(STUB) A method of computing homology groups by taking a sequence of approximations. Particularly useful in algebraic topology.

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Reference request: a Künneth spectral sequence map from equivariant K-theory to cohomology

The analogue Künneth formula for $G$-equivariant cohomology can be obtained as the Eilenberg–Moore spectral sequence of a pullback $\require{AMScd}$ \begin{CD} (X \times Y)_G @>>> ...
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Estimating pseudo-periodicity of signals

I have pressure data which are measured at a given point in a standing wave. These data(signals) are 'almost' sinusoidal in nature. Each cycle may slightly vary from the original signal i.e the ...
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Computing the “limit” of a SSeq with $E_{\infty}^{*,*}$ a free graded $\Gamma$-module.

I'm trying to prove the following proposition from Kochman's book. For completion I will write it here the relevant part: Let $E$ be an oriented spectrum with orientation class $x\in ...
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Rational homology of $\Omega^{n+1}\Sigma^{n+1}X$

I want to know how compute, by induction and using the Serre spectral sequence for homology, $H_*(\Omega^{n+1}\Sigma^{n+1}X, \mathbb{Q})$. I know that I have to use the path-loop fibration $$ ...
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When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
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unboundness of an infinite series $f(t)\cos(tx)\sim t^{-1}\cos(tx)$

If $\lim_{t\to \infty} f(t)t=1$, i.e., $f(t)\sim t^{-1}$, then \begin{equation} {\text{ess}\sup}_{x\in [-\pi,\pi]}\sum_{t=1}^{\infty} f(t)\cos(tx)=\infty ? \end{equation} Here ${\text{ess}\sup}$ is ...
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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 ...
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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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33 views

Hochschild homology of dgas with nontrivial differential

In this question, we see how to compute the Hochschild homology of a dga with zero differential: it's just the same as computing its Hochschild homology as a graded algebra. I want to know about ...
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36 views

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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Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
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26 views

Finding the rate of convergence

We know that $$‎‎‎\frac{a^k}{\Gamma(k\alpha+1)}‎‎\longrightarrow0‎‎$$ when $k$ tends to $\infty$ and $\alpha ,a\in R$. I want to find rate of convergence for this sequence. thanks
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107 views

Surjectivity of Edge morphism in A-H cohomology Spectral Sequence

As the title suggests, I'm interested in proving the following claim: Recall the AH-spectral sequence:$$ E_2^{pq}=H^p(X,\mathcal{H}^q(\ast)) \Longrightarrow \mathcal{H}^{p+q}(X)$$ and since ...
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1answer
33 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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$E_{p, 0}^2$ 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 ...
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What is the filtration on cohomology of total space?

What is the filtration on cohomology of total space in Leray spectral sequence associated to fibration? The cohomology of total space is filtered by subgroups whose successive quotients are stable ...
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28 views

$E_{p,q}^r$ 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} ...
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51 views

An algebra problem from spectral sequence [duplicate]

Recently, I am reading the article "You Could Have Invented Spectral Sequences" by Timothy Y. Chow. Link: http://www-math.mit.edu/~tchow/spectral.pdf In page 17, he used the following splitting which ...
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Proving that Tor is a balanced functor using the derived category

At this end of this expository article on derived categories, R.P. Thomas says the following. There are two main advantages of this approach. Firstly that we have managed to make the complex ...
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1answer
51 views

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 ...
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1answer
22 views

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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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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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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74 views

$h^{0, 1}$ of K3 surface (a priori non-Kahler)

I am trying to understand paper by Siu "Every K3 surface is Kahler". Let $M$ be a K3 surface. Siu wrote $H^1 ( M , \mathscr{O}_M ) =0$ without any references. It is written on fifth page. Maybe I ...
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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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59 views

Grothendieck spectral sequence from the hypercohomology spectral sequence

Is it possible to write a proof of the convergence of the Grothendieck spectral sequence of the composition of two functors only using the convergence of the hypercohomology spectral sequences ...
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38 views

Leray sheaf being constant - what does it mean in terms of singular cohomology?

Let me first admit that I know next to nothing about sheaf cohomology, but I might have encountered a good reason to learn it. Suppose that I have a fibration $F \to E \to B$ and I know that its ...
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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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24 views

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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Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy ...
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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 ...
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Calculation with Leray spectral sequence

The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$ for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to ...
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86 views

Calculate the cohomology group of $U(n)$ by spectral sequence.

Here $U(n)$ is the unitary group, consisting of all matrix $A \in M_n (\mathbb{C})$ such that $AA^*=I$ Problem How to calculate the integer cohomology group $H^*(U(n))$ of $U(n)$? What if $O(n)$ ...
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67 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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1answer
55 views

Computation of Ext as a cohomologies of certain complex

Let $R$ be a ring and $K^\bullet$ be a complex of $R$-modules such that $K^\bullet$ has only one nontrivial cohomology $H^0(K^\bullet)=M$. Suppose that $R$-module $N$ is such that ...
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1answer
66 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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170 views

Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is ...
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146 views

Group cohomology of dihedral groups

If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by $$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n ...
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Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title ...
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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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145 views

What is spectral sequence?

Can anyone explain me what is spectral sequence? What is the motivation behind this? Is it a natural tool? Why should we study spectral sequences? Pardon me for asking too many question.Actually I ...
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463 views

Long exact sequence for a triple follows from long exact sequence for a pair?

In homology theory, the long exact sequence for a pair $(X,Y)$ is just $H(Y)\to H(X)\xrightarrow{\partial(X,Y)}H(X,Y)\to H(Y)[-1]$. The long exact sequence for a triple $(X,Y,Z)$ is $H(Y,Z)\to ...
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218 views

Vector space identity from Chow's “You Could Have Invented Spectral Sequences”

In Chow's You Could Have Invented Spectral Sequences (3rd page, left column) appears the following isomorphism of vector spaces: $$\frac{Z_d}{B_d}\cong \frac{Z_d+C_{d,1}}{B_d+C_{d,1}}\oplus ...
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100 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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63 views

A form of Künneth formula?

Problem (Exercise 15.12 in Bott & Tu's Differential Forms in Algebraic Topology) If $X$ is a space having a good cover, e.g., a triangularizable space, and $Y$ is any topological space, prove ...
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120 views

A seemingly wrong definition of convergence of spectral sequences in Bott & Tu?

After introducing exact couples, Bott & Tu defines spectral sequences as follows: A sequence of differential groups $\{E_r,d_r\}$ in which each $E_r$ is the homology of its predecessor ...
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66 views

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 ...
4
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1answer
105 views

Does every elliptic cohomology theory represent a complex-orientable $E_\infty$-ring spectra and vice-versa?

The last paragraph in Two-Vector Bundles and Forms of Elliptic Cohomology remarks that neither the spectrum $K(ku)$ nor tmf is complex orientable. In the case of $K(ku)$: "...the unit map for $K(ku)$ ...
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150 views

Homotopy of double chain complexes

Consider complexes $(A,d_1), (A',d_1)$, $(C,d_2), (C',d_2)$ and morphisms $f_1,f_2: (A,d_1)\to (A',d_1)$ and $g_1,g_2: (C,d_2)\to (C',d_2)$ of degrees $0$. Consider the functor $(-\otimes-)$, then ...