Questions tagged [spectral-sequences]
The spectral-sequences tag has no usage guidance.
388 questions
1
vote
0
answers
184
views
A naturality question concerning the universal coefficient spectral sequence
I am reading Hillman's book "algebraic invariants of links" and on page 20 he mentions the following universal coefficient spectral sequence.
Let X be a connected finite CW complex.Let $H$ be a ...
- 1,033
3
votes
2
answers
319
views
cohomology algebra of braid spaces, configuration spaces
In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for $p$...
- 4,133
6
votes
1
answer
2k
views
cohomology version of Cartan-Leray spectral sequence that deduces cup product
On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:
If $X$ is a connected pace on which the group $\...
- 40.3k
5
votes
1
answer
366
views
Is the space of real conics with a singular point an orientable manifold?
Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set
of such a polynomial gives a real curve ...
- 3,245
4
votes
0
answers
576
views
generalized Atiyah-Hirzebruch spectral sequence from Postnikov truncation
The Atiyah-Hirzebruch spectral sequence
\begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*}
computes the generalized homology $h$ of a total space $E$ of a Serre fibration ...
- 1,219
1
vote
0
answers
135
views
Spectral sequence and HOM functor
I work with the category $A-{\rm Mod}$ of left modules over a unital ring $A$, but I could ask the same question for any abelian category with enough projectives. Let $M$ and $N$ be two $A$-modules ...
- 6,349
2
votes
0
answers
747
views
What is the abutment filtration of the second spectral sequence of hypercohomology?
I have been recently learning about spectral sequences, following mainly Illusie's notes and EGA, and I am about to write some expository notes, but there are still some points that I was not able to ...
- 142
10
votes
0
answers
813
views
On functoriality of the Leray spectral sequence
The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
- 40.3k
4
votes
2
answers
514
views
stability results for the Atiyah-Hirzebruch spectral sequence
For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
- 103
9
votes
0
answers
517
views
extension problem for the Atiyah-Hirzebruch spectral sequence
For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
- 1,219
7
votes
2
answers
2k
views
Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$
I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them.
It can be shown that $H^d[U(1), Z]$ is $Z$ for ...
- 1,219
7
votes
1
answer
523
views
Hochschild-Serre spectral sequence and non-trivial action on coefficients
Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi_k: a\mapsto k\cdot a$. Define a $G$-action $a\...
- 35.9k
4
votes
1
answer
754
views
spectral sequence with non-trivial action on coefficients
Set-up:
Consider the trivial extension, where $p$ is the projection onto the $\mathbb{Z}_2$ component,$$1\rightarrow N\rightarrow N\times\mathbb{Z}_2\xrightarrow{p}\mathbb{Z}_2\rightarrow 1$$
Define ...
- 3,051
3
votes
0
answers
223
views
spectral sequence differential for cobordism
From page 6 of these solutions:
the differential\begin{equation}d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})\end{equation}connecting the 1-st and the 2-nd row is the $\textbf{...
- 1,219
6
votes
3
answers
546
views
Adams Spectral Sequence for Triangulated Categories
We have the Adams SS with
$$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$
where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.
I was wondering if there is a SS for ...
- 3,184
3
votes
1
answer
317
views
Spectral sequence associated to elliptic fibration degenerates?
Let $\phi:S\rightarrow \mathbb{CP}^1$ be an elliptic fibration of a K3 surface. When is the Leray spectral sequence associated to the fibration $E_2$-degenerate? Are there any good criteria for the $...
- 35.2k
3
votes
1
answer
816
views
When does the filtration in the limit of the Leray spectral sequence split?
Let $\ell$ be a prime, and $k$ a field of characteristic $\ne \ell$. Let $f \colon X \to Y$ be a proper map of smooth projective $k$-varieties. The Leray spectral sequence says
$$
E_{2}^{pq} = H^{p}(\...
- 5,504
5
votes
1
answer
2k
views
Natural morphism appearing in Grothendieck spectral sequence
Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now ...
CommunityBot
- 1
8
votes
1
answer
1k
views
Convergence of spectral sequences of cohomological type
Following the first chapter of Hatcher's great book "Spectral Sequences in Algebraic Topology", I got into problems with spectral sequences of cohomological type. Fix a ring $R$ once and for all. ...
- 6,210
6
votes
2
answers
358
views
Differentials in the Lyndon-Hoschild-Serre Sequence for p=0
I'm interested in whether there is a simple description of the differentials in the first column of the LHS spectral sequence (the column with $E_2^{0,q}=H^0(BK,H^q(BG))=H^q(BG)^K$ for a short exact ...
