M_p \to B$ from the mapping cylinder
of $p$ also a fibration?
I know that it is if $p$ is trivial, or locally trivial; and I know (from Strøm's "The Ho …
- 12.5k
12
votes
2
answers
635
views
Spaces that invert weak homotopy equivalences.
Are there any nontrivial spaces $Y$ so that for all weak homotopy equivalences
$A\to B$, the induced map $[B, Y]\to [A,Y]$ is bijective?
This would be a property of the homotopy type of $Y$, and
…
- 12.5k
10
votes
2
answers
740
views
Model categories of simplicial objects
If $\mathcal{C}$ is a category, then surely the category of simplicial
objects $s\mathcal{C}$ is not automatically a model category. What conditions
must $\mathcal{C}$ satisfy in order for $s\mathcal …
- 12.5k
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 …
- 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
9
votes
1
answer
1k
views
Are there homotopy equivalences that are not weak homotopy equivalences?
I can imagine a map $f: X\to Y$ which is a homotopy equivalence of unpointed spaces, but which is not a homotopy equivalence of pointed spaces, no matter what basepoint is chosen. That being the case …
- 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
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