Questions tagged [spectral-sequences]

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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 ...
Tim Campion's user avatar
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17 votes
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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. ...
Fernando Muro's user avatar
14 votes
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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 ...
Lukas Miaskiwskyi's user avatar
14 votes
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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 ...
Jonathan Beardsley's user avatar
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 ...
Matt's user avatar
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11 votes
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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 $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
Borromean's user avatar
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10 votes
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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 ...
Francis Baer's user avatar
10 votes
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313 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 ...
Luuk Stehouwer's user avatar
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 ...
Dan Petersen's user avatar
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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 $K(2)$-...
Drew Heard's user avatar
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9 votes
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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 ...
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9 votes
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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 ...
Laura's user avatar
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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 ...
wonderich's user avatar
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9 votes
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249 views

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 ...
Riccardo's user avatar
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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}...
Alex Turzillo's user avatar
9 votes
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673 views

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 $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ ...
tj_'s user avatar
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8 votes
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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 ...
gualterio's user avatar
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8 votes
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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 ...
annie marie cœur's user avatar
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 ...
qqqqqqw's user avatar
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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} \...
Federico Barbacovi's user avatar
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 ...
Samarkand's user avatar
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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{...
wonderich's user avatar
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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 ...
SashaP's user avatar
  • 7,027
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. $E$ ...
Thi's user avatar
  • 326
7 votes
0 answers
1k views

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 ...
David Hansen's user avatar
6 votes
0 answers
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 ...
Diego95's user avatar
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6 votes
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266 views

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 $...
Vitolo's user avatar
  • 81
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 ...
wonderich's user avatar
  • 10.3k
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 ...
S. carmeli's user avatar
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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 ...
Dominic Else's user avatar
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 $[...
Rui's user avatar
  • 61
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 ...
Just Me's user avatar
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6 votes
0 answers
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 ...
Zhipeng Duan's user avatar
6 votes
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^...
user136725's user avatar
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 ...
asv's user avatar
  • 21.1k
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\...
Mariano Suárez-Álvarez's user avatar
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}=\...
Gabriel's user avatar
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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) ...
Eduardo de Lorenzo's user avatar
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 ...
prefix.crm114's user avatar
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(...
Anh Dũng Lê's user avatar
5 votes
0 answers
217 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 $...
Saal Hardali's user avatar
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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 ...
wonderich's user avatar
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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 ... \...
BrianT's user avatar
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5 votes
0 answers
479 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 ...
yoyostein's user avatar
  • 1,219
5 votes
0 answers
505 views

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 $...
Mikhail Bondarko's user avatar
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 @...
jdc's user avatar
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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 ...
Roberto Frigerio's user avatar
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 $...
Daniel Grady's user avatar
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}) \...
user avatar
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 ...
Ralph's user avatar
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