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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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Well-definedness of the morphism for the chain complex $\text{Hom}(V,W)$

Let $R$ be a unital commutative ring. Suppose that for the $\mathbb{Z}$-graded $R$-modules $V$ and $W$, $(V,d)$ and $(W,d)$ are chain complexes. Let $\text{Hom}(V,W)$ be the module of homogeneous ...
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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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45 views

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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445 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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47 views

Exhaustive filtration of a torsion abelian group

Let $p$ be a prime integer and $A$ a torsion abelian group. Define a filtration on $A$ by $F_{-1}A=0$ and $F_s A=\ker ( p^{s+1}:A\to A)$ for $s\geq 0$. Why is this filtration exhaustive, i.e. why is $...
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1answer
159 views

fibrations and map of spectral sequences.

I have a concern regarding maps of serre fibrations and the induced map into the respective spectral sequence. The situation is the following. Consider two serre fibrations $F \rightarrow E \...
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Calculating the extraordinary cohomology of $\mathbb{C}P^n$

Let $E$ be a ring spectrum with an orientation. Now I want to calculate $E^*(\mathbb{C}P^n)$. The definition of orientation I am using is: There is an element $x \in E^*(\mathbb{C}P^{\infty})$ such ...
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43 views

Mapping (in/sur/bi-jective) between a group to its (normal subgroup, quotient group)

Say $G/N=Q$ and $G,N,Q$ are all groups. $N$ is the normal subgroup and $Q$ is the quotient group of $G$. Let us say we require group homormophism for every possible map below. Is it true that We ...
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41 views

Examples of group extension $G/N=Q$ with continuous $G$ and $N$, but finite $Q$

Can we have some examples of group extensions $G/N=Q$ with continuous (e.g. topological groups or Lie groups) $G$ and $N$, but a finite discrete $Q$? Note that $1 \to N \to G \to Q \to 1$. What else ...
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Examples of group extension $G/N=Q$ with continuous $G$ and $Q$, but finite $N$

Can we have some (new) examples of group extensions $G/N=Q$ with continuous (e.g. Lie groups) $G$ and $Q$, but a finite discrete $N$? Note that $1 \to N \to G \to Q \to 1$. What I know already ...
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Can isomorphic spectral sequences converge to non-isomorphic limits?

Overview: I'm learning about spectral sequences by working through exercises in Chapter 10 of Bourbaki's Algebra book. One of the exercises makes me suspect that the notion of isomorphism of spectral ...
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376 views

Are long exact sequences in homology a special case of spectral sequences?

I want to start by saying that I only have very basic notions about spectral sequences. Consider a short exact sequence of chain complexes $$0\longrightarrow A\longrightarrow B\longrightarrow C\...
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1answer
133 views

Why is the limit of a spectral sequence unnatural?

Fix an Abelian category $\mathscr{A}$, and a homological spectral sequence $(E_{p,q}^r)$, so that $E^{r+1}\cong H(E^r)$, then, according to the ncatlab, we say that $E^r$ converges to $E^\infty$ if ...
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1answer
61 views

How does an augmentation of a cosimplicial space give an augmentation of its realization tower?

Let $\mathbf{C}$ be a cosimplicial space and let the realization tower be a collection of simplicial sets $Tot_s{C}$ fitting into a tower $ ... \to Tot_s{C} \to Tot_{s-1}{C}...$. I will now define ...
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140 views

What can we say about the category of spectral sequences?

For simplify, let's just consider the following category first: object: "spectral sequences", that is a sequence of pages $(E_r)_{r\ge0}$, where each page $E_r$ is a differential object in an abelian ...
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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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1answer
314 views

Applying the Lyndon-Hochschild-Serre spectral sequence

I would like to clarify the assumptions under the Lyndon-Hochschild-Serre spectral sequence can be applied for a group extension $1 \rightarrow N \rightarrow G \rightarrow G/N \rightarrow 1$ In many ...
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1answer
199 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
35 views

SES associated to a spectral sequence

this is exercise 5.1.1 in Weibel. Suppose we have a double Complex $E_{\bullet,\bullet}$ with only the p and p-1 columns nonzero. Show that there is a SES: 0 $\rightarrow$ $E^2_{p-1,q+1}$ $\...
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132 views

Bockstein homomorphism for $\mathbb{Z}_n$ and “Steenrod” $n$th power

The Bockstein homomorphism can be generalized for $\mathbb{Z}_n$ values, $$\beta_n: H^m(M^d,\mathbb Z_n) \to H^{m+1}(M^d,\mathbb Z_n),$$ and $$\beta_n x =\frac1n d x \text{ mod } n,$$ $$x \in H^m(M^...
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1answer
336 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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Inverse of $1-ba$ when $a,b$ are the elements of unital algebra $A$

I know if $a,b$ are elements of a unitral algebra $A$, then $1-ab$ is invertible if and only if $1-ba$ is invertible. And it follows from the observation that if $1-ab$ has inverse $c$ then $1-ba$ has ...
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1answer
80 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
52 views

Showing that filtered complexes induce a spectral sequence

Let $(F_p C_\bullet)_p$ be a filtered complex of $R$-modules, with chain maps $$\cdots \xrightarrow{}C_n\xrightarrow{d_n}C_{n-1}\xrightarrow{}\cdots$$ such that $$\cdots \hookrightarrow F_pC_n\...
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Spectral sequence of non-cohomologically trivial fibration.

