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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Local-to-Global Spectral Sequence for locally free resolution

Consider two coherent sheaves $\mathcal{E},\mathcal{F}$, with locally free resolutions \begin{align} &0 \longrightarrow \mathcal{E}_1 \longrightarrow \cdots \longrightarrow \mathcal{E}_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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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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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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computing $\pi_1 S^1$ from a spectral sequence

In most of my calculations that I have done based off of mosher and tangora, calculations have proceeded by knowing for example that the fiber of some fibration is say a $1-sphere$. From this we are ...
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Comparison Criterion for Atiyah - Hirzebruch Spectral Sequence

Let us denote with $E(X)$ the A-H spectral sequence associated to a CW complex $X$ and homology theory $h_*$: $$ E(X)_{pq}^2 = H_p(X, h_q(\ast))\Longrightarrow h_{p+q}(X)$$ and with $E(Y)$ the one ...
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Mapping Lemma for Spectral Sequences in Weibel's book - Help with the proof

We start by recalling the definition of a morphism in the category of Spectral Sequences: a morphism $f \colon A \to E$ is a family of maps $f^r_{pq}\colon A^r_{pq}\to E^r_{pq}$ in the abelian ...
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Basic questions about convergence of spectral sequences.

I have a fairly rudimentary understanding of spectral sequences. I have a couple questions though. In Algebra, Lang states what seems to me, to be two slightly different notion of convergence of a ...
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How networks with high largest eigenvalues are more robust?

In the literature, it sometimes indicates that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust. Robustness here is relevant to link/...
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Gysin sequence and Serre spectral sequence

Given an oriented $S^k$ bundle $E$ over a compact manifold $M$ we get the Gysin Sequence (I am interested in the DeRham cohomology). We can obtain this sequence from the Serre Spectral Sequence if the ...
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Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
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Reference request: where can I find illustrative, concrete examples of the use of the Eilenberg–Moore spectral sequence?

Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle $S^3 \to S^2$...
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117 views

Serre spectral sequence and locally constant coefficients

I have a brief question - In the Serre Spectral sequence for a fibration $$F \rightarrow E \rightarrow B$$ one can require, to avoid using local system of coefficients, that the action $\pi_1(B)$ on $...
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185 views

Spectral Sequence and Stiefel Manifold

Let $Spin(3)$ be embedded in $Spin(5)$ by the spin embedding then we have a fibration: $$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3)$$ Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel ...
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What is the difference between multiplication and direct sum on homotopy groups of spheres?

In Allen Hatcher's book Algebraic topology he states that factoring out 2-torsion $\pi_{i}(S^{2n})\cong\pi_{i-1}(S^{2n-1})\times\pi_{i}(S^{4n-1})\:\forall n$ but in his book Spectral Sequences in ...
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196 views

Wang Sequence for the circle $S^1$

Let $F\stackrel i \to E\stackrel \pi\to S^1$ a fiber bundle over the circle $S^1$. There is a long exact sequence sequence in cohomology, called Wang: $$\dots\to H^k(E)\stackrel {i^*}\to H^k(F)\...
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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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Serre's Exact Sequence in Homology

I am trying to derive the following result of Serre's: Let $F \hookrightarrow E \stackrel{p}{\to} B$ be a fibration with $B$ simply connected. Suppose $H_i(B)=0$ for $0 < i < p $ and that $...
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208 views

Hochschild-Serre spectral sequence

I have a hopefully simple question about the Hochschild-Serre spectral sequence (which may just be a simple question about general spectral sequences). Whenever I see the sequence written down, it has ...
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1answer
63 views

A spectral sequence with only one index in Atiyah's paper?

I would like to read Atiyah's paper Characters and cohomology of finite groups; but when I started reading the introduction, Atiyah mentions that he will prove that there is a "spectral sequence $\{E^...
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$\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$?

From the computation of some lower dimension $N$ of $Sp(N)$ group, we see that the homotopy groups are: $\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$, at ...
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166 views

Künneth formula and Leray spectral sequence

I want to prove the Künneth formula with $\mathbb R$ coefficients using the Leray spectral sequence. Let $f: X \times Y \to Y$ the projection map. Then we get a Leray spectral sequence $E^{p,q}_r \...
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58 views

Homology with coefficients from homology.

My main goal is to understand the computations behind the cohomology ring of $\mathbb{C}P^n$ as done in Bott & Tu. To this ends, I am reading a set of notes about Spectral Sequences (here) by ...
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Cohomological AHSS for projective space $\mathbb{C}P^n$ ALWAYS collapses at the second page

While I was computing the cohomology ring $E^*(\mathbb{C}P^n)$ for $E$ an oriented ring spectrum, via AHSS I realised that orientation is not a necessary hypothesis in order to compute the cohomology ...
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1answer
242 views

Tensor product of homology equivalences

Let $f : C \to C'$ and $g : D \to D'$ be chain maps of non-negative chain complexes of $R$-modules, where $R$ is any commutative ring. Assume that $f$ and $g$ are homology equivalences. Is the same ...
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286 views

An exact homology sequence associated with a principal SO(n) bundle

Suppose $P$ is a principal $SO(n)$ bundle, X is its base space. Why is there an exact sequence in homology groups $$ 0 \to H^1(X;\mathbb{Z}_2) \to H^1(P;\mathbb{Z}_2) \to H^1(SO(n);\mathbb{Z}_2)\to H^...
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131 views

