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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.
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
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
2
votes
0
answers
94
views
Cellular or acyclic inequalities for homotopy fibers of suspension maps
Is it true that (modulo connectivity hypotheses perhaps)
$$
\mathrm{Fib}(f) < \mathrm{Fib}(g)
$$
implies
$$
\mathrm{Fib}(\Sigma f) < \mathrm{Fib}(\Sigma g)?
$$
A class $\mathcal{C}$ of pointed spaces …
- 12.5k
3
votes
3
answers
268
views
uniqueness of $f$-localization
The $f$-localization I mean is the one described and studied in detail in the book by E. D. Farjoun; $L_f$ is a homotopy idempotent functor which associates to each space $X$
an $f$-equivalence $X\to …
- 12.5k
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 …
- 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
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
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
1
vote
3
answers
2k
views
Reference for intersection and linking in algebraic topology
I have a feeling that I have seen some kind of theory of linking and intersection that applies in spaces that are not manifolds. I've found two kinds of theories in the books I've checked:
1) inter …
- 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
3
votes
1
answer
298
views
Eilenberg-Mac Lane spaces for groups that can't see $p$-groups
All groups here are abelian and $p$ is a prime number; I'll say $P$ is a $p$-group if every element
of $P$ has finite order which is a power of $p$.
Suppose $\mathrm{Hom}(G,P) = 0$ for every $p$-gr …
- 12.5k
2
votes
1
answer
197
views
Reference for an automorphism in a paper of Toda
In Selick's very pretty paper "Odd primary torsion in $\pi_*(S^3)$" he makes use of an automorphism which was established by Toda in his paper "On the double suspension $E^2$".
Unfortunately, Selick …
- 12.5k
3
votes
2
answers
238
views
Fitting desired weak equivalences and cofibrations into a model category
Suppose I have a category $\mathbf{C}$ and classes of morphisms $\mathcal{W}$ and
$\mathcal{C}$, and I would like to know that $\mathcal{W}$ and $\mathcal{C}$ are
the weak equivalences and the cof …
- 12.5k
6
votes
0
answers
78
views
Examples of maps with nontrivial Hopf invariant but Lusternik-Schnirelmann category of the c...
Let $A$ be a suspension and $X$ be a space with Lusternik-Schnirelmann category $n$ and let $\alpha: A\to X$. It is easy to see that the cofiber $C_\alpha$ has $\mathrm{cat}(C_\alpha) \leq n+1$. One …
- 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