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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Bott and Tu Proposition 12.1

Question about the proof of Bott and Tu's Proposition 12.1: Given any double complex $K$, if $H_\delta H_d(K)$ has entries only in one row, then $H_\delta H_d$ is isomorphic to $H_D$. Remark: $\delta$...
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Bott and Tu Spectral Sequence of a Double Complex

This page has confused me a great deal, I have a few questions: Is $A^k$ the same thing as $C^k = \oplus_{p + q = n} K^{p,q}$ as defined in example 14.2 or is it different because $A = \oplus K_p$ ...
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Different definitions of the $E^{\infty}$ page of a spectral sequence.

In Lecture Notes in Algebraic Topology by Davis and Kirk it seems that given a (convergent, bigraded) spectral sequence $(E^r, d^r)$ one defines the $E^{\infty}$-page as $$E^r_{p, q} = \operatorname{...
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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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Does the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence go under a different name?

This is more of a naming convention question than anything else. I am reading up on spectral sequences mainly from the book Lecture Notes in Algebraic Topology by Davis and Kirk. Now in this book the ...
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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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Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence

Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$. In the beginning he states that there exisits ...
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Why does spectral sequence $E^\infty$ need AB4*

So in Weibel, he states Warning: In an unbounded spectral sequence, we will tacitly assume that $B^{\infty}$, $Z^{\infty}$, and $E^{\infty}$ exist! The reader who is willing to only work in the ...
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Isomorphism between Dolbeault Cohomology groups

Let $M^n$ be a compact Kahler manifold and consider the product complex manifold $M \times \mathbb{C}^m$. By the Leray spectral sequence associated to the projection $\pi_{\mathbb{C}^m} : \mathbb{C}^m ...
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Derived proofs of elementary homological algebra theorems?

I know basically the definition and very general-not-so-useful properties of derived categories, and to build a deeper understanding of them, I'd like to see if it can help in re-thinking some basic ...
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differential in AHSS for spin cobordism

According to these solutions, the differential $d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})$ is the dual of $Sq^2$. Why? This MO post asks a similar question (but about $d_3$ in ...
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Colimit of a sequence of objects in a category

In the book "Lecture notes in algebraic topology" by Davis and Kirk I came across the following colimit $$E^{\infty} = \operatorname{colim}_{r \to \infty}E^r$$ where each of the $E^r$ were $R$-...
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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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Can there be nonisomorphic functorial Serre spectral sequences?

The Serre spectral sequence is a very useful tool in algebraic topology, but as often with these beasts, the differentials can be hard to compute. Of course, the proof that this sequence exists is ...
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Chapter 1 in A User's Guide to Spectral Sequences, reconstruction

In Chapter 1 of John McCleary's book A User's Guide to Spectral Sequences the condition assumed for the filtrations is that they are bounded below. In pg. 4 it is said that $H^*$ (if good enough) can ...
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Serre classes and the Serre spectral sequence

Let $C$ be a Serre class which satisfies the additional axioms about $\otimes, \mathrm{Tor}, K(A,1)$'s. It is then easy to check that if $F\to X\to B$ is a Serre fibration with $\pi_1(B)$ acting ...
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Convention for the grading in spectral sequences

Usually spectral sequences are defined together with a grading on it, the differential $d_r$ on the page $r$ taking $E_r^{p,q}$ to $E_r^{p-r, q+r-1}$. This convention on the grading on the ...
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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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Proof of existence of spectral sequence.

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $\tilde{\Phi_*}$ in the diagram it's written that: ...
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Serre Spectral sequence [Hatcher]

I'm having trouble understanding Serre Spectral sequence given in Hatcher. https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf In the beginning of the section, (Page 526 of the book, or rather, Pg 9 of ...
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Prerequisite of understanding a topological construction in Spectral Sequences

This is a post on the construction of a spectral sequence. I am in fact lost in the first paragraph. Let $B$ be a CW complex and $\pi\colon X\to B$ a Serre fibration. Put $X^k=\pi^{-1}(B^k)$. A ...
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Serre Spectral Sequence I

I am trying to follow the proof in Kochman's Introduction to Stable Homotopy Theory, page 59. This will be first part of a series of post. (Serre Spectral Sequence) Let $R$ be a commutative ring ...
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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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Spectral sequence of filtered complex.

