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.

Filter by
Sorted by
Tagged with
4
votes
0answers
307 views

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 ...
7
votes
1answer
229 views

“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, ...
3
votes
1answer
118 views

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 ...
12
votes
1answer
878 views

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 T,...
7
votes
1answer
296 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 cases....
2
votes
1answer
194 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 $F_k:=\mathrm{Im}\,(...
3
votes
1answer
154 views

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 paragraph : ...
4
votes
1answer
193 views

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 ...
2
votes
1answer
98 views

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 ...
9
votes
2answers
710 views

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: ...
4
votes
0answers
107 views

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. ...
6
votes
2answers
985 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 ...
6
votes
0answers
394 views

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 \...
4
votes
1answer
1k views

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 (p,q\in\...
11
votes
1answer
195 views

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 ...
6
votes
2answers
241 views

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 ...
11
votes
3answers
3k views

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 ...
10
votes
1answer
862 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 ...
2
votes
1answer
1k views

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 $...