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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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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Product structure in Cohomology Spectral sequence

In the Serre Spectral sequence, we know, the cup product structure induces a canonical product in all $E_{r}$ pages which is compatible with respect to the differential. I am trying to understand ...
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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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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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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 ...
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57 views

functoriality of Grothendieck spectral sequence

I am looking for a reference which treats the functoriality of the Grothendieck spectral sequence for elements of the derived category of an abelian category.
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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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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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What does it mean that if the cover of $k$-fold intersections is not contractible it takes the form of a spectral sequence of cohomology?

I asked the following question in a previous post: Suppose a CW complex $M$ is given by the union of $n$-spheres, namely $M=\bigcup_{\alpha\in A}S^n$, without knowledge of intersections. The only ...
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147 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 ...
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Reference for the construction of the Leray spectral sequence from filtration

So far I have found only two possible constructions of the Leray spectral sequence for a continuous map $f : X \to Y$ between topological spaces with CW-complex structures: one through Cech complexes, ...
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On the limit of a directed system of spectral sequences

Suppose that we have a directed system $(E_N^{pq}, f_{N,N'})_{N,N' \in \mathbb{N}}$ of spectral sequences, and that, moreover, for any $N$, the spectral sequence $E_N^{**}$ collapses at its $E_2$-page ...
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A question related to group cohomology and spectral sequences

It is actually a follow-up question of a mathoverflow question. I don't quite understand the answer there. I tried to compute the group cohomology of $H^n(\mathbb{Z}_4,\mathbb{Z})$ via the Lyndon-...
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Understanding Adams spectral sequence and Pontryagin-Thom isomorphism intuitively

The question is about understanding Adams spectral sequence intuitively and some of the meanings of its relations. In Adams spectral sequence, $$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTG), \...
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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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Truncating columns of a double complex to get a filtration of the homology of the total complex

My question refers to the document found here. Specifically page 394 of the book (page 14 of the pdf). Theorem 2.5 on that page refers to "the filtration of $H_{m}(Tot)$ obtained by truncating columns ...
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Differentials on the second page of the spectral sequence of a first quadrant double complex

Suppose we have some (homological) double complex $\{E_{pq}\}$ with $p$ labelling the row and $q$ the column (is this standard or not?). Taking the homology of the vertical maps, it's easy enough to ...
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Thom space, homotopy group and cohomology group

In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
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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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1answer
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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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174 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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Connecting homomorphism and fibre transfer

I have a very special situation: Consider an orientable surface bundle $F_g\to E\stackrel{\pi}{\to} B$ with fibre a closed orientable surface of genus $g$. We have the “integration along the fibre” $$\...
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Maps between two Leray spectral sequences

Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
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108 views

Trivial action of fundamental group in Serre spectral sequence

Recently I'm studying Serre spectral sequence in Hatcher's book. Let $\pi : X \to B$ is a fibration, it's an easy exercise to check that when B is path-connected then all the fibers are homotopy ...
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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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$E_{p,q}^r$ 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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How to show that $ 0 \to E_{0,n}^2 \to H_n \to E_{1,n-1}^2 \to 0 $ is exact?

Suppose that a spectral sequence converging to $ H_\ast$ has $ E_{pq}^r = 0$ for all $ p\neq 0,1 $. Show that there are exact sequences $$ 0 \to E_{0,n}^2 \to H_n \to E_{1,n-1}^2 \to 0 \,. $$ ...
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Easy spectral sequence

This is a question in Weibel's Homological Algebra. Suppose that a spectral sequence converging to $H_\ast$ has $E^2_{pq} = 0$ unless $p = 0,1$. Show that there are exact sequences $$0 \rightarrow E^...
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Homology of a trivial fibre bundle via a spectral sequence

We have a fibration $F \rightarrow X \rightarrow B$. If $X = B \times F$, then combining Künneth formulas and universal coefficient theorem gives an isomorphism $H_n(X;G) \simeq \oplus H_p (B;H_{n-p}(...
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170 views

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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1answer
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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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301 views

Leray spectral sequence

Let $f : X\to Y$ be a continuous map of topological spaces, $A$ an abelian sheaf on $X$. We have the Leray spectral sequence $$E_2^{p,q} := H^p(Y, R^qf_*A)\Rightarrow H^{p+q}(X, A).$$ Could someone ...
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Adams operations and an artificial grading on K-theory

In this article by Snaith (p. 575) appears the following comment: ... these transgressive elements [...] can be located by means of the Adams operations [...]. These operate (unstably) in both the ...
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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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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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Low-degree integral cohomology of $K(\mathbb{Z}/n,2)$

Consider the Serre spectral sequence for the fibration $K(\mathbb{Z}/n,1)\rightarrow * \rightarrow K(\mathbb{Z}/n,2)$, $$ E^{pq}_{2}=H^{p}\bigl(K(\mathbb{Z}/n,2);H^{q}(K(\mathbb{Z}/n,1);\mathbb{Z})\...
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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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Construction of a sequence

Can I find a sequence $(f_j)_{j\in\Bbb{N}}\in C^{\infty}(\Bbb{R^+})$ such that : $$ \lim_{j\to\infty}\int^\infty_0 \big(\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\big)^2 rdr=0$$ and $$ \...
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1answer
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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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Cohomology of double complex with exact rows

Let $(C^{p,q},d_h,d_v)$ be a double complex of modules and let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that (1) the horizontal rows are exact, (2) the columns ...
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Showing when the spectral sequence associated to a filtered complex $C$ collapses using a similar complex $C\otimes\mathbb{F}[t]$

Let $\mathbb{F}$ be a field, and let $C=\bigoplus_{i,j\in\mathbb{Z}} C^{i,j}$ be a bigraded $\mathbb{F}$-vector space of finite total dimension. Suppose there are two differentials $d$ and $\widetilde{...
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Contravariant Grothendieck Spectral Sequence

I'm currently getting confused about indices in some spectral sequences. Assume we work in the category of modules for simplicity. Let $A^\cdot$ be a (bounded on the right) complex and let $B^\cdot$ (...
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Explicit formula for Higher Bockstein

The formula for the Bockstein $\beta:H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p\mathbb{Z})$ is $$\beta[c\otimes 1]=[\frac{1}{p}\partial c\otimes 1]$$ (McCleary page 456) How about for ...
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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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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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Times p map in Bockstein (Quite confused)

The following is from McCleary's book pg 461. Q1) I am confused about the map $$-\times p^{r-1}:H_n(X;\mathbb{Z}/p^r\mathbb{Z})\to H_n(X;\mathbb{Z}/p^r\mathbb{Z})$$ Just one page earlier, I saw this ...
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Problem understanding this presentation of Bockstein Spectral Sequence

I have some problems understanding this presentation of the Bockstein Spectral Sequence (McCleary pg 460). Q1) Firstly, how does this short exact sequence of coefficients work? $$0\to\mathbb{Z}/p\...
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163 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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(Bockstein Spectral Sequence) Why is $v_1$ also $p$-divisible?

Why if $u=pv_1$, then $v_1$ is also $p$-divisible? I am quite puzzled by the above. I can see that $v_1$ must also generate a copy of $\mathbb{Z}$, thus $v_1\notin\ker p^r$. I can't seem to proceed ...
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Bockstein Spectral Sequence (Why is this a direct sum)

(from McCleary's User guide pg 459). I am curious why is $pH_n(X)+\ker p^r$ a direct sum? (Or is it?) I tried to reason that $\ker p^r$ is equal to $\text{Im}\ \partial:H_{n+1}(X;\mathbb{F}_p)\to ...