- 35.9k
17
votes
0
answers
757
views
The spectral sequence of a tower of principal fibrations
Assume we have a tower of fibrations (of simplicial sets, let's say):
$$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$
Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...
- 15.2k
12
votes
2
answers
523
views
A question on some computation of group cohomologies
Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...
CommunityBot
- 1
20
votes
5
answers
3k
views
Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?
One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
- 6,210
3
votes
0
answers
175
views
"Cut-off" of the Adams exact couple
(This question has been asked on Math.StackExchange where it attracted a few upvotes, but - unfortunately - no answer.)
I have been reading Chapter 2. of A. Hatcher's draft of "Spectral Sequences in ...
CommunityBot
- 1
7
votes
0
answers
557
views
Motivic homotopy spectral sequence
I would like to have a question about the re-index convention. Let us consider a spectrum $E$ (I am mainly interested in motivic setting, however let's consider the simplicial case firstly, i.e. $E$ ...
- 326
10
votes
0
answers
494
views
Differentials in the Adams-Novikov spectral sequence and the geometric boundary theorem
$\newcommand\Ext{\mathrm{Ext}}
\newcommand\Z{\mathbb{Z}}
\newcommand\G{\mathbb{G}}$
The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-...
- 3,784
12
votes
1
answer
2k
views
Some calculations with the Adams spectral sequence and the cobar complex
I am trying to 'get my hands dirty', so to speak, with some of the calculations with the Adams spectral sequence in Ravenel's Complex Cobordism book, and I have a few questions (I hope it is OK to ask ...
- 3,784
9
votes
2
answers
1k
views
H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory
Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...
CommunityBot
- 1
4
votes
1
answer
371
views
Hodge classes and Leray filtration
Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers $p$ and $q$ such ...
- 35.2k
0
votes
2
answers
226
views
sequence, such that sum of any combinations in the sequence does not equal another [closed]
Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.
...
CommunityBot
- 1
31
votes
1
answer
2k
views
K(r)-localization and monochromatic layers in the chromatic spectral sequence
While preparing some lecture notes, I had a basic point of confusion come up that I haven't been able to settle.
The $BP$-Adams spectral sequence (or $p$-local Adams-Novikov spectral sequence) for ...
- 6,069
7
votes
1
answer
1k
views
Explicit 2-Cocycles of G=Z2×Z2xZ2 over U(1)
We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\beta(A,B)$ satisfies 2-...
- 10.5k
9
votes
2
answers
1k
views
Khovanov-Rozansky homology and spectral sequences
In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and ...
- 2,273
10
votes
3
answers
2k
views
Serre Spectral Sequence of Representations
Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...
- 6,210
5
votes
0
answers
675
views
Do exact functors commute with spectral sequences ?
Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let
$$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral ...
- 16.2k
2
votes
1
answer
1k
views
Transgression maps in group cohomology and group homology / duality of spectral sequences
I am interested in whether the transgression maps for group cohomology and group homology are related via a version of the universal coefficient theorem.
Let $G$ be a group, $H$ a normal subgroup of $...
- 16.2k
0
votes
1
answer
800
views
Spectral sequence for composition of global sections and tensor product of sheaves
Hi all,
on the forum page http://www.groupsrv.com/science/about506648.html one can read the following (i cut out nonimportant parts):
Question: Does anyone know any condition (non trivial) that ...
- 42.9k
2
votes
2
answers
708
views
Leray spectral sequence of the inclusion of an open subvariety
Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let $U=X-Z$. I'm trying to understand what information do the Leray spectral sequences attached to the ...
CommunityBot
- 1
7
votes
2
answers
966
views
Computation of stable homotopy groups of $RP^2$
I would like to compute the first few stable homotopy groups of $RP^2$.
I first thought to use the Atiyah-Hirzebruch Spectral Sequence, (see Davis & Kirk, pg. 242). Here is what I computed for ...
- 2,023
2
votes
0
answers
478
views
Morphisms of Spectral Sequences and alternating products
Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms $...
CommunityBot
- 1
7
votes
1
answer
1k
views
Do people still use Massey Products for computations in the Adams Spectral Sequence
Hey everyone,
It seems to me like in the literature of the Adams Spectral Sequence, older publications (Toda, May, Tengora+Mahowald) make heavy and explicit use of Massey Products for computations.