The spectral-sequences tag has no wiki summary.
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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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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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Complex expression for periodic binary sequences
We have infinite binary sequences of type
$$\langle g_n \rangle_{j=4}=\{0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,...\} \,;\, n\to\infty$$
where $j$ indicates the length of a period that starts/ends with a $1$; ...
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Trivial question in the spectral sequence
From Bott&Tu, Differential Forms in Algebraic Topology, from bottom of page 161 to 162.
Bott&Tu claimed that since $b$ represents an element of $E_{1}=H_{d}(K)$, we have $db=0,Db=\delta b$. ...
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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
$$0 \rightarrow ...
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Hopefully an easy question on spectral sequences
I'm trying to understand Proposition 4.3 (page 562) in S. Morita's article Characteristic Classes of Surface Bundles, which can be found on Andy Putman's website here. I don't think that my question ...
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Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions
I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$
$0 ...
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A spectral sequence for Tor
Suppose $R \to T$ is a ring map such that $T$ is flat as an $R$-module. Then for $R$ an $A$-module, $C$ a $T$-module there is an isomorphism
$$\text{Tor}^R_n(A,C) \simeq \text{Tor}^T_n(A \otimes_R ...
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Spectral Sequence and Stiefel Manifold
Let $Spin(3)$ be embedded in $Spin(5)$ by the spin embedding then we have a fibration:
$$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3)$$
Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel ...
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Confused about Hypercohomology terminology and meaning
check this: Given a sheaf complex $F^\bullet$, let's say I want to compute the hypercohomology of this complex, if we consider the bicomplex of sheaves
$C^\bullet(F^\bullet) = (C^p(F^q))\quad ...
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An Exercise in Allen Hatcher's book on Spectral Sequences
anyone knows how to solve Exercise 3 of Chapter 1 of Allen Hatcher's book on Spectral Sequences? The question is as follows:
For a fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$ associated to ...
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Adams spectral sequence for computing 3-torsion in $\pi_*(S)$
A novice to the Adams spectral sequence, I am attempting to follow a computation in McCleary's book in the mod 3 Adams spectral sequence for $\pi_*(S)$. By working out part of a minimal resolution of ...
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General convergence criterion for a spectral sequence associated to a filtration of simplicial groups
Let $\cdots\subset F_r \subset \cdots \subset F_1 \subset F_0=G$ be a filtration of simplicial groups. If $\bigcap F_r$ is trivial, does the associated spectral sequence converge? If so, what does ...
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cohomology ring structure of conf($\mathbb{R}^m$, 3)
I am attempting to compute the (integral) cohomology ring structure of the 3 configuration space of $\mathbb{R}^m$ and have run into a few doubts.
Using a result of Fadell and Neuwirth, we have that ...
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274 views
The Atiyah Hirzebruch Spectral Sequence
I am learning about the AHSS (for complex K-theory) by trying to compute the K-theory of some spaces. I have heard that the AHSS is functorial (maps of spaces induce maps of spectral sequences). Is ...
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1answer
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Serre's Exact Sequence in Homology
I am trying to derive the following result of Serre's:
Let $F \hookrightarrow E \stackrel{p}{\to} B$ be a fibration with $B$ simply connected. Suppose $H_i(B)=0$ for $0 < i < p $ and that ...
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Spectral Sequence proof of the five lemma
The five lemma is an extremely useful result in algebraic topology and homological algebra (and maybe elsewhere). The proof is not hard - it is essentially a diagram chase.
Exercise 1.1 in McCleary's ...
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Serre Spectral Sequence and Fundamental Group Action on Homology
I am looking at my algebraic topology notes right now, and I am looking at our definition for the Serre Spectral Sequence and it requires that the action of the fundamental group of the base space of ...
