All Questions
Tagged with spectral-sequences stable-homotopy
24 questions
3
votes
1
answer
421
views
Spectral sequence in Adams's book, Theorem 8.2
I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
- 224
4
votes
0
answers
170
views
infinite families in stable homotopy groups
The question is about infinite families in stable homotopy groups. Yes, there are some Q&A about the topic.
But I wonder if the order of Mahowald's elements is known?
in Green Book it mentioned ...
- 171
7
votes
1
answer
261
views
Relation between cohomology operations and the Adams spectral sequence
$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Cone}{Cone}$
I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...
- 71
4
votes
1
answer
177
views
$BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$
What is know about the homotopy groups of $S/3$ where $S/3 = \mathrm{hocofib}(S \xrightarrow{\cdot 3} S)$? Otherwise, is there some reference I can consult for the $BP$-ANSS for $S/3$?
- 151
10
votes
0
answers
325
views
Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
- 101
5
votes
0
answers
219
views
Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in the $n$-line of the $E_2$ of the ASS of the sphere?
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element $...
- 7,799
4
votes
1
answer
639
views
Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum
This is a following up question of Sphere spectrum, Character dual and Anderson dual.
What are the differences and the significances of the following:
(1). Homotopy classes of maps from a Thom ...
- 10.5k
10
votes
2
answers
2k
views
Sphere spectrum, Character dual and Anderson dual
The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
What is the significance of ...
- 10.5k
9
votes
0
answers
131
views
Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...
- 10.5k
6
votes
0
answers
562
views
The $E_2$-page of the May spectral sequence
I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS.
At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...
- 4,189
4
votes
1
answer
227
views
How are p-primary parts determined for odd p?
When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away.
How are odd primary part calculations done in relation ...
- 41
8
votes
1
answer
615
views
Functorial construction of ("pre"-)spectral sequences? (Or - what is the "higher structure" underlying spectral sequences?)
Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the ...
- 7,799
8
votes
1
answer
363
views
Adams spectral sequence and short exact sequences. Some clarifications
as the title suggests I'm looking for some clarifications in the computations of the ext charts of some $A(1)$-modules arising as extensions of other modules. In particular, I've the following example ...
- 503
10
votes
1
answer
370
views
Adams Spectral sequence for computing some $B$-bordism groups
As the title suggests, I'm trying to apply the Adams Spectral sequence to get some insights of the bordism group
$$ \Omega_4(\xi)= \pi_4(M\xi)$$
where $\xi \colon BSpin \times K(D_{2n},1) \to BSO$ is ...
- 2,018
6
votes
3
answers
545
views
Adams Spectral Sequence for Triangulated Categories
We have the Adams SS with
$$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$
where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.
I was wondering if there is a SS for ...
- 163
7
votes
2
answers
966
views
Computation of stable homotopy groups of $RP^2$
I would like to compute the first few stable homotopy groups of $RP^2$.
I first thought to use the Atiyah-Hirzebruch Spectral Sequence, (see Davis & Kirk, pg. 242). Here is what I computed for ...
- 1,162
31
votes
1
answer
2k
views
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 ...
- 6,308
7
votes
1
answer
613
views
Image of J in the classical Adams Spectral Sequence
Hey all,
I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of ...
- 1,147
12
votes
2
answers
1k
views
Differentials in the Adams Spectral Sequence for spheres at the prime p=2
How does one compute the differentials in the Adams Spectral Sequence for spheres at the prime 2 in the range $13\le t-s\le 20$? There seem to be 6 nonzero differentials, and at this point I only ...
- 1,147
4
votes
1
answer
373
views
How do you know when something must die in the Adams Spectral Sequence for $\pi_*^s$
Hey everybody,
I think this question might be just a simple oversight on my part, but this has been bugging me a few days.
I am reading Hatcher's Spectral Sequences book, and trying to understand ...
- 1,147
12
votes
1
answer
961
views
What is the first interesting matric Toda bracket in the stable homotopy of the sphere?
Feel free to gloss ‘interesting’ as you see fit. One way:
1. What is the lowest degree matric Toda bracket in $\pi_\ast(S)$ that doesn't contain zero?
By ‘degree’ I mean total homotopical degree, ...
- 3,103
19
votes
2
answers
826
views
Is the 4-line of the E_2 term of the classical Adams spectral sequence known?
In other words:
What is $\mathrm{Ext}_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?
If the 4-line is not known, how much is known about it?
Here, $\mathcal{A}$ is the 2-primary Steenrod ...
- 3,103
11
votes
1
answer
957
views
Technology for various models of spectra
There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of $S$-...
- 3,726
6
votes
1
answer
890
views
Serre spectral sequence with spectra
A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than ...
- 3,726