Questions tagged [spectral-sequences]
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388 questions
3
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Degeneration of relative Hodge-de Rham spectral sequence
$$\require{AMScd}$$
$$\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\Hdr}{H_{\mathrm{dRh}}}
\newcommand{\tensor}{\otimes}
\newcommand{\Ohol}{\mathcal{O}}$$
Please excuse that ...
- 708
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
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
2
votes
1
answer
142
views
Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group
Let $M^d$ be a nontrivial bordism generator for the bordism group
$$
\Omega_d^G= \mathbb{Z}_n,
$$
suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...
- 2,147
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
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
8
votes
1
answer
503
views
Cohomology ring of a fiberwise join
I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...
6
votes
0
answers
300
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Degeneracy of the Serre Spectral Sequence
I am learning the Serre spectral sequence and I am intrigued about the degeneracy of such at the $E_2$-page. Assuming field coefficients in cohomology for simplicity.
In fact, for a Serre fibration $...
- 81
3
votes
1
answer
260
views
Strong convergence of whole-plane spectral sequences
I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-...
- 205
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
7
votes
1
answer
331
views
Does a filtered A_N algebra give rise to a multiplicative spectral sequence?
The question is pretty much in the title. It is a classical fact that a filtered dga gives rise to a multiplicative spectral sequence. It is claimed in Remark 4.1 of https://arxiv.org/pdf/1410.6728....
6
votes
1
answer
609
views
References for properties of Atiyah-Hirzebruch Spectral Sequence for a spectrum $X$ and generalised homology theory $MSpin_*$
Currently I'm working on the following version of the AHSS $$ E^2_{pq}\cong H_p(M\eta; MSpin_q(\ast))\Rightarrow MSpin_{p+q}(M\eta)$$
where $\eta \colon B \to BSO$ is a stable vector bundle, and $M\...
- 503
15
votes
2
answers
1k
views
Where does the primary obstruction of a fibration show up in its spectral sequence?
Let $f\colon\thinspace E\to B$ be a Serre fibration whose fibre $F$ is $k-1$-connected, $k\geq 1$. Assume $B$ is a connected CW complex. Then the primary obstruction to the existence of a cross ...
- 35.9k
3
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0
answers
226
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Does Leray Spectral sequence degenerates at $E_2$ over product of curves
Let $C$ be a smooth, projective curve (can assume to be rational) and $X:=C \times C$. Denote by $p:X \to C$ one of the two natural projections. Let $E$ be a vector bundle on $X$. Is it true that,
$$...
- 2,032
15
votes
1
answer
1k
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Multiplicative Structure of the Atiyah-Hirzebruch/Leray-Serre spectral sequence
This is related to this question (edit: now answered, see below). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ ...
- 6,813
9
votes
1
answer
455
views
Is this sequence of Lie algebra cohomology a part of spectral sequence?
There is an exact sequence
$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \...
- 1,980
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
11
votes
1
answer
495
views
Identification of a Serre Spectral Seq. via Thom Isomorphism with the Atiyah-Hirzebruch Spectral Seq
Let $\xi_n$ be an orientable $n$-dimensional vector bundle over a pointed space $B_n$. We can consider the relative Serre Spectral Sequence $$ H_p(B_n; h_q(D(\xi_n|\ast),S(\xi_n|\ast))\Rightarrow h_{p+...
- 2,018
7
votes
2
answers
4k
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Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective
I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective.
I am a little confused about when maps between cohomology groups are ...
- 1,211
6
votes
0
answers
237
views
A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?
This question is a follow-up to my previous question:
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
In that question, I discussed two different spectral sequences for ...
- 863
7
votes
0
answers
436
views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
- 353
3
votes
1
answer
260
views
non-simple local coefficient system on a fibration of classifying spaces
Long story short; I posted in MSE
https://math.stackexchange.com/questions/2500745/local-system-of-coefficients-on-a-fibration-of-classyfing-spaces
It is well known that if $G$ is a lie group ...
- 283
31
votes
1
answer
2k
views
K(r)-localization and monochromatic layers in the chromatic spectral sequence
While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...
- 6,308
10
votes
1
answer
474
views
Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?
Given a fiber square of simplicial sets
$$\begin{array}{cc}
& \hspace{-7mm} E \\
&\hspace{-7mm}\downarrow \\
\ast\longrightarrow &\hspace{-7mm} B
\end{array}$$
...
- 10.5k
7
votes
1
answer
506
views
$G$ cocycle split to a coboundary in $J$, via a group extension
Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
- 10.5k
3
votes
0
answers
166
views
Edge map in derived categories
Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...
