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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
5
votes
Whitehead for maps
A simple example with finite complexes would be the collapse map $q:X\times Y\to X \wedge Y$. Since the inclusion $X \vee Y \hookrightarrow X\times Y$ is surjective on $\pi_*$,
$q_* = 0$. But $q$ is …
- 12.5k
5
votes
Two H-space structures on S^3 and [X,S^3] different as groups for each: Explicit Example?
This is semiexplicit. For any H-space $G$ with multiplication $\mu$,
the projection maps $p_1, p_2: G\times G\to G$
have the property that
$$
[p_1] \cdot [p_2] = [\mu] \in [G\times G, G].
$$
So if …
- 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
0
votes
Killing the torsion in homotopy
Zabrodsky does this in a paper on phantom maps. It's not functorial, but you can do it coherently for all the spaces and maps in a diagram that is finite (in the appropriate sense).
- 12.5k
3
votes
Accepted
How can I prove that the derived couple of the homotopy exact couple is an invariant?
Let me start by making a definition: an $n$-skeleton of a space $X$ is an
$n$-equivalence
$X_n \to X$, where $X_n$ is an $n$-dimensional (at most) CW complex
($X$ itself need not be a CW complex). …
- 12.5k
1
vote
Classifying maps into homogeneous spaces up to homotopy
Homogeneous spaces tend to be buildable from spheres in finitely many steps by extensions by fibrations. If you pretend you understand $[X, S^n]$ for all $n$, then you can try to
analyze the exact …
- 12.5k
8
votes
Computing homotopies
In my experience, the vast majority of homotopies come from some combination of (1) homotopies guaranteed by cofibrations or fibrations and (2) straight-line homotopies.
For example, a standard appro …
- 12.5k
1
vote
Lifting relative homotopies between maps $X\to Y/B$ when $B$ is contractible
This example is wrong (sorry!):
Try $(X,A) = (D^n, S^{n-1})$, $Y = D^n$, $f_A = \mathrm{in}_{S^{n-1}}$, and $B = Y$.
Then we have plenty of nonequivalent (rel. $A$) maps $f, g: X\to Y$, and they …
- 12.5k
3
votes
$[\nu_4,\iota_4]=?$, $\nu_4$ is the Hopf map in $\pi_7(S^4)$ and $\iota_4$ is the generator ...
This is how far I can get without checking a book.
Since $v_4 = \pm {1\over 2} [i_4, i_4]$, you are interested in the triple Whitehead product
$\alpha = {1\over 2}[ [i_4, i_4],i_4]$. Since it is a W …
- 12.5k
2
votes
The most general context of Mather's Cube Theorems
The second cube theorem (if base and sides are ok, then so is the top) is a straightforward consequence of the formally crazy fact that if $p:E\to B$ is a fibration, $i:A\to B$ is a cofibration, and $ …
- 12.5k
1
vote
uniqueness of $f$-localization
An $f$-equivalence is a map $\alpha:X\to Y$ which induces a weak equivalence $\alpha^\ast:map_\ast(Y,Q)\to map_\ast (X,Q)$ for all $f$-local spaces $Q$; equivalently, $L_f(\alpha)$ is a weak equivale …
- 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
17
votes
An abstract nonsense proof of the Hurewicz theorem
I’d argue that it boils down to the generator $S^n\to K(\mathbb{Z},n)$ being an $(n+1)$-equivalence.
More detail: If you take the represented version of homology, it is given by
$$
H_n(X;\mathbb{Z} …
- 12.5k