I am trying to understand the construction of a spectral sequence of a filtered complex. After reading through the entry in the nLab I came up with an example, that I don't understand: Consider the ...
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induced map in homology on a fiber bundle

Let $F \rightarrow E \rightarrow B$ be a fiber bundle of compact connected smooth manifolds and $B$ simply connected. Suppose that there is a map $f: E \rightarrow E$ that covers a map $g: B \...
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Spectral sequence $B^1_{p,q}= B^\infty_{p,q}$ definition, nlab

In an introductory notes to spectral sequences, nlab, Definition 1.26 we define $$B^r_{p,q} = \partial(F_{p+r-1} C_{p+q+1})$$ and $$B^\infty_{p,q}= \partial(F_{p}C_{p+q+1})$$ So what is the ...
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Restrict differential forms that vanish along fibres to the base

I am reading Jean-Luc Brylinski's book on loop space. At the end of section 1.6 Leray Spectral Sequence, he claims without proof that Let $f: Y\to X$ be a smooth bundle of paracompact manifolds. A $...
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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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First and third homology of $S^5/Z_q$ and Leray spectral sequence

I read from an article that the space $X=S^5/Z_q$ is not a Lens space because the orbifold action is not compatible with the action of the Hopf fibration $S^1\longrightarrow S^5\longrightarrow CP^2$. ...
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Tate construction of the Anderson dual of the sphere spectrum

I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
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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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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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Bartels Test for Cycle Significance for uneven samples

The Cycles Institute has published an article on Bartels Test for cycle significance. The article refers to data from an even sample space (12 segments of 41 months each). Bartels Test of Cycle ...
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Cohomology Leray-Serre spectral sequence in case fiber is not connected?

Consider a fibration $F \to E \to B $, where $B $ is path connected. My question is can we use the cohomology Leray-Serre spectral sequence in case fiber is not connected? For example, when fiber is ...
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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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Question on a proof about spectral sequences from exact couples

I am going through Proposition 2.9 in User's guide in spectral sequences (2nd edition) by McCleary. This is a proof on defining spectral sequences using the language of exact couples. Towards the end ...
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Ring isomorphism in Leray-Serre spectral sequence

The Leray-Hirsch theorem: let $k$ be a field. Given a fibration $F \to E \to B $ with $F, B$ path connected and suppose system of local coefficient is zero and the following condition satisfied (a) $...
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spectral sequence example diagonal map confusion

I'm attempting to wrap my head around spectral sequences, so constructed a really basic example to apply the definitions and go through the motions. My filtered chain complexes are: $F_2C_*: 0 \...
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Homology of $K(\mathbb{Z}/n,1)$ via Fibrations.

I would like to calculate the $\mathbb{Z}$-Homology of $K(\mathbb{Z}/n,1)$ via the Fibration $$K(\mathbb{Z},1)\hookrightarrow K(\mathbb{Z},1)\twoheadrightarrow K(\mathbb{Z}/n,1)$$ and the Serre-...
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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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Wang exact sequence with base space homology sphere

Let $F\rightarrow E\rightarrow S^n$, $n\geq 2$, be a fibration. Then we have the Wang exact sequence, $$ \cdots\rightarrow H_q(F)\rightarrow H_q(E)\rightarrow H_{q-n}(F)\rightarrow H_{q-1}(F)\...
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Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
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Understanding why the Adams Spectral Sequence works

I am trying to learn about the Adams Spectral Sequence and my question is basically summed up in the title. More precisely, let $X$, $Y$, and $E$ be spectra. We have a homomorphism $[X,Y] \to Hom_{E^...
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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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Homotopy groups $\pi_{n+1}(S^{n})$

In section $5.1$ of Hatcher's note about spectral sequences, he starts to compute stable homotopy group $\pi_{n+k}(S^{n}),k \leq 3$. Particularly for $\pi_{n+1}(S^{n})$, by Freudenthal suspension ...
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Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence

For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
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Cohomology groups $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$

I'm trying to compute $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$. Here the semi-direct product is given by the outer automorphism of $PSU(4)$. By Serre spectral sequence, we have $...
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Structure of module over Eilenberg MacLane spectrum

Let $HR$ be the Eilenberg-Maclane spectrum for a commutative ring $R$ and $M$ be a module over $HR.$ Then I want to prove that $M$ is a product of Eilenberg-Mac Lane spectra. Construction: Let $\...
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degeneracy of the Serre spectral sequence

The following are well-known facts on the Serre spectral sequence For a fibration $F \rightarrow E \rightarrow B$ we have the Serre spectral sequence (in cohomology with a coefficients in a field ...
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Spectral sequences on their own right

I have begun to read a bit about Spectral Sequences. I have been studying some topics of algebraic topology and this "tool" came up from time to time. I didn't need to go deep, but I felt I had to ...