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.
4
votes
0answers
51 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 ...
3
votes
1answer
88 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 ...
0
votes
1answer
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 ...
1
vote
0answers
24 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 ...
6
votes
0answers
52 views
induced map in homology on a fiber bundle
Let $F \rightarrow E \rightarrow B$ be a fiber bundle of compact connected smooth manifolds and $B$ simply connected. Suppose that there is a map
$f: E \rightarrow E$ that covers a map $g: B \...
3
votes
0answers
170 views
Spectral sequence $B^1_{p,q}= B^\infty_{p,q}$ definition, nlab
In an introductory notes to spectral sequences, nlab, Definition 1.26 we define
$$B^r_{p,q} = \partial(F_{p+r-1} C_{p+q+1})$$
and
$$B^\infty_{p,q}= \partial(F_{p}C_{p+q+1})$$
So what is the ...
0
votes
0answers
15 views
Restrict differential forms that vanish along fibres to the base
I am reading Jean-Luc Brylinski's book on loop space. At the end of section 1.6 Leray Spectral Sequence, he claims without proof that
Let $f: Y\to X$ be a smooth bundle of paracompact manifolds. A $...
2
votes
0answers
52 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 ...
0
votes
3answers
61 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 $S^1\longrightarrow S^5\longrightarrow CP^2$. ...
1
vote
0answers
30 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 ...
2
votes
0answers
39 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 $E_2^{p,q} = H^p(BG;H^q(X))$ ...
2
votes
1answer
61 views
A spectral sequence with only one index in Atiyah's paper?
I would like to read Atiyah's paper Characters and cohomology of finite groups; but when I started reading the introduction, Atiyah mentions that he will prove that there is a "spectral sequence $\{E^...
2
votes
0answers
26 views
Why does spectral sequence $E^\infty$ need AB4*
So in Weibel, he states
Warning: In an unbounded spectral sequence, we will tacitly assume that $B^{\infty}$,
$Z^{\infty}$, and $E^{\infty}$ exist! The reader who is willing to only work in the ...
0
votes
0answers
8 views
Bartels Test for Cycle Significance for uneven samples
The Cycles Institute has published an article on Bartels Test for cycle significance. The article refers to data from an even sample space (12 segments of 41 months each). Bartels Test of Cycle ...
0
votes
0answers
48 views
Cohomology Leray-Serre spectral sequence in case fiber is not connected?
Consider a fibration $F \to E \to B $, where $B $ is path connected. My question is can we use the cohomology Leray-Serre spectral sequence in case fiber is not connected? For example, when fiber is ...
2
votes
0answers
34 views
Computing maps in Leray spectral sequence
Let $f:X_{s1} \to X_{s2}$ be morphism of sites ( Here $X$ is some scheme $X_{s1}$ refers to the site on $X$).
Now using the Leray spectral sequence one gets the following exact sequence
$0 \to H^1(...
0
votes
0answers
17 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 ...
0
votes
0answers
43 views
Ring isomorphism in Leray-Serre spectral sequence
The Leray-Hirsch theorem: let $k$ be a field. Given a fibration $F \to E \to B $ with $F, B$ path connected and suppose system of local coefficient is zero and the following condition satisfied (a) $...
3
votes
1answer
63 views
Homology of $K(\mathbb{Z}/n,1)$ via Fibrations.
I would like to calculate the $\mathbb{Z}$-Homology of $K(\mathbb{Z}/n,1)$ via the Fibration
$$K(\mathbb{Z},1)\hookrightarrow K(\mathbb{Z},1)\twoheadrightarrow K(\mathbb{Z}/n,1)$$
and the Serre-...
0
votes
1answer
27 views
spectral sequence example diagonal map confusion
I'm attempting to wrap my head around spectral sequences, so constructed a really basic example to apply the definitions and go through the motions. My filtered chain complexes are:
$F_2C_*: 0 \...
2
votes
1answer
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 ...
0
votes
0answers
13 views
Wang exact sequence with base space homology sphere
Let $F\rightarrow E\rightarrow S^n$, $n\geq 2$, be a fibration. Then we have the Wang exact sequence,
$$
\cdots\rightarrow H_q(F)\rightarrow H_q(E)\rightarrow H_{q-n}(F)\rightarrow H_{q-1}(F)\...
6
votes
0answers
143 views
Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?
Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...
1
vote
0answers
44 views
Understanding why the Adams Spectral Sequence works
I am trying to learn about the Adams Spectral Sequence and my question is basically summed up in the title.
More precisely, let $X$, $Y$, and $E$ be spectra. We have a homomorphism $[X,Y] \to Hom_{E^...
1
vote
1answer
141 views
Homotopy groups $\pi_{n+1}(S^{n})$
In section $5.1$ of Hatcher's note about spectral sequences, he starts to compute stable homotopy group $\pi_{n+k}(S^{n}),k \leq 3$. Particularly for $\pi_{n+1}(S^{n})$, by Freudenthal suspension ...