It is a well known that in case when we have $\xi: E\to B$ - a Serre fibration, with the trivial action of $\pi_1(B)$ on a fiber $F$, we have standart Atiyah - Hirzebruch spectral sequence. What we ...
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1answer
58 views

Contradiction in spectral sequence for $K(\mathbb{Z},3)$

$\newcommand{\Z}{\mathbb{Z}}$ Take the fibration $K(\Z,2) \hookrightarrow * \to K(\Z,3)$. Then $d_3^{0,2}$ is an isomorphism since this is the only way to get rid of $H^2(K(\Z,2))$ and to kill $H^3(K(...
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2answers
148 views

Another way to compute $\pi_4(S_3)$: contradiction in spectral sequence calculation

$\newcommand{\Z}{\mathbb{Z}}$ I decided that I would try another way of computing $\pi_4(S_3)$. Take the fibration $S_3 \to K(\Z,3)$ with fiber defined to be $X_4$. I want to directly use this ...
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1answer
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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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1answer
278 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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results in commutative algebra that use spectral sequences

I am trying to convince my professor to spend some time on the Grothendieck spectral sequence in class(she asked us what we want to do in the last weeks). She told me that she needs a result that ...
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1answer
72 views

Some clarifications about the Secondary Cohomology Operation associated to $Sq^2\circ Sq^2=0$

As explained in the title, I'm looking for some clarifications about the secondary cohomology operation associated to the relation $Sq^2\circ Sq^2=0$. I've just started reading the relevant chapter in ...
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159 views

Equivariant cohomology with respect to Lie group and its maximal torus

Let $G$ be a compact connected Lie group, $T \subset G$ its maximal torus and $W=N(T)/T$ its Weyl group. The formula 2.11 of Atiyah, Bott The Moment Map and Equivariant Cohomology states that for any ...
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Value of a left exact functor on a filtered object

If I have a filtered object $A = A_0 \supset A_1 \supset A_2 \supset \cdots$ such that the filtration is complete and a left exact functor $F$, is there a spectral sequence with inputs $F(A_i/A_{i+1})$...
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1answer
329 views

Vanishing terms in Leray spectral sequence

Let $f:X\to Y$ be a continuous map of complex manifolds. Consider $H^p(Y;R^qf_*V)$, where $V$ is a vector bundle on $X$, and $R^qf_*V$ are the higher direct images. (This is the $E_2^{p,q}$ term in ...
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141 views

Logarithmic de Rham complex on a curve and the Hodge spectral sequence

Given a smooth projective curve $C$ over a field $k$ of characteristic $0$, let $D$ be a divisor. Let $\Omega_C^{\bullet}(D)$ be the logarithmic de Rham complex of $C$, with log poles along $D$. Is ...
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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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91 views

Isomorphic total complexes, spectral sequence argument

I would like to proof the following claim: Let $f: C_{\ast,\ast}\to C'_{\ast,\ast}$ be a map of bicomplexes (differentials anticommute) that is a quasi-isomorphism restricted to each column. Then $f$...
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96 views

Product structure on cohomological Serre spectral sequence

Multiple sources state that the cohomological Serre spectral sequence has a multiplicative structure $E^{p,q}_r \times E^{s,t}_r \to E^{p+s,q+t}_r$. For $r=1$, $E^{p,q}_1$ is $H^q(X_p,X_{p-1})$ and $E^...
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142 views

Bott & Tu - Spectral Sequences

I have been reading this book with the main purpose of introducing myself to spectral sequences. Although so far I've been enjoying it quite a lot, I've had a problem I cannot handle: In theorem 14.6 ...
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Edge homomorphisms for Atiyah-Hirzebruch spectral sequence for a spectrum $X$

I'm interested in some nice identifications (together with explanations) about the edge homomorphisms in the AHSS for a spectrum $X$. $$\times \times \times $$ Let $X$ be a connective spectrum, as ...
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1answer
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Computing oriented Bordism groups of a manifold $X$ for $n\leq 4$

I'm following the book of Davis and Kirk "Lecture notes in Algebraic Topology" where, at pages $246-247$ there is a computation of the aforementioned groups via the Atiyah- Hirzebruch Spectral ...
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A problem with a definition of homology of a spectrum

I'm currently reading some notes about the James spectral sequence (here) and there is a passage which is bothering me (page 749 bottom): $$ colim_n h_{p+q}(T(\xi_n))\cong h_{p+q}(M\xi)$$ where $M\...
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Edge morphisms coincide cup-products in the Tate spectral sequence

In the Tate spectral sequence, the edge morphism coincides with the cup product. The proof is written in Neukirch-Schmidt-Wingberg's book: Cohomology of Number Fields (Theorem 2.5.5,p125). https://www....
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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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1answer
96 views

Reference Request: The Atiyah-Hirzebruch Spectral Sequence

I have just finished learning the Serre spectral sequence and I would like to learn about the Atiyah-Hirzebruch spectral sequence. Could someone suggest an accessible reference? Thank you in advance.
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Calculating torsion in $\pi_i(S^{2n})$

Let $p$ be an odd prime and $n>1$. I want to prove that $\pi_i(S^n)$ has no $p$-torsion for $i<n+2p-3$. For odd $n$ this is Proposition 6.26 in McCleary's book (p.206). He mentions afterward ...
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1answer
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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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0answers
96 views

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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1answer
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Grading of Cech-de Rham cohomology

I am currently self-studying Bott and Tu. In chapter 2 the Cech-de Rham cohomology is introduced and I thought I had understood it well enough. However when I got to chapter 3 on spectral sequences I ...
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1answer
186 views

Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth's theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,...