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

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Question about spectrum versus spectral sequences

What is the difference between spectrum sequences and spectral sequences? Are they considered to be the same? I know that the spectrum sequence of a real number $\alpha$ is the sequence that has ...
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Module structure in the Serre spectral sequence of the Borel construction

Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its ...
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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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Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
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Exact sequence from Serre spectral sequence

let me say first that I don't know homological algebra very well, so I apologize in advance if my question is stupid.. It regards the Serre spectral sequence associated to a fibration $0\rightarrow F ...
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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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extension problem of a spectral sequence

From Hatcher's SSAT, If the coefficient group $G$ is a field, then $H_n(X;G)$ is the direct sum $\oplus_p E^\infty_{p,n-p}$ of the terms along the $n^\text{th}$ diagonal of the $E^\infty$ page. ...
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The product of $E_2$-degenerate spectral sequences also $E_2$-degenerates?

Assume the Leray spectral sequence of a map $f_i:X_i\rightarrow B_i$ $E_2$-degenerates for $i=1,2$. Is it true that the Leray spectral sequence of the map $f_1\times f_2:X_1 \times X_2 \rightarrow B_1 ...
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what is the natural map from BP to $HF_p$?

what is the natural map from BP to $HF_p$? BP is the Brown-Peterson spectrum and $HF_p$ is Eilenberg Maclane spectrum. I am trying to learn the connections between ASS and ANSS. This map should ...
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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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Derived functor vs. spectral sequence

I heard many times that because of introducing derived category, we can avoid cumbersome spectral sequence. However, I don't quite understand its meaning. Here is a precise example people talking ...
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Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
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What is the topology of $E^*(E)$ where $E$ is a ring spectrum?

In Adams' lectures on generalised cohomology Page 51, it states $E^*(E)$ is a topology ring. I do not know the topology on it. the reference that Adams offered there is by Novikov in Russian. Can ...
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why $HF_p$(Eilenberg Mac Lane spectrum) smash X (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$?

why $HF_p$(Eilenberg Mac Lane spectrum) smash $X$ (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$? This is Prop 2.1.2 (g) in Ravenel's green book. Can ...
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37 views

Examples of spectral sequences

Where can I find geometric examples of spectral sequence calculation? For instance, calculation of the spectral sequences of a filtration on torus induces by an enough nice Morse function is quite a ...
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54 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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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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The differentials of a spectral sequence

Suppose we are on the $E_r$ page and the lattice either consists of 0 or $\mathbb{Q}[x,y]$ in each entry. Suppose in particular that the points $(p,q)$ and $(r, s)$ (and "their codomains") are equal ...
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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 ...
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The Poincare series for a bigraded vector space

I don't understand this computation (this is from McCleary's book on spectral sequences, p.15): The Poincare series of a (locally finite) bigraded vector space $E^{\ast,\ast}$ is defined as ...
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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 ...
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Are there spectral sequences for calculating homology or cohomology of homotopy (co)limits?

Suppose my nice topological space $X$ is the homotopy colimit $$\operatorname{hocolim}D\cong X$$ of a diagram $D\colon I\to \mathbf{Top}$ and the homotopy limit $$\operatorname{holim}E\cong X$$ of a ...
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Exact sequences and spectral sequences

We have the well-known theorem for cohomological spectral sequences as follows: Theorem: Let $(E_r , d_r )$ be a third quadrant spectral sequence and let $E^{p,q}_2‎\Rightarrow‎ H^n(Tot(M)$. a) If ...
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Spectral sequences to involve together two ideals of a ring

I'm looking for spectral sequences to involve together two ideals of a ring. For instance, let $I,J$ be two ideals of Noetherian ring $R$ and $M$ be a finite $R$-module then we have the following ...
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Example to show that a graded algebra with a stable filtration is not determined by its associated bigraded algebra