2
votes
1answer
155 views
Wang Sequence for the circle $S^1$
Let $F\stackrel i \to E\stackrel \pi\to S^1$ a fiber bundle over the circle $S^1$. There is a long exact sequence sequence in cohomology, called Wang:
$$\dots\to H^k(E)\stackrel {i^*}\to H^k(F)\...
5
votes
0answers
66 views
Cohomology groups $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$
I'm trying to compute $H^*(B(\mathbb{Z}_2\ltimes PSU(4)),\mathbb{Z}_2)$ for $*\le3$.
Here the semi-direct product is given by the outer automorphism of $PSU(4)$.
By Serre spectral sequence, we have
$...
3
votes
0answers
71 views
Structure of module over Eilenberg MacLane spectrum
Let $HR$ be the Eilenberg-Maclane spectrum for a commutative ring $R$ and $M$ be a module over $HR.$ Then I want to prove that $M$ is a product of Eilenberg-Mac Lane spectra.
Construction: Let $\...
1
vote
0answers
50 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 ...
3
votes
1answer
97 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-...
3
votes
2answers
84 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 ...
0
votes
0answers
51 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 ...
1
vote
0answers
82 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 ...
1
vote
0answers
48 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 ...
1
vote
0answers
23 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 ...
2
votes
0answers
34 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 ...
0
votes
0answers
45 views
functoriality of Grothendieck spectral sequence
I am looking for a reference which treats the functoriality of the Grothendieck spectral sequence for elements of the derived category of an abelian category.
1
vote
1answer
45 views
What's the meaning of “decreasing filtration”?
Page 152-153, "Algebraic Geometry" by Lei Fu.
The condition (e) of the definition of spectral sequence is listed as follows:
(e) A family of objects $H^n(n\in \Bbb Z)$ in $\mathcal C$ and each $H^...
1
vote
0answers
43 views
Group actions on spectral sequences of group cohomology
Suppose I have a group extension $1 \rightarrow N \rightarrow H\rightarrow K\rightarrow 1$, and we have a group $G$ which acts on $H$, and $K$ by automorphisms and it does not have action on $N$. ...
3
votes
1answer
91 views
Product structure in Cohomology Spectral sequence
In the Serre Spectral sequence, we know, the cup product structure induces a canonical product in all $E_{r}$ pages which is compatible with respect to the differential. I am trying to understand ...
2
votes
0answers
34 views
What is the filtration in Leray's spectral sequence?
Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration
$$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
0
votes
0answers
68 views
What does it mean that if the cover of $k$-fold intersections is not contractible it takes the form of a spectral sequence of cohomology?
I asked the following question in a previous post: Suppose a CW complex $M$ is given by the union of $n$-spheres, namely $M=\bigcup_{\alpha\in A}S^n$, without knowledge of intersections. The only ...
1
vote
0answers
115 views
Deligne's theorem on the Leray spectral sequence and weights
Motivation : If $f : X \to Y$ is a smooth projective map between algebraic varieties, then there is a theorem by Deligne which says that the Leray spectral sequence degenerates at $E_2$.
The proof I ...
1
vote
0answers
54 views
Reference for the construction of the Leray spectral sequence from filtration
So far I have found only two possible constructions of the Leray spectral sequence for a continuous map $f : X \to Y$ between topological spaces with CW-complex structures: one through Cech complexes, ...
0
votes
0answers
26 views
On the limit of a directed system of spectral sequences
Suppose that we have a directed system $(E_N^{pq}, f_{N,N'})_{N,N' \in \mathbb{N}}$ of spectral sequences, and that, moreover, for any $N$, the spectral sequence $E_N^{**}$ collapses at its $E_2$-page
...
4
votes
1answer
72 views
Understanding Adams spectral sequence and Pontryagin-Thom isomorphism intuitively
The question is about understanding Adams spectral sequence intuitively and some of the meanings of its relations.
In Adams spectral sequence,
$$E_2^{s,t}=\text{Ext}_{\mathcal{A}}^{s,t}(H^*(MTG), \...
3
votes
0answers
65 views
A question related to group cohomology and spectral sequences
It is actually a follow-up question of a mathoverflow question. I don't quite understand the answer there.
I tried to compute the group cohomology of $H^n(\mathbb{Z}_4,\mathbb{Z})$ via the Lyndon-...
2
votes
1answer
50 views
$\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$?
From the computation of some lower dimension $N$ of $Sp(N)$ group,
we see that the homotopy groups are:
$\pi_3(Sp(N))=\mathbb{Z}$,
$\pi_4(Sp(N))=\mathbb{Z}_2$,
$\pi_5(Sp(N))=\mathbb{Z}_2$,
at ...
0
votes
1answer
25 views
Truncating columns of a double complex to get a filtration of the homology of the total complex
My question refers to the document found here. Specifically page 394 of the book (page 14 of the pdf). Theorem 2.5 on that page refers to "the filtration of $H_{m}(Tot)$ obtained by truncating columns ...
4
votes
0answers
34 views
Differentials on the second page of the spectral sequence of a first quadrant double complex
Suppose we have some (homological) double complex $\{E_{pq}\}$ with $p$ labelling the row and $q$ the column (is this standard or not?). Taking the homology of the vertical maps, it's easy enough to ...
