Questions tagged [spectral-sequences]

Spectral sequences compute homology groups by taking a sequence of approximations, and generalise exact sequences. They find application in algebraic topology.

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Thom spectrum $MSpin$ and $E_2$-page for a large degree $i\geq 8$

Let the $MTH$ is Madsen-Tillman spectrum (which is a close cousin of the more usual Thom spectrum $MH$) associated to tangential structure $H$. For a computation involving no odd torsion, the Adams ...
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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 ... \...
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95 views

Collapse of Serre spectral sequence in the presence of a cross-section

I was under the impression that if a Serre fibration $f: E \rightarrow B$ has a right inverse $s: B \rightarrow E$, then the associated Serre spectral sequence would collapse on the second page. This ...
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Cohomolgoy of $S^1$-bundle over genus $g$ surface

Suppose $\Sigma_g$ is the closed, orientable surface of genus $g\ge 1$. Isomorphism classes of principal $S^1$-bundle on $\Sigma_g$ is then classified by $H^2(\Sigma_g,\mathbb{Z})=\mathbb{Z}$. Suppose ...
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Spectral sequence associated to the stupid filtration

In Methods of Homological Algebra by Gelfand and Manin, they define the spectral sequence associated to a filtered complex $(K^{\bullet},d^{\bullet})$. For example, the stupid filtration is defined as ...
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local system of coefficients on a fibration of classyfing spaces

It is well known that if $G$ is a lie group and $H$ is a closed subgroup of $G$, the inclusion $H \hookrightarrow G$ induces a fiber bundle on the classifying spaces $$ G/H \rightarrow BH \rightarrow ...
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McCleary Spectral Sequence: Is this a definition or a result of a theorem?

(McCleary's User Guide to Spectral Sequences pg 458) I refer to the statement "We denote the $E^1$-term by $B_n^1\cong H_n(X;\mathbb{F}_p)$". Is $B_n^1\cong H_n(X;\mathbb{F}_p)$ just a definition, ...
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Leray-Hirsch theorem for Dolbeault cohomology

In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this: Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$...
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Cover and extension of a Lie group

We know that $SU(2)$ is a double cover of $SO(3)$, such that $$SU(2)/Z_2=SO(3),$$ through a finite extension $N=Z_2$. For other examples of simply-connected Lie groups such as $SU(2)$, $SU(N)$ or $...
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Group homomorphism for cohomology group

Inspired by this post, I wonder, for a short exact sequence for some finite groups $A,B,C$, (I am happy to consider Lie groups too) $$1\to A \to B \to C \to 1,$$ whether there are generic group ...
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Comparing spectral sequences

Given a double complex $C_{\ast,\ast}$ concentrated in the first quadrant. Then I understand how one can associate two spectral sequences to this complex. One that first takes horizontal and then ...
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Is there a spectral sequence for Ext associated to an $n$-term exact sequence of modules?

I ask this question because of the following fact: Let $R$ be a ring and $0 \to M_1 \to M_2 \to M_3 \xrightarrow{\alpha} M_4 \to 0$ be an exact sequence of $R$-modules. Suppose we know $Ext_R(M_1,R)$,...
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What is the action on cycles

Let $A$ be a graded decreasing filtered chain complex, with cohomology differential $d$. Let $^nA^p=A^{p,n-p}$ be the elements of $A$ with total degree $n$, filter degree $p$, (and therefore with ...
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Extension problem for serre spectral sequence of trivial fibration

Consider the cohomology spectral sequence of the trivial fibration $F \hookrightarrow F \times B \to B$. I showed in my answer here Why all differentials are $0$ for Serre Spectral Sequence of ...
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Pairing on the AHSS induced by cap product: why does it exists

This is my setting: Let $\xi \colon X \to B$ be a vector bundle. Let $E$ be a ring spectrum. Suppose given a natural cap product $$ \frown \colon E^s(X,X\setminus B)\otimes E_{s+t}(X,X\setminus B)\...
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Reference on Weibel's Homological Algebra: “$G/H$ acts by conjugation in LHS-spectral sequence”

I'm studying the Lyndon-Hochschild-Serre spectral sequence for $H\triangleleft G$: $$ H_p(G/H;H_q(H;A))\Rightarrow H_{p+q}(G;A) $$ where $A$ is a $G$-module. I was told (w/o giving a proof) that ...
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Spectral Sequence associated to a filtration abuts because we can find closed representatives