Information about the total complex from the second page of a spectral sequence

This is an exercise from Ravi Vakil's notes on spectral sequences. Suppose you have a spectral sequence $E^{\bullet\,\bullet}_\bullet$ such that $E^{i\,j}_0$ is zero if either $i$ or $j$ is ...
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162 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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1answer
134 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
59 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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1answer
406 views

Spectral sequence page isomorphism

Suppose we have a map of spectral sequences $\{E_{p,q}^r,d^r\}\to \{{E'}_{p,q}^r,d'^r\}$, both generated from total chain complexes, $C$ and $C'$ respectively, such that for some $r$ the map between ...
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52 views

What does the $\Omega$ represent in $\Omega S^{n}$?

To put my question in context, I'm reading Hatcher's book on Spectral sequences is which is say " The suspension homomorphism $E$ is the map on $pi_{i}$ induced by the natural inclusion map $S^{n}\...
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1answer
24 views

A different approach towards spectral sequences.

I am somewhat acquainted with spectral sequences as in Weibel's book(the usual definition with many indices and pages), but I have found a different approach in the Stacks project.Link Although I can ...
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1answer
104 views

The spectral sequence for hyper-derived functors

Let $F$ be a right exact functor between two abelian categories $A$ and $B$.Suppose that $C_\bullet$ is a complex in $A$,then there is a convergent spectral sequence $$E_{p,q}^2 = ({L_p}F)({H_q}({...
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Showing induced action of G by conjugation on Hochschild-Serre $H_i(G/N, H_j(N,M))$ is trivial

Given a group extension: $$ 0 \rightarrow N \rightarrow G \rightarrow \frac{G}{N} \rightarrow 0 $$ I need to show that the induced action of $G$ by conjugation is trivial on the Hochschild-Serre ...
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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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1answer
139 views

Generalized cohomology groups of torus

Let $\tilde h^\bullet$ be a reduced generalized cohomology theory, and let $T^2$ be the torus. For what theories $\tilde h^\bullet$ is $\tilde h^\bullet(T^2)$ known (or easily computable)? For ...
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1answer
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$E_{p,q}^r$ spectral sequence. Find a l.e.s. $\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} E_{p-1,1}^2\to A_p\to E_{p,0}^2\to\cdots$

Let $E_{p,q}^r$ be a spectral sequence which converges to $A_n$. Let $E_{p,q}^r=0$ for $q\ge 2$. How to construct a long exact sequence $$\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} E_{...
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1answer
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Using the Bockstein spectral sequence to identify direct summands

I have a question about demonstrating part 2 of corollary 5.9.12 in Weibel's An Introduction to Homological Algebra. Here is the setup. Fix a prime $p$ and suppose I have a long exact sequence of ...
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1answer
71 views

The spectral sequence of the path fibration of $S^2$

Bott and Tu use the fibration $\Omega S^2 \to PS^2 \to S^2$ to compute the cohomology of $\Omega S^2$. They do this by looking at the (Serre?) spectral sequence. Then $E_2^{p,q} = H^p(S^2,H^q(\Omega ...
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1answer
86 views

Example leading to spectral sequence

I am reading A User's Guide To Spectral Sequences and I don't understand the example in the informal introduction chapter: We want to compute $H^*$, where $H^*$ is a graded $R$-module or a graded $...
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1answer
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filtration on the (co)homology of a space from the filtration of a space

Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let $F_k:=\mathrm{Im}\,(...
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1answer
98 views

Elementary (?) question on differentials in a spectral sequence

Suppose I have a chain complex $C$ with differential $D$ and filtration $F$. Suppose further that I can decompose $D$ by its action on the filtration, i.e. there are maps $D = D_1 + D_2 + \cdots$ so ...
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What's wrong with this argument that the Atiyah-Hirzebruch spectral sequence always degenerates?

Let $E$ be a spectrum and let $X$ be a space or connective spectrum. Then the cohomological Atiyah-Hirzebruch spectral sequence is of the form: $H^s(X,E^t) \Rightarrow E^{s+t}(X)$ This is a half-...
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Cohomology of spectra as free modules over Steenrod Algebra

I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement". Let $X$ be a connective CW-spectrum of finite ...
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Group cohomology and singular cohomology of classifying space

Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ ...
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Computing maps in Leray spectral sequence

Let $f:X_{s1} \to X_{s2}$ be morphism of sites ( Here $X$ is some scheme $X_{s1}$ refers to the site on $X$). Now using the Leray spectral sequence one gets the following exact sequence $0 \to H^1(...
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Serre spectral sequence of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2=\mathbb{R}P^\infty$

As the title shows, we have a fibration of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2\sim\mathbb{R}P^\infty$. I am trying to check my understanding of Serre spectral sequence with ...
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55 views

Why intermediate map in Gysin sequence is multiplication by Euler class

I am reading Bott and Tu, Differential Form in Algebraic Topology. At page 178, they constructed Gysin sequence of Sphere Bundle. I am having trouble understanding the argument, $d_{k+1}$ is ...
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35 views

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 ...