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Results tagged with homotopy-theory
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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
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Milnor's exact sequence and a certain proof
First take a space $X$, and try the fibration $Hom(\Delta_1,X)\to X\times X$. The fiber over $(x,x')\in X\times X$ is the space of paths in $X$ which go from $x$ to $x'$ (which can sometimes be empty …
- 27.2k
8
votes
questions on steenrod algebra
For part 2: The two "units" are very nearly always distinct. You should regard it as an accident that they coincide for $H\mathbb{F}_p$.
In fact, if $k$ is a field which is not a prime field, then t …
- 27.2k
5
votes
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The homotopy fibre of an map $f \colon S^{2n-1} \to S^n$
In the following, any identity may actually be up to a sign. I don't want to keep track of that.
We are thinking about the fibration sequence
$$ \Omega S^n \to F \to S^{2n-1} \xrightarrow{f} S^n.$$
…
- 27.2k
15
votes
Accepted
Finite spectrum annihilated by multiplication by two
This doesn't seem very hard. Am I missing something?
Let $X$ be a non-trivial finite spectrum of characteristic $2$. Then let $R=\mathrm{Hom}(X,X)$, the function spectrum of maps $X$ to $X$. This …
- 27.2k
14
votes
Accepted
BU with tensor product H-space structure
I'll write $U(X)=[X,BU_\otimes]$. So $U(X)\subset K(X)=[X,Z\times BU]$ is the multiplicatively closed subset corresponding to "virtual bundles of rank 1".
Likewise, I'll write $I(X)=[X,BU]\subset K(X …
- 27.2k
14
votes
Group completion theorem
Well, if $\pi_0=\pi_0(M)$ is already a group, then $H_*(M)\approx H_*(M)[\pi_0^{-1}]$. So $M$ and $\Omega B M$ have the same homology in this case. This isn't quite enough on its own, but if you can …
- 27.2k
7
votes
Accepted
Simple examples of homotopy colimits
Here's an answer to question 2: A sufficient condition for $\mathrm{colimit}(X \leftarrow A\rightarrow Y)$ to be weakly equivalent to the homotopy colimit, is (a) for one of the maps (say $A\to X$) t …
- 27.2k
36
votes
Accepted
A possible generalization of the homotopy groups.
There's always information to be got. But in this case:
Based homotopy classes of maps $T^2\to X$ don't form a group! To define a natural function $\mu\colon [T,X]_*\times [T,X]_*\to [T,X]_*$, you …
- 27.2k
9
votes
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Transversality in the proof of the Blakers-Massey Theorem. Is it necessary?
Take a look at the proof (attributed to Puppe) given in tom Dieck's new algebraic topology texbook (section 6.9). (I believe it also appears in tom Dieck, Kamps, Puppe (Lecture Notes in Mathematics 1 …
8
votes
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Homotopy Extension Property involving mapping cylinder
Neil has given an explicit retraction. But it may be useful to note that you can obtain results like this from a combination of some "easier" facts:
The pair $(I,\{0,1\})$ has the HEP.
If $(L,K)$ h …
- 27.2k
5
votes
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free homotopy groups -- when do they exist?
For the last part of your question: given a group π1 which acts on an abelian group πn, there is always as space X with these homotopy groups with this action, and you can manufacture one using Eilenb …
- 27.2k
13
votes
Accepted
Computing homotopy (co)limits in a nice simplicial model category?
In practice, one tends to "compute" arbitrary homotopy colimits as bar constructions, especially when you have a simplicial model category.
If $X:J\to P$ is a simplicially enriched functor, where $ …
- 27.2k
20
votes
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Quasifibrations and homotopy pullbacks
The definition of quasifibration (according to Dold & Thom, 1958) is: a map $f:E\to B$ such that for all $b$ in $B$, the canonical map from the fiber to the homotopy fiber is a weak equivalence. Pull …
- 27.2k
27
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What is the intuitive meaning of the coskeleton of a simplicial set?
It should be added that if $X$ is a Kan complex, then $X\to \mathrm{cosk}_n X$ computes a model for the $(n-1)$-truncation of the homotopy type of $X$. That's a fact you can easily read off using ele …
- 27.2k
26
votes
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Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
I believe that CGWH spaces were first used in a systematic way in the work of Lewis-May-Steinberger on spectra. It is certainly the case that Gaunce Lewis's (unpublished) thesis contains the best ref …
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