Let $(K,D)$ be a differential complex of abelian groups, and $K = K_0 \supset K_1 \supset K_2 \supset \cdots \supset K_{p+1} = 0$ a filtration of $K$ by sub-complexes. Let $(E^{r},d^r)_{r\ge 1}$ be ...
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cohomology of orbit space by a free group action

Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$. Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get $H^*(B;...
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120 views

Exact sequence from Serre spectral sequence

let me say first that I don't know homological algebra very well, so I apologize in advance if my question is stupid.. It regards the Serre spectral sequence associated to a fibration $0\rightarrow F ...
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what is the natural map from BP to $HF_p$?

what is the natural map from BP to $HF_p$? BP is the Brown-Peterson spectrum and $HF_p$ is Eilenberg Maclane spectrum. I am trying to learn the connections between ASS and ANSS. This map should ...
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The differentials of a spectral sequence

Suppose we are on the $E_r$ page and the lattice either consists of 0 or $\mathbb{Q}[x,y]$ in each entry. Suppose in particular that the points $(p,q)$ and $(r, s)$ (and "their codomains") are equal ...
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filtration on the cohomology of a complex

Let $K^\bullet$ be a complex and let $F_I$ and $F_{II}$ be two filtrations on it. suppose $F_I^i K^n$ intersects $F_{II}^i K^n$ trivially. It then follows that in the induced filtration on the ...
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326 views

turning a map into a fibration

In Allen Hatcher's book Spectral Sequence page 29 Example 1.18, What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a ...
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343 views

Bockstein homomorphism and Steenrod square

question: What is the relation between Bockstein homomorphism and Steenrod square? For example, can one explain why the following relation works in the case of cohomology group with $\mathbb{Z}_2$ ...
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1answer
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Definition of $E^{\infty}_{pq}$ terms in a spectral sequence. Something strange seems to happen

I'm trying to prove the following assertion in Weibel's Homological Algebra page 125, 5.2.8 Given a homology spectral sequence, we see that each $E^{r+1}_{pq}$ is a subquotient of the previous term ...
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1answer
47 views

What's the meaning of “decreasing filtration”?

Page 152-153, "Algebraic Geometry" by Lei Fu. The condition (e) of the definition of spectral sequence is listed as follows: (e) A family of objects $H^n(n\in \Bbb Z)$ in $\mathcal C$ and each $H^...
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Example of a nongraded chain complex

A chain complex is an abelian group $A$ with an endomorphism $d \in End(A)$, s.t. $d^2=0$ called the differential. I am trying to come up with an example of a nongraded chain complex with nonzero ...
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462 views

Short exact sequence and cohomology group

Given a short exact sequence for some finite groups $A,B,C$, $$1\to A \to B \to C \to 1,$$ how could we construct an exact sequence of their cohomology group out of it? One version of the story I ...
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1answer
98 views

Two basic questions regarding spectral sequences

I am reading the wonderful article Spectral sequences : friend or foe ? by Ravi Vakil. However I do not understand the notation used when on p.4 he says : More precisely, there is a filtration $E_\...
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71 views

$E_{p, 0}^2$ and $E_{0, 1}^2$ terms in sequence, in terms of homology of $K(G, 1)$, homology of $K(R, 1)$, and action of $G$ on $R$ by conjugation?

This is a followup to my previous question, reproduced here. Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R, ...
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1answer
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What is the topology of $E^*(E)$ where $E$ is a ring spectrum?

In Adams' lectures on generalised cohomology Page 51, it states $E^*(E)$ is a topology ring. I do not know the topology on it. the reference that Adams offered there is by Novikov in Russian. Can ...
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1answer
47 views

Colimit of a sequence of objects in a category

In the book "Lecture notes in algebraic topology" by Davis and Kirk I came across the following colimit $$E^{\infty} = \operatorname{colim}_{r \to \infty}E^r$$ where each of the $E^r$ were $R$-...
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146 views

Homotopy groups $\pi_{n+1}(S^{n})$

In section $5.1$ of Hatcher's note about spectral sequences, he starts to compute stable homotopy group $\pi_{n+k}(S^{n}),k \leq 3$. Particularly for $\pi_{n+1}(S^{n})$, by Freudenthal suspension ...
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First page of a spectral sequence

I have a very basic problem when trying to understand "You could have invented spectral sequences" by Timothy Chow (but I will index cohomologically since I'm more interested in cohomology). This is ...
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2answers
252 views

Cohomology of augmented double complex with exact rows using spectral sequences

I am learning about spectral sequences through Vakil's notes and wanted to try out a simple spectral sequence argument on a double complex. I recalled this little paragraph in Bott & Tu, p.97: ...
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157 views

How does complete knowledge of Bockstein spectral sequences allow complete description of integral homology?