I'm reading McCleary's book on spectral sequences, and his Example 1.J (reproduced below) seems to be incorrect. I'm looking for two graded algebras $H_1^*$ and $H_2^*$ with the following properties: ...
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Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
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57 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 ...
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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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68 views

Question About Transgression

I have been working on this question here. Here is the setup: First, all cohomology groups are assume to be with $\mathbb{Q}$ coefficients. We assume that $H^*(K(\mathbb{Q},n))=\mathbb{Q}[x]$, with ...
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Spectral sequences from Cartan-Eilenberg systems

This is an exercise from Mcleary's book on Spectral sequence which I have been stuck with for some time. Let us recall what a Cartan-Eilenberg system is: IT consists of a module $H(p,q)$ for each pair ...
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filtration on the cohomology of a complex

Let $K^\bullet$ be a complex and let $F_I$ and $F_{II}$ be two filtrations on it. suppose $F_I^i K^n$ intersects $F_{II}^i K^n$ trivially. It then follows that in the induced filtration on the ...
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Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
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definition of convergence of a spectral sequence

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant and ...
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“Cut-off” of the Adams exact couple in A. Hatcher's “Spectral Sequences in Algebraic Topology”

I have been reading Chapter 2. of A. Hatcher's "Spectral Sequences in Algebraic Topology", which is freely available at the author's website. I have trouble understanding the Adams exact couple, ...
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filtered modules (LNAT, Davis & Kirk)

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written: Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a ...
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How to define the natural map on the second page of a spectral sequence?

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
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107 views

Is there a nice list of spectral sequences that don't come from particular constructions?

When you first learn about rings, it's important to have examples of, say, a PID which is not a Euclidean domain, a UFD which is not a PID, and so forth, to help build intuition and provide test ...
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80 views

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 ...
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Example 1.K in A User's Guide to Spectral Sequences

I'm having trouble with Example 1.K, p.25, of John McCleary's book A User's Guide to Spectral Sequences. Specifically, I don't understand how he defines the "obvious map" in the second ...
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Spectral sequences: equivalence of exact couples and classic (?) method

By the 'classic' method I mean the construction of the spectral sequence associated to a filtration as found in Weibel's book p. 133-134. There is also the method of construction through exact couples ...
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71 views

Kernel of Hurewicz map using the spectral sequence of the universal cover

In Davis & Kirk's Lecture Notes in Algebraic Topology, Exercise 159 reads: Use the spectral sequence of the universal cover to show that for a path-connected space $X$ the sequence ...
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Confusion of grading in spectral sequences

From Bott&Tu, Differential Forms in Algebraic Topology, Page 160. Bott&Tu asserted that considering a "graded filtered complex" $K=\oplus K^{n}$, with a filtration given by $K\supset ...
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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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Homology of the fiber of a fibration

I was wondering whether the following conjecture is true and, if so, how one would proof this. All spaces are assumed to be pointed spaces but we drop the base point from notation. Conjecture: ...
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Generating morphisms of spectral sequences

When we define spectral sequnces (as Weibel's book) for example in the abelian category $R$-mod, they are a collection of objects $E_{pq}^r$ for $p,q$ and $r\geq a$ integers with a collection of ...
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Examples of exact couples of abelian groups

Exact couples are really important when defining spectral sequences. However, I have never really seen a simple non-trivial example of two exact couples of abelian group with a morphism between them. ...
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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 ...
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342 views

Hopefully an easy question on spectral sequences

I'm trying to understand Proposition 4.3 (page 562) in S. Morita's article Characteristic Classes of Surface Bundles, which can be found on Andy Putman's website here. I don't think that my question ...
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Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions

I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$ $0 ...
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A spectral sequence for Tor

Suppose $R \to T$ is a ring map such that $T$ is flat as an $R$-module. Then for $A$ an $R$-module, $C$ a $T$-module there is an isomorphism $$\text{Tor}^R_n(A,C) \simeq \text{Tor}^T_n(A \otimes_R ...