All Questions
40 questions
6
votes
0
answers
141
views
Are the $K(n)$-local $E_n$-Adams spectral sequences isomorphic to the Adams-Novikov spectral sequences?
Let $H$ be a closed subgroup of the Morava stabilizer group $\mathbb G_n$. [Devinatz-Hopkins, Prop. 6.7] identifies the $K(n)$-local $E_n$-Adams spectral sequence for $E_n^{hH}$ as the homotopy fixed ...
- 155
2
votes
1
answer
201
views
Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\...
- 508
4
votes
2
answers
290
views
Loop-space functor on cohomology
For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$.
More concretely, $\omega$ is given by the Puppe sequence
$$\...
- 663
13
votes
5
answers
2k
views
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
5
votes
1
answer
472
views
Two spectral sequences arising from a simplicial spectrum
Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization.
Let's assume each $X_n$ is connective.
From this situation, we can form two filtrations on $X$: the ...
- 339
16
votes
1
answer
776
views
The second stable homotopy group
I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\...
- 663
27
votes
0
answers
1k
views
Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 64k
10
votes
1
answer
321
views
$\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}_{p^{r+1}}$ summand
I am reading Neisendorfer's paper Samelson products and exponents of homotopy groups and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}_{p^{r+1}}$ ...
- 1,263
4
votes
0
answers
201
views
Using the Serre spectral sequence - moving between $\mathbb{Z}/2$ and $\mathbb{Z}$ information
I am trying to understand the computation of $\pi_5(S^3)$ and $\pi_6(S^3)$ using the Serre spectral sequence. I know already that $\pi_5(S^3)$ is only 2-torsion and $\pi_6(S^3)$ is 2-torsion together ...
- 5,349
5
votes
0
answers
328
views
Bousfield-Kan and Generalized Eilenberg-Moore spectral sequences
Building on the work of Anderson and Rector, Bousfield's paper "On the homology spectral sequence of a cosimplicial space" constructs a spectral sequence which takes in a cosimplicial space (here ...
- 131
14
votes
0
answers
404
views
Computations using "Stover's spectral sequence"
In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.
The second ...
- 208
11
votes
1
answer
862
views
What is the relationship between spectral sequences and obstruction theory?
Let $X,Y$ be objects of some category $\mathcal C$, and suppose I want to study homotopy classes of maps from $X$ to $Y$ (almost everything one does in algebraic topology can be viewed this way). It ...
- 64k
11
votes
0
answers
266
views
Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper
In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
- 1,329
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
15
votes
1
answer
988
views
Why is it difficult to obtain the next differential in a spectral sequence?
I am following Hatcher's notes on spectral sequences. On page 522 an exact couple $(A, E, i, j, k)$ is defined. You can construct a derived exact couple easily from this to get your desired $E'$ ...
- 151
7
votes
0
answers
168
views
Adams spectral sequence for loop spaces
Let $X = \Omega_0^3S^3$ a connected component of $\Omega^3S^3$. I am interested in explicit construction of spectral sequence converging to odd prime torsion in homotopy groups of $X$.
There is a ...
- 1,129
3
votes
0
answers
310
views
Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
1
vote
0
answers
109
views
Empty regions on the second list of unstable Adams spectral sequence
Define $\phi(n) = 4n - 2$. Is there a proof that on the second list of unstable Adams spectral sequence (for all spheres) there are no elements in squares $(n, m)$ such that $m < \phi(n)$. The ...
- 1,129
2
votes
0
answers
326
views
A version of Leray Hirsch better for local coefficients
Let $F \hookrightarrow E \to B$ be a fibration, and let everything be over a field $k$.
The leray hirsch theorem in its usual form says that the (homology) cohomology spectral sequence degenerates at (...
- 579
10
votes
1
answer
657
views
Cap product on Leray-Serre spectral sequences
Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and $H^*(...
- 1,484
14
votes
0
answers
830
views
What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of the totalization?
Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and ...
- 10.5k
2
votes
0
answers
216
views
completion and convergence of spectral sequence
I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly ...
- 21
10
votes
1
answer
474
views
Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?
Given a fiber square of simplicial sets
$$\begin{array}{cc}
& \hspace{-7mm} E \\
&\hspace{-7mm}\downarrow \\
\ast\longrightarrow &\hspace{-7mm} B
\end{array}$$
...
- 10.5k
1
vote
1
answer
293
views
Unordered configuration space of $\mathbb{R}P^1$
In the paper
GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
159-165,
Theorem 2. (b): $TP^n(\mathbb{R}P^1)$ is ...
- 1,990
3
votes
2
answers
319
views
cohomology algebra of braid spaces, configuration spaces
In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for $p$...
- 2,223
17
votes
0
answers
757
views
The spectral sequence of a tower of principal fibrations
Assume we have a tower of fibrations (of simplicial sets, let's say):
$$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$
Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...
- 15.2k
10
votes
3
answers
2k
views
Serre Spectral Sequence of Representations
Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...
- 4,920
5
votes
1
answer
407
views
spectral sequence for cobordism without leaving smooth category
In Bott & Tu's marvelous book there is a derivation of the spectral sequence for a (smooth) fiber bundle for deRham cohomology done entirely in the realm of the smooth category. Unfortunately, as ...
- 13.5k
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
10
votes
0
answers
494
views
Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem
$\newcommand\Ext{\mathrm{Ext}}
\newcommand\Z{\mathbb{Z}}
\newcommand\G{\mathbb{G}}$
The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-...
- 3,784
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
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
827
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
12
votes
1
answer
2k
views
Some calculations with the Adams spectral sequence and the cobar complex
I am trying to 'get my hands dirty', so to speak, with some of the calculations with the Adams spectral sequence in Ravenel's Complex Cobordism book, and I have a few questions (I hope it is OK to ask ...
- 3,784
17
votes
1
answer
1k
views
Multiplicativity in the descent spectral sequence
For a homotopy sheaf $\mathcal{F}$ of ring spectra over some space (/ site / whatever) $X$ with a cover $U_i$, we can build a "descent spectral sequence" with signature $$E^1_{p, q} = \pi_{p+q} \...
- 6,308
19
votes
1
answer
2k
views
Slick Proof of Kudo Transgression Theorem
The Kudo Trangression Theorem has to do with the transgression in the Leray-Serre spectral sequence for cohomology in $\mathbb{Z}/p$ ($p$ odd). It can be proved by the method of the universal example,...
- 12.5k
7
votes
2
answers
1k
views
Proof of the ''trangression theorem''
Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...
- 20.9k
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
13
votes
2
answers
713
views
How do you compute the space of lifts of an E-infinity map?
Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g:...
- 3,103
20
votes
5
answers
3k
views
Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?
One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
- 25.2k