- 5,670
6
votes
1
answer
564
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a question about Bockstein spectral sequence
I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459:
Question. for a fixed $k$, if $\beta$ does not hit $H_k(X;\...
- 1,990
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0
answers
352
views
Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?
For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
- 357
4
votes
1
answer
182
views
The converse of Vietoris-Begle theorem
It is well known the following result:
Lemma: Let $F\rightarrow E\rightarrow B$ be a fibration with $B$ connected and
simply connected. Suppose that $F$ is $n$-acyclic, i.e. $H^{p}\left( F;%
%...
- 1,367
6
votes
1
answer
399
views
Leray-Serre spectral sequence for algebraic groups
Let $G$ be a semisimple, simply-connected, complex algebraic group. Fix a Borel subgroup $B$ and let $P$ be a parabolic subgroup properly containing $B$. If $M$ is a $B$-module, then we have the Leray-...
- 422
5
votes
2
answers
651
views
Inflate a finite-group cocycle into coboundary in non-Abelian groups
Edit: In case that there is no solution for the original question, I modify to enrich the question.
We like to ask a possible specific inflation a $H^3(Q, \mathbb{R} /\mathbb{Z})$ cocycle with a ...
- 755
3
votes
0
answers
174
views
Induced Homomorphism on Cohomology of Symmetric Group 3
For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
- 39
2
votes
1
answer
594
views
Leray spectral sequence for lowest weight part of a smooth morphism
Let me assume everything in sight is as nice as possible, probably if the result I want is true then these conditions are too restrictive. All spaces will be smooth algebraic varieties over the ...
- 40.3k
4
votes
0
answers
419
views
Cohomology of double complex with exact rows
Let $(C^{p,q},d_h,d_v)$ be an unbounded double complex of modules of some algebra $A$ in an abelien category. Let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that
(1) ...
- 205
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votes
0
answers
299
views
Local coefficients system
Let $G$ be a compact group. Then there exists a universal principal $G$-bundle $G\rightarrow E_{G}\rightarrow B_{G}$. Let $X$ be a paracompact $G$-space and suppose that $X\rightarrow X_{G}\rightarrow ...
- 1,367
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
3
votes
0
answers
165
views
Reference for specific detail on Serre spectral sequence
In "A primer on spectral sequences" http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf (by J.P.May apparently, although no name is given in the pdf) I found a very detailed version of the ...
- 404
2
votes
1
answer
212
views
Leray spectral sequence for continuous functions on pairs of topological spaces
Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$.
The Leray spectral sequence (with complex ...
- 1,227
8
votes
0
answers
125
views
Relating bordism generators in d and d+2 dimensions --- an explicit example
This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
- 2,147
5
votes
0
answers
102
views
Group cohomology of "twisted" projective SU(N) with various coefficients
Given a group
$$
G= PSU(N) \rtimes \mathbb{Z}_2,
$$
where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then
$$
c a c= a^*,
$$
which $c$ flips $a$ to its ...
- 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
2
answers
503
views
Interpretations of differentials in hypercohomology spectral sequences as Yoneda products
I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups.
More ...
- 2,160
6
votes
1
answer
242
views
For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My ...
- 16.9k
5
votes
0
answers
544
views
Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple explanation?)
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than ...
- 1,229
3
votes
0
answers
149
views
Gel'fand's and Fuks' calculation of cohomology of formal vector fields - isomorphic spectral sequences yield isomorphic cohomology?
In this book (and, in what seems to be an equivalent fashion, in this article), Gel'fand and Fuks calculate the Lie algebra cohomology of the formal vector fields $W_n$ on $\mathbb{R}^n$ with trivial ...
user126256
6
votes
0
answers
122
views
Bordism groups and a short exact sequence
Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...
- 10.5k
3
votes
0
answers
122
views
Twisted spin cobordism v.s. KO theory in low dimensions
Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like
$${\Omega_d^{(\mathrm{spin} \times G)/N}},...
- 10.5k
1
vote
1
answer
379
views
Spectral sequence of Galois cohomology over local fields
On page 530 in his paper,
Notes on etale cohomology of number fields, Ann. Scient. ENS (1973),
Mazur insisted that
$$
\text{Ext}^q_{G_K} (M,~\bar{K}^*) \...
- 367
1
vote
0
answers
91
views
non zero differential in a spectral sequence
This is the situation:
Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...
- 81
6
votes
0
answers
163
views
Spectral Sequence for Twisted K-theory
Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[...
- 61