# 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.

265 questions
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### On the limit of a directed system of spectral sequences

Suppose that we have a directed system $(E_N^{pq}, f_{N,N'})_{N,N' \in \mathbb{N}}$ of spectral sequences, and that, moreover, for any $N$, the spectral sequence $E_N^{**}$ collapses at its $E_2$-page ...
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### Maps between two Leray spectral sequences

Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
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### Trivial action of fundamental group in Serre spectral sequence

Recently I'm studying Serre spectral sequence in Hatcher's book. Let $\pi : X \to B$ is a fibration, it's an easy exercise to check that when B is path-connected then all the fibers are homotopy ...
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### Collapse of Serre spectral sequence in the presence of a cross-section

I was under the impression that if a Serre fibration $f: E \rightarrow B$ has a right inverse $s: B \rightarrow E$, then the associated Serre spectral sequence would collapse on the second page. This ...
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### Showing induced action of G by conjugation on Hochschild-Serre $H_i(G/N, H_j(N,M))$ is trivial

Given a group extension: $$0 \rightarrow N \rightarrow G \rightarrow \frac{G}{N} \rightarrow 0$$ I need to show that the induced action of $G$ by conjugation is trivial on the Hochschild-Serre ...
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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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### Hochschild-Serre spectral sequence

I have a hopefully simple question about the Hochschild-Serre spectral sequence (which may just be a simple question about general spectral sequences). Whenever I see the sequence written down, it has ...
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### Explicit formula for Higher Bockstein

The formula for the Bockstein $\beta:H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p\mathbb{Z})$ is $$\beta[c\otimes 1]=[\frac{1}{p}\partial c\otimes 1]$$ (McCleary page 456) How about for ...
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### Times p map in Bockstein (Quite confused)

The following is from McCleary's book pg 461. Q1) I am confused about the map $$-\times p^{r-1}:H_n(X;\mathbb{Z}/p^r\mathbb{Z})\to H_n(X;\mathbb{Z}/p^r\mathbb{Z})$$ Just one page earlier, I saw this ...
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### How to show that $0 \to E_{0,n}^2 \to H_n \to E_{1,n-1}^2 \to 0$ is exact?

Suppose that a spectral sequence converging to $H_\ast$ has $E_{pq}^r = 0$ for all $p\neq 0,1$. Show that there are exact sequences $$0 \to E_{0,n}^2 \to H_n \to E_{1,n-1}^2 \to 0 \,.$$ ...
I have some problems understanding this presentation of the Bockstein Spectral Sequence (McCleary pg 460). Q1) Firstly, how does this short exact sequence of coefficients work? $$0\to\mathbb{Z}/p\... 1answer 150 views ### How does complete knowledge of Bockstein spectral sequences allow complete description of integral homology? In this notes (pg 4): http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf, it is written that "complete knowledge of the Bockstein spectral sequences of C for all primes p allows a complete ... 0answers 18 views ### (Bockstein Spectral Sequence) Why is v_1 also p-divisible? Why if u=pv_1, then v_1 is also p-divisible? I am quite puzzled by the above. I can see that v_1 must also generate a copy of \mathbb{Z}, thus v_1\notin\ker p^r. I can't seem to proceed ... 1answer 28 views ### Bockstein Spectral Sequence (Why is this a direct sum) (from McCleary's User guide pg 459). I am curious why is pH_n(X)+\ker p^r a direct sum? (Or is it?) I tried to reason that \ker p^r is equal to \text{Im}\ \partial:H_{n+1}(X;\mathbb{F}_p)\to ... 0answers 23 views ### Is connected necessary for this theorem on Bockstein spectral sequence? 1) I am curious if the condition that X is a connected space is necessary for the above theorem for Bockstein Spectral Sequence? The proof is from McCleary's User Guide to Spectral Sequence, and I ... 0answers 45 views ### McCleary Spectral Sequence: Is this a definition or a result of a theorem? (McCleary's User Guide to Spectral Sequences pg 458) I refer to the statement "We denote the E^1-term by B_n^1\cong H_n(X;\mathbb{F}_p)". Is B_n^1\cong H_n(X;\mathbb{F}_p) just a definition, ... 1answer 116 views ### How to use spectral sequences? [closed] How to compute and to use spectral sequences ?. I dont know the steps to follow to compute a spectral sequence. Do we need to compute all pages :  E_r  and all spacs  E_r^{pq}  and differentials  ... 0answers 20 views ### Path-connectedness of the fiber ine the LSAH spectral sequence I've read some proofs of the identification of the second term of the Leray-Serre-Atiyah-Hirzebruch spectral sequence, and I don't quite seem to understand where in the proof do we need the fiber of ... 1answer 65 views ### Questions regarding Bockstein Spectral Sequence (McCleary's book) I have some questions regarding Bockstein homomorphism in John McCleary's book (pg 455-456). Q1) Is there a typo, is it supposed to be \bar{u}\in H_n(X;\mathbb{Z}/r\mathbb{Z})? Q2) How do we see ... 0answers 97 views ### Cohomology of genus g surface fibration Let F\to E\to X be a Serre fibration with connected base space X and let the fibre F be an orientable surface of genus g. Is there anything we can say about how H^n(E) depends on H^n(X)? ... 1answer 113 views ### Convergence of spectral sequences Consider a double complex C^{*,*}, where each C^{*,*} are bigraded modules over R (in fact they are vector spaces). Let the horizontal and vertical differentials be d_1,d_2, so that d_1,d_2 ... 1answer 94 views ### Reference for Bockstein Homomorphisms / Spectral Sequence for Homology Is there any book for Bockstein Homomorphisms specialising in the case of homology? So far the books I read (Hatcher, Munkres) discuss it for cohomology. I am aware that it is said to be similar for ... 0answers 92 views ### Leray-Hirsch theorem for Dolbeault cohomology In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this: Theorem (Leray-Hirsch): Let E be a fiber bundle over M... 2answers 229 views ### How can I determine the Steenrod Square Sq^2 for complex projective space? I am trying to learn about Steenrod Squares for algebraic varieties so that I can compute examples of complex topological K-theory using the Atiyah-Hirzebruch Spectral Sequence (AHSS). One of the key ... 1answer 141 views ### What is the integral cohomology of \mathrm{PGL}_n(\mathbb{C}) as a space? The computation of the cohomology of \mathrm{GL}_n(\mathbb{C}) is one of the basic applications of the Serre spectral sequence, using the fiber bundle \mathrm{GL}_{n-1}(\mathbb{C})\to \mathrm{GL}_n(... 1answer 165 views ### Morphisms in the derived category vs. morphisms on cohomology Let A and B be complexes R-modules. Assume that A^\bullet is bounded above and B^\bullet is bounded below. Then there is a convergent spectral sequence$$\prod_{r\in \mathbb{Z}} \...
Consider two coherent sheaves $\mathcal{E},\mathcal{F}$, with locally free resolutions \begin{align} &0 \longrightarrow \mathcal{E}_1 \longrightarrow \cdots \longrightarrow \mathcal{E}_n \...