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Results tagged with stable-homotopy
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user 2039
Stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.
18
votes
Accepted
Does an H-space have at most one delooping?
Another example is $S^3$. If I am not mistaken, there are exactly $12$ H-space structures on $S^3$. Indeed, we can consider the long exact sequence
$$[S^4\vee S^4, S^3] \to [S^6, S^3] \to [S^3\times S …
- 12.1k
17
votes
Accepted
Stable homotopy groups of complex projective plane
The question is equivalent to asking what the multiplication-by-$\eta$-map $\pi_4\mathbb{CP}^2 \to \pi_5 \mathbb{CP}^2$ is (which can be rewritten as $\pi_2\mathbb{S}/\eta \to \pi_3\mathbb{S}/\eta$). …
- 12.1k
9
votes
Second homotopy group of the mod 2 Moore spectrum
Often these extension problem are solved using Toda brackets (as Peter May already mentioned). I will first give the general statement, not only for spectra but also for module spectra. The statement …
- 12.1k
22
votes
3
answers
816
views
Boardman's thesis or mimeographed notes
I would like to know if there is some online source where Boardman's 1964 thesis is available or his Warwick mimeographed notes. This is because by what I've heard Boardman's construction has a more m …
- 12.1k
7
votes
Finiteness of stable homotopy groups of spheres
Let me phrase a proof of Serre's computation of the rational stable homotopy groups of spheres as stably as I can:
For every spectrum $X$, we can define its rationalization $X_{\mathbb{Q}}$ as the ho …
- 12.1k
28
votes
Latest results in chromatic homotopy theory
I want to mention five directions where in the last years significant progress has been made in chromatic homotopy theory. This is of course not exclusive!
Unstable chromatic homotopy theory
Among the …
- 12.1k
14
votes
Modern source for spectra (including ring spectra)
There is probably no ideal source for this. The canonical choice for symmetric spectra is probably Stefan Schwede's book project http://www.math.uni-bonn.de/~schwede/SymSpec.pdf . There you will find …
16
votes
Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
The original question has been answered in the sense that there are people who are confident to prove every statement about spectra they care about without recourse to models or model categories of sp …
23
votes
Accepted
References and resources for (learning) chromatic homotopy theory and related areas
I am not sure whether it is in the spirit of the original question, but let me add a wordy version of Emily's extensive and excellent bibliography -- a bit more of a road map. Let me divide to this pu …
51
votes
What is modern algebraic topology(homotopy theory) about?
While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still a …