In this notes (pg 4): http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf, it is written that "complete knowledge of the Bockstein spectral sequences of $C$ for all primes $p$ allows a complete ...
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1answer
203 views

is it possible to derive the künneth theorem from serre spectral sequences?

In case it is any easier, we can consider $\mathbb Q$ coefficients, so that there is no torsion. In this case it should be enough to prove that in the second page of the spectral sequence, all ...
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1answer
81 views

Cohomology of $S^1 \rtimes \mathbb{Z}/2$

I am trying to compute the mod2 cohomology of the semi direct product $G = S^1 \rtimes \mathbb{Z}/2$ using the extension $$ 1\rightarrow S^1 \rightarrow G \rightarrow\mathbb{Z}/2\rightarrow 1$$ ...
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1answer
291 views

Local-Global Ext sequence

Let $F,G$ be two sheaves of $\mathcal{O}_X$-modules, where $X$ is a scheme. The local-global Ext exact sequence starts like this: $$0\to H^1(\mathcal{Hom}(F,G))\to Ext(F,G)\to H^0(\mathcal{Ext}(F,G))\...
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H-spaces act trivially on fiber?

If $F\to E\to B$ is a fibration and $B$ is a path-connected H-space, is the action of $\pi_1(B)$ on $H^*(F)$ by fiber transport always trivial? The reason I am asking is that I would like to consider ...
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Contradiction in spectral sequence calculation of $H_*(BO(2))$

$\newcommand{\Z}{\mathbb{Z}}$ For this post I am going to assume the answer namely $H_*(BO(2))=\Z_2[w_1,w_2]$. Consider the fibration $S^1 \hookrightarrow BO(1) \to BO(2)$. The $E^2$ page has $E^2_{i,...
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Relation between long exact sequence and associated graded

I'm reading these notes by Hutchings on spectral sequences. In the first section, he motivates spectral sequences with the long exact sequence in relative homology. Given a chain complex $C_*$ and a ...
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1answer
167 views

Multiplicative spectral sequence

I have a simple question regarding the definition of a multiplicative spectral sequence, which I couldn't answer myself by looking at the definitions in various texts: Is the product assumed to be ...
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Spectral sequence of filtered complex.

I am trying to understand the construction of a spectral sequence of a filtered complex. After reading through the entry in the nLab I came up with an example, that I don't understand: Consider the ...
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Tate construction of the Anderson dual of the sphere spectrum

I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
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Understanding why the Adams Spectral Sequence works

I am trying to learn about the Adams Spectral Sequence and my question is basically summed up in the title. More precisely, let $X$, $Y$, and $E$ be spectra. We have a homomorphism $[X,Y] \to Hom_{E^...
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degeneracy of the Serre spectral sequence

The following are well-known facts on the Serre spectral sequence For a fibration $F \rightarrow E \rightarrow B$ we have the Serre spectral sequence (in cohomology with a coefficients in a field ...
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Functoriality of Lyndon-Hochschild-Serre spectral sequences in coefficients.

It is a question about group cohomology. Supposing that I have a short exact sequence of $G$-modules $1\rightarrow A_1 \rightarrow A_2\rightarrow A_3\rightarrow 1$, I know that there will be a long ...
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47 views

Group actions on spectral sequences of group cohomology

Suppose I have a group extension $1 \rightarrow N \rightarrow H\rightarrow K\rightarrow 1$, and we have a group $G$ which acts on $H$, and $K$ by automorphisms and it does not have action on $N$. ...
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138 views

Deligne's theorem on the Leray spectral sequence and weights

Motivation : If $f : X \to Y$ is a smooth projective map between algebraic varieties, then there is a theorem by Deligne which says that the Leray spectral sequence degenerates at $E_2$. The proof I ...