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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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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1answer
76 views

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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71 views

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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1answer
51 views

$\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$?

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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1answer
25 views

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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40 views

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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83 views

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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1answer
163 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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40 views

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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31 views

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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102 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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92 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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1answer
101 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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1answer
113 views

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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1answer
162 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
45 views

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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1answer
280 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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180 views

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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23 views

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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1answer
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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28 views

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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1answer
121 views

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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2answers
42 views

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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1answer
205 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
206 views

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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166 views

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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32 views

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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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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1answer
130 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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102 views

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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1answer
106 views

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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22 views

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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1answer
91 views

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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1answer
27 views

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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1answer
150 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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1answer
28 views

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 ...
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23 views

Is connected necessary for this theorem on Bockstein spectral sequence?

1) I am curious if the condition that $X$ is a connected space is necessary for the above theorem for Bockstein Spectral Sequence? The proof is from McCleary's User Guide to Spectral Sequence, and I ...
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45 views

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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116 views

How to use spectral sequences? [closed]

How to compute and to use spectral sequences ?. I dont know the steps to follow to compute a spectral sequence. Do we need to compute all pages : $ E_r $ and all spacs $ E_r^{pq} $ and differentials $ ...
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Path-connectedness of the fiber ine the LSAH spectral sequence

I've read some proofs of the identification of the second term of the Leray-Serre-Atiyah-Hirzebruch spectral sequence, and I don't quite seem to understand where in the proof do we need the fiber of ...
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1answer
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Questions regarding Bockstein Spectral Sequence (McCleary's book)

I have some questions regarding Bockstein homomorphism in John McCleary's book (pg 455-456). Q1) Is there a typo, is it supposed to be $\bar{u}\in H_n(X;\mathbb{Z}/r\mathbb{Z})$? Q2) How do we see ...
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97 views

Cohomology of genus g surface fibration

Let $F\to E\to X$ be a Serre fibration with connected base space $X$ and let the fibre $F$ be an orientable surface of genus $g$. Is there anything we can say about how $H^n(E)$ depends on $H^n(X)$? ...
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1answer
113 views

Convergence of spectral sequences

Consider a double complex $C^{*,*}$, where each $C^{*,*}$ are bigraded modules over $R$ (in fact they are vector spaces). Let the horizontal and vertical differentials be $d_1,d_2$, so that $d_1,d_2$ ...
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1answer
94 views

Reference for Bockstein Homomorphisms / Spectral Sequence for Homology

Is there any book for Bockstein Homomorphisms specialising in the case of homology? So far the books I read (Hatcher, Munkres) discuss it for cohomology. I am aware that it is said to be similar for ...
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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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2answers
229 views

How can I determine the Steenrod Square $Sq^2$ for complex projective space?

I am trying to learn about Steenrod Squares for algebraic varieties so that I can compute examples of complex topological K-theory using the Atiyah-Hirzebruch Spectral Sequence (AHSS). One of the key ...
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1answer
141 views

What is the integral cohomology of $\mathrm{PGL}_n(\mathbb{C})$ as a space?

The computation of the cohomology of $\mathrm{GL}_n(\mathbb{C})$ is one of the basic applications of the Serre spectral sequence, using the fiber bundle $\mathrm{GL}_{n-1}(\mathbb{C})\to \mathrm{GL}_n(...
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1answer
165 views

Morphisms in the derived category vs. morphisms on cohomology

Let $A$ and $B$ be complexes $R$-modules. Assume that $A^\bullet$ is bounded above and $B^\bullet$ is bounded below. Then there is a convergent spectral sequence $$\prod_{r\in \mathbb{Z}} \...
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144 views

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 \...