Questions tagged [spectral-sequences]
The spectral-sequences tag has no usage guidance.
148
questions with no upvoted or accepted answers
27
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answers
1k
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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 ...
17
votes
0
answers
719
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. ...
14
votes
0
answers
355
views
Gel'fand and Fuks' "Globalizing" of cohomology of formal vector fields
I apologize for the length of this question. If anybody already spent some time with cohomology of (formal) vector fields and the results of Gel'fand and Fuks, I imagine a lot can be skipped. I do ...
14
votes
0
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784
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 ...
13
votes
0
answers
393
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 ...
11
votes
0
answers
263
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 $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
10
votes
0
answers
309
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 ...
10
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0
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313
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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 ...
10
votes
0
answers
787
views
On functoriality of the Leray spectral sequence
The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
10
votes
0
answers
482
views
Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem
$\newcommand\Ext{\mathrm{Ext}}
\newcommand\Z{\mathbb{Z}}
\newcommand\G{\mathbb{G}}$
The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-...
9
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252
views
Mahowald uncertainty outside of homotopy theory
In homotopy theory there is the following informal idea:
The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named ...
9
votes
0
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385
views
Hochschild-Serre spectral sequence via explicit filtration
Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
9
votes
0
answers
129
views
Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...
9
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0
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249
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Building examples of elements of $\Omega_4(\xi)$ via surgery theory: how to do it?
When computing special bordism groups, I often need to determine existence of (singular) smooth $4$-manifolds with fixed fundamental group and certain properties like the spin behaviour (i.e. being ...
9
votes
0
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487
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extension problem for the Atiyah-Hirzebruch spectral sequence
For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
9
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673
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Can the Bockstein spectral sequence be used to compute cohomology rings ?
If $G$ is a finite group then there is the so-called Bockstein spectral sequence $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ ...
8
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0
answers
262
views
A diagram in the proof of Theorem 2.5.5 of 'Cohomology of Number Fields' and the Tate Spectral Sequence
I've been reading the book 'Cohomology of Number Fields' for years.
But I couldn't check the commutativity of the diagram
on page 126 until now. So I ask for help.
The diagram is induced by taking ...
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 ...
7
votes
0
answers
257
views
Differentials in spectral sequences and Massey products
Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
7
votes
0
answers
448
views
Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
7
votes
0
answers
162
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 ...
7
votes
0
answers
147
views
Cohomology of Lie group $E_8$, e.g. $H^d(E_8,\mathbb{R}/\mathbb{Z})$
What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient?
I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{...
7
votes
0
answers
348
views
Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
7
votes
0
answers
538
views
Motivic homotopy spectral sequence
I would like to have a question about the re-index convention. Let us consider a spectrum $E$ (I am mainly interested in motivic setting, however let's consider the simplicial case firstly, i.e. $E$ ...
7
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0
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1k
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An example computation of etale cohomology
(edited for clarity)
In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...
6
votes
0
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201
views
$\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence
I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the ...
6
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0
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266
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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 $...
6
votes
0
answers
122
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Bordism groups and a short exact sequence
Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...
6
votes
0
answers
470
views
The $E_2$-page of the May spectral sequence
I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS.
At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...
6
votes
0
answers
225
views
A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?
This question is a follow-up to my previous question:
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
In that question, I discussed two different spectral sequences for ...
6
votes
0
answers
150
views
Spectral Sequence for Twisted K-theory
Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[...
6
votes
0
answers
389
views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
6
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0
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167
views
A spectral sequence problem in Alejandro Adem's Paper
I am reading Adem's paper Periodic Complexes and Group actions. But I can't give an argument about a statement on spectral sequences.
Suppose you have an orientable fibration of CW-complexes like ...
6
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0
answers
358
views
Transgression map spectral sequence of Ext
Let $X$ be a scheme over $k$ and $p:\ X \to Spec(k)$ the structure morphism. If $M$ is an étale sheaf of abelian groups over $Spec(k)$ I have a Grothendieck spectral sequence $$E^{p,q}_2=Ext^p_k(M,R^...
6
votes
0
answers
710
views
On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
6
votes
0
answers
413
views
The Hochschild-Serre spectral sequence relative to an ideal containing the derived subalgebra
Is the Hochschild-Serre spectral sequence $$H_\bullet(\mathfrak g/\mathfrak h,H_\bullet(\mathfrak h,k))\Rightarrow H_\bullet(\mathfrak g,k)$$ for an extension of Lie algebras $$0\to\mathfrak h\to\...
5
votes
0
answers
401
views
Hypercohomology spectral sequence from the derived category point of view
Let $F\colon \mathsf{A}\to\mathsf{B}$ be an additive functor between abelian categories and let $M$ be a complex on $\mathsf{A}$. There's a "hypercohomology spectral sequence"
$$E_1^{i,j}=\...
5
votes
0
answers
616
views
Spectral sequence from a stratification by closed subvarieties
I am looking for a reference for the following result: If $X$ is an algebraic variety and
$$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$
is a stratification (edit: filtration) ...
5
votes
0
answers
305
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 ...
5
votes
0
answers
160
views
multiplication in spectral sequence
I am trying to understand this paper. Let $M$ be a compact Kaehler manifold of dimension $n$, $X$ is a holomorphic vector field, $i_X$ the contraction operator, i.e. for $\alpha$ a $p$-form, then $i_X(...
5
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0
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217
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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 $...
5
votes
0
answers
101
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 ...
5
votes
0
answers
613
views
What is the filtration in Leray's spectral sequence?
Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration
$$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
5
votes
0
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479
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Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple explanation?)
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than ...
5
votes
0
answers
505
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Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
5
votes
0
answers
358
views
Reference request: a Künneth spectral sequence map from equivariant K-theory to cohomology
The analogue of the Künneth formula for Borel $G$-equivariant cohomology can be obtained as the Eilenberg–Moore spectral sequence of a pullback
$\require{AMScd}$
\begin{CD}
(X \times Y)_G @...
5
votes
0
answers
282
views
Two natural maps asssociated with the nerve of a cover
Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...
5
votes
0
answers
210
views
Bockstein morphism of spectral sequences
Given an omega spectrum $E$, there is a type of chern character map given by its rationalization
$$r:E\to E\wedge M\mathbb{R}\;,$$
where $M\mathbb{R}$ denotes a Moore spectrum. The cofiber of the map
$...
5
votes
0
answers
258
views
Spectral sequences and hypercohomology for projective space
Suppose we are given a complex of sheaves on $\mathbb P^N$ in which every term is direct sum of invertible sheaves:
$$
\mathcal F^\bullet = \dots \to \oplus_{j=1}^{n_{p-1}} \mathcal O (k_j^{p-1}) \...
5
votes
0
answers
647
views
Do exact functors commute with spectral sequences ?
Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let
$$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral ...