Spectral sequences compute homology groups by taking a sequence of approximations, and generalise exact sequences. They find application in algebraic topology.
Questions tagged [spectral-sequences]
464 questions
23
votes
0 answers
Bousfield–Kan spectral sequence for homotopy colimits
Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ following Bousfield and Kan; more precisely,…
Zhen Lin
- 97,105
22
votes
4 answers
Useful fibrations
What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to S^{2n+1}\to \mathbb{C}\textrm{P}^n$. Anything else…
asdf
- 745
20
votes
1 answer
Group cohomology of dihedral groups
If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by
$$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n > 0 \\ \mathbb{Z}/2 & n \equiv 2 \bmod 4 \\ 0 & n \text{…
Martin Brandenburg
- 181,922
18
votes
1 answer
Computing cohomology of Cech-De Rham Complex
Bott & Tu use what they call the "Cech-de Rham complex" a lot, which is a double complex that uses the Cech differential horizontally and the de Rham differential vertically, with cochains being the algebras of differential forms on finite…
yakmacker
- 183
15
votes
3 answers
Spectral Sequence proof of the five lemma
The five lemma is an extremely useful result in algebraic topology and homological algebra (and maybe elsewhere). The proof is not hard - it is essentially a diagram chase.
Exercise 1.1 in McCleary's "Users Guide to Spectral Sequences" has the…
Juan S
- 10,526
15
votes
1 answer
Vector space identity from Chow's "You Could Have Invented Spectral Sequences"
In Chow's You Could Have Invented Spectral Sequences (3rd page, left column) appears the following isomorphism of vector spaces:
$$\frac{Z_d}{B_d}\cong \frac{Z_d+C_{d,1}}{B_d+C_{d,1}}\oplus \frac{Z_d\cap C_{d,1}}{B_d\cap C_{d,1}}$$
The context is a…
user153312
14
votes
2 answers
Serre Spectral Sequence and Fundamental Group Action on Homology
I am looking at my algebraic topology notes right now, and I am looking at our definition for the Serre Spectral Sequence and it requires that the action of the fundamental group of the base space of a fibration $F\to E\to B$ be trivial on all…
Jonathan Beardsley
- 4,658
13
votes
1 answer
Are there spectral sequences for calculating homology or cohomology of homotopy (co)limits?
Suppose my nice topological space $X$ is the homotopy colimit
$$\operatorname{hocolim}D\cong X$$
of a diagram $D\colon I\to \mathbf{Top}$ and the homotopy limit
$$\operatorname{holim}E\cong X$$
of a diagram $E\colon J\to \mathbf{Top}$.
Are there…
howard
- 137
13
votes
1 answer
Calculation with Leray spectral sequence
The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$
for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to Y$. Is there an example of a concrete calculation…
Martin Brandenburg
- 181,922
12
votes
1 answer
A spectral sequence for Tor
Suppose $R \to T$ is a ring map such that $T$ is flat as an $R$-module. Then for $A$ an $R$-module, $C$ a $T$-module there is an isomorphism
$$\text{Tor}^R_n(A,C) \simeq \text{Tor}^T_n(A \otimes_R T,C)$$
This can be proved directly by choosing an…
Juan S
- 10,526
11
votes
1 answer
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 $\mathbb{Z}/3\mathbb{Z}$ over the mod 3 Steenrod…
Vitaly Lorman
- 1,850
11
votes
1 answer
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 there a nice example of spaces $X$, $Y$ and a map…
Vitaly Lorman
- 1,850
11
votes
4 answers
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 results in homological algebra.
More…
Maxime Ramzi
- 45,086
11
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1 answer
Why are spectral sequences called "spectral"?
Why are spectral sequences called "spectral"? Is that use of "spectral" related to other uses in math, such as spectra in homotopy theory, the spectrum of a ring in algebraic geometry or the spectrum of an operator? Why are those things called…
Omar Antolín-Camarena
- 5,654
11
votes
1 answer
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 $O(n)$ replace $U(n)$?
My primitive idea is that: as for…
Hang
- 2,982