All Questions
Tagged with homotopy-theory stable-homotopy
261
questions
123
votes
4
answers
8k
views
What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
$\mathbb{Z}\...
- 22.3k
70
votes
6
answers
8k
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third stable homotopy group of spheres via geometry?
It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...
- 20.8k
63
votes
5
answers
17k
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What is modern algebraic topology(homotopy theory) about?
At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
50
votes
5
answers
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What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...
- 39.5k
43
votes
4
answers
5k
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Integral cohomology (stable) operations
There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...
- 3,696
41
votes
1
answer
9k
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Why not a Roadmap for Homotopy Theory and Spectra?
MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...
- 1,162
33
votes
2
answers
2k
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What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
32
votes
2
answers
10k
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Open problems in algebraic topology and homotopy theory
Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...
31
votes
4
answers
4k
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Fibrations and Cofibrations of spectra are "the same"
My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:
"For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
- 1,538
31
votes
2
answers
3k
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Conceptual explanation for the relationship between Clifford algebras and KO
Recall the following table of Clifford algebras:
$$\begin{array}{ccc}
n & Cl_n & M_n/i^{*}M_{n+1}\\
1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\
2 & \mathbb{H} & \mathbb{Z}/2\...
- 635
31
votes
1
answer
2k
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K(r)-localization and monochromatic layers in the chromatic spectral sequence
While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...
- 6,258
31
votes
0
answers
849
views
The central insight in the proof of the existence of a class of Kervaire invariant one in dimension 126
I understand from a helpful earlier MO question that the techniques leading to the celebrated resolution of the Kervaire invariant one problem in the other candidate dimensions yield no insight on ...
- 2,984
30
votes
1
answer
2k
views
Morava K-theories for dummies?
Professor Urs Würgler passed away one year ago, and his wife engraved his tombstone with "the formula he was the most proud of" :
$B(n)_*(X)\cong P(n)_*(K(n))\square_{\Sigma_n}K(n)_*(X)$
However ...
- 403
30
votes
1
answer
766
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Is a filtered colimit of rational spaces again rational?
Let me first explain the statement of the question and then give some indication why the answer might be 'yes'.
By a space I mean, say, a simplicial set and by rational I mean rational in the sense of ...
- 383
28
votes
5
answers
4k
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Modern source for spectra (including ring spectra)
I am looking for a modern introduction to spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what ...