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
33 views

Can there be nonisomorphic functorial Serre spectral sequences?

The Serre spectral sequence is a very useful tool in algebraic topology, but as often with these beasts, the differentials can be hard to compute. Of course, the proof that this sequence exists is ...
31 views

Chapter 1 in A User's Guide to Spectral Sequences, reconstruction

In Chapter 1 of John McCleary's book A User's Guide to Spectral Sequences the condition assumed for the filtrations is that they are bounded below. In pg. 4 it is said that $H^*$ (if good enough) can ...
62 views

Serre classes and the Serre spectral sequence

Let $C$ be a Serre class which satisfies the additional axioms about $\otimes, \mathrm{Tor}, K(A,1)$'s. It is then easy to check that if $F\to X\to B$ is a Serre fibration with $\pi_1(B)$ acting ...
4 views

Convention for the grading in spectral sequences

Usually spectral sequences are defined together with a grading on it, the differential $d_r$ on the page $r$ taking $E_r^{p,q}$ to $E_r^{p-r, q+r-1}$. This convention on the grading on the ...
35 views

What's wrong with this argument that the Atiyah-Hirzebruch spectral sequence always degenerates?

Let $E$ be a spectrum and let $X$ be a space or connective spectrum. Then the cohomological Atiyah-Hirzebruch spectral sequence is of the form: $H^s(X,E^t) \Rightarrow E^{s+t}(X)$ This is a half-...
60 views

Proof of existence of spectral sequence.

I'm trying to understand the proof of existence of spectral sequence from Ch5 of Allen Hatcher's Spectral Sequence notes. While constructing the iso $\tilde{\Phi_*}$ in the diagram it's written that: ...
52 views

Serre Spectral sequence [Hatcher]

I'm having trouble understanding Serre Spectral sequence given in Hatcher. https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf In the beginning of the section, (Page 526 of the book, or rather, Pg 9 of ...
94 views

Serre Spectral Sequence I

I am trying to follow the proof in Kochman's Introduction to Stable Homotopy Theory, page 59. This will be first part of a series of post. (Serre Spectral Sequence) Let $R$ be a commutative ring ...
42 views

Prerequisite of understanding a topological construction in Spectral Sequences

This is a post on the construction of a spectral sequence. I am in fact lost in the first paragraph. Let $B$ be a CW complex and $\pi\colon X\to B$ a Serre fibration. Put $X^k=\pi^{-1}(B^k)$. A ...
30 views

Spectral sequence of filtered complex.

I am trying to understand the construction of a spectral sequence of a filtered complex. After reading through the entry in the nLab I came up with an example, that I don't understand: Consider the ...
57 views

62 views

Cohomology of spectra as free modules over Steenrod Algebra

I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement". Let $X$ be a connective CW-spectrum of finite ...
69 views

First and third homology of $S^5/Z_q$ and Leray spectral sequence

I read from an article that the space $X=S^5/Z_q$ is not a Lens space because the orbifold action is not compatible with the action of the Hopf fibration $S^1\longrightarrow S^5\longrightarrow CP^2$. ...
32 views

Tate construction of the Anderson dual of the sphere spectrum

I am trying to understand example I.2.3 of the article "On Topological Cyclic Homology", i.e. im trying to see the necesity of the bounded below assumption in the Tate Orbit Lemma (Lemma I.2.1) by ...
48 views

Group cohomology and singular cohomology of classifying space

Let $G$ be a finite group, and denote by $BG = K(G,1)$ the classifying space. For any fibration $X \rightarrow E \xrightarrow{\pi} BG$, the Serre spectral sequence $E_2^{p,q} = H^p(BG;H^q(X))$ ...
62 views

18 views

Question on a proof about spectral sequences from exact couples

I am going through Proposition 2.9 in User's guide in spectral sequences (2nd edition) by McCleary. This is a proof on defining spectral sequences using the language of exact couples. Towards the end ...
44 views

24 views

A different approach towards spectral sequences.

I am somewhat acquainted with spectral sequences as in Weibel's book(the usual definition with many indices and pages), but I have found a different approach in the Stacks project.Link Although I can ...
23 views

67 views

53 views

degeneracy of the Serre spectral sequence

The following are well-known facts on the Serre spectral sequence For a fibration $F \rightarrow E \rightarrow B$ we have the Serre spectral sequence (in cohomology with a coefficients in a field ...
105 views

Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence

For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
85 views

Spectral sequences on their own right

I have begun to read a bit about Spectral Sequences. I have been studying some topics of algebraic topology and this "tool" came up from time to time. I didn't need to go deep, but I felt I had to ...
59 views

spectral sequence and a comparison theorem

I am wondering how the comparison theorem can be useful for resolving extension problem. Here is a quote from the book An Introduction to Homological Algebra by Weibel. (Weibel) Comparison Theorem ...
90 views

Serre spectral sequence of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2=\mathbb{R}P^\infty$

As the title shows, we have a fibration of $\mathbb{Z}_2\rightarrow E\mathbb{Z}_2\rightarrow B\mathbb{Z}_2\sim\mathbb{R}P^\infty$. I am trying to check my understanding of Serre spectral sequence with ...
55 views

Why intermediate map in Gysin sequence is multiplication by Euler class

I am reading Bott and Tu, Differential Form in Algebraic Topology. At page 178, they constructed Gysin sequence of Sphere Bundle. I am having trouble understanding the argument, $d_{k+1}$ is ...
27 views

Functoriality of Lyndon-Hochschild-Serre spectral sequences in coefficients.

It is a question about group cohomology. Supposing that I have a short exact sequence of $G$-modules $1\rightarrow A_1 \rightarrow A_2\rightarrow A_3\rightarrow 1$, I know that there will be a long ...
35 views

Thom spectrum $MSpin$ and $E_2$-page for a large degree $i\geq 8$

Let the $MTH$ is Madsen-Tillman spectrum (which is a close cousin of the more usual Thom spectrum $MH$) associated to tangential structure $H$. For a computation involving no odd torsion, the Adams ...