Questions tagged [spectral-sequences]
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388 questions
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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 ...
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On the Leray spectral sequence and sheaf cohomology
I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...
- 35.2k
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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 ...
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Torsion in the integral cohomology of $BPU_{n}$
I would like to prove that the integral cohomology of $BPU_{n}$ the classifying space of the projective unitary group of order $n$ has $n-$primary torsion.
We have a fiber sequence of the form $BSU_{...
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fibrations of classifying spaces - Leray Hirsch Theorem converse
Let $G$ be a topological group and let $H$ be a closed subgroup. Assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces
$$G/H \rightarrow BH \rightarrow ...
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Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?
For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...
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Relating two different approaches to the Atiyah-Hirzebruch Spectral Sequence
Given a (for simplicity connective) spectrum $E$ and a pointed CW-space $X$ there is "the" (homological) Atiyah-Hirzebruch spectral sequence
$$E_{pq}^2 = \tilde{H}_p( X, \pi_q(E)) \Rightarrow \pi_{p+...
- 16.5k
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Étale cohomology of tensor product
Let $X$ be a smooth projective variety over a field $k$.
Suppose we have étale abelian sheaves $A, B$ on $X_{\rm ét}$ such that
$$H^j(X_{\rm ét}, A),\ H^j(X_{\rm ét}, B)$$
are finitely generated ...
- 149k
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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 ...
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Calculating topological index
Consider the space $X=BSL(8,\mathbb{C})/(\mathbb{Z}/2)$. The topological Brauer group of $X$ is given by $Br_{top}(X)=Tor(H^{3}(X;\mathbb{Z}))=\mathbb{Z}/2$. I'm studying concepts of topological ...
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Hochschild-Serre filtration and etale cohomology
I encountered the Hochschild-Serre spectral sequence in étale cohomology
$$H^i(\text{Gal}(\overline{k}/k), H^j_{et}(X_{\overline{k}}, F))\Rightarrow H^{i+j}_{et}(X_{{k}}, F)$$
How is the filtration ...
- 16.5k
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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-...
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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....
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Hodge Numbers and Leray Spectral Sequence
Mark Gross' notes survey of SYZ fibrations and toric degenerations begin by explaining why dual torus fibrations interchange Hodge numbers. But he defined the Hodge numbers in an unusual way
$$h^{p,q}(...
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Multiplicative structure on spectral sequence
Let $E$ be a spectral sequence and assume that there is a product
$E^{r}_{p_1,q_1} \times E^r_{p_2,q_2} \to E^r_{p_1+p_2,q_1+q_2}$
which satisfies the Leibniz rule (for all $p_i,q_i$, but $r$ fixed)....
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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 ...
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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) ...
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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 ...
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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 ...
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Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?
I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.
How about for multiple summands in the ...
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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 ...
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Pullback and homology
Suppose I have two maps of topological spaces, $f:X\rightarrow B$ and $g:Y\rightarrow B$, such that $f$ induces a homology isomorphism and $g$ is a fibration and $B$ is connected. Is it true that the ...
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Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$
Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω_{n,O}(pt)$ and $Ω_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$?
Here are some ...
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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'$ ...
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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 ...
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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 ...
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hirzebruch spectral sequence for a cohomology theory on a subcategory of TOP
The answer to my question is probably going to be 'yes, sometimes'. So I'll give my motivation first.
I am trying to give a short argument that for a fibration $F \hookrightarrow E \to B$ and a $\...
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Spectral sequences: opening the black box slowly with an example
My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials.
...
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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 ...
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How can I find the differential in the Serre spectral sequence for this sphere fibration?
Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials
$$
d_4^{p,m}:...
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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-...
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Monodromy and simple system of local coefficients
I was interested in the following question: if one has a fibration
$F\to E\to B$
there is associated a monodromy map, that is basically an action of the fundamental group $\pi_1(B)$ on the ...
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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 ...
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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, ...
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Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
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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 ...
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Question about the precise statement of Leray spectral sequences and a simple example
On Bott's paper "Homogeneous vector bundles" there is the following statement of Leray spectral sequence:
Let $X$ and $Y$ be paracompact and locally compact spaces and $f : Y \to X$ be a proper map....
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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{...
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isomorphic spectral sequences => quasi-isomorphic filtered chain complexes?
Let $(C,\partial)$ and $(C',\partial')$ be chain complexes of $R$-modules where $R$ is a (commutative) ring. Let $F$ and $F'$ be finite filtrations of $C$ and $C'$ respectively, i.e., $$\varnothing = ...
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Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space
Let $G$ be a compact, connected Lie group. There is an Atiyah–Hirzebruch spectral sequence
$$H^*(BG;K^*) \implies K^*(BG)$$
connecting $H^*BG$, which generally contains torsion, with $K^*BG \cong \...
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Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
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Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective
I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective.
I am a little confused about when maps between cohomology groups are ...
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521
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Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
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Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
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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
$$
\text{Ext}^q_{G_K} (M,~\bar{K}^*) \...
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108
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Relating inflation maps from spectral sequences in lower and higher dimensions
The spectral sequence has some nice property.
Consider $ N \to G \overset{R}{\to} Q $ and $G/N=Q$. There is a spectral sequence $\{E^{p,q}_n, d_n\}$ with: (i) The differential is defined as a map $...
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Is this sequence of Lie algebra cohomology a part of spectral sequence?
There is an exact sequence
$$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}, \...
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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 ...
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526
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The inability to continue a fibration sequence even when a delooping exists
$\newcommand{\i}{\iota}$
The general notion that I am trying to disprove is that if we are given a fibration $X \to Y$ with fiber $F$ such that the delooping $BF$ exists, that there is a map $Y \to BF$...
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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 ...
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