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Results tagged with stable-homotopy
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user 8824
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.
7
votes
Accepted
Image of J in the classical Adams Spectral Sequence
The image of $J$ is pretty easy to see in the Adams $E_2$ term: it consists of the elements along the vanishing line, plus, in dimensions 8k-1, of the towers that end near the vanishing line.
This ide …
- 2,821
5
votes
Realizing $\mathcal{A}(2)//\mathcal{A}(1)$ by a finite spectrum
As pointed out by John Rognes, this answer is not correct (it misses the $Sq^4$ from the bottom class to the class in dimension $4$). Sorry for the confusion.
=========== previous answer ============ …
- 1,554
11
votes
Accepted
Adams-Novikov spectral sequence at p = 2
I don't think anybody knows how to compute this $E_2$-term efficiently (not just at the prime $2$). I would love to be proved wrong on this, of course.
So far the only documented, algorithmic method …
- 1,554
12
votes
Geometric interpretation of families in the stable homotopy groups of spheres
This has been a burning question for quite some time, but not much is known. Surely, people believe that the next layer (i.e. the $\beta$-family) should also admit a geometric description, although as …
- 1,554
15
votes
What are the best known results for the stable homotopy groups of spheres?
Computing $\pi_\ast(S)$ is a tedious business that to this day can only be done "by hand", i.e. by humans. The $p=2$ computation up to dimension 64 was completed by Kochman (see his SLNM book) with la …
- 1,554
4
votes
Accepted
Computing squaring operations in the Adams spectral sequence
I think for $k>0$ currently no-one knows an efficient algorithmic way to compute the $Sq^k$, e.g., from a minimal resolution. (The $Sq^0$ is easy since it is induced by the "Frobenius" map on $A_\ast$ …
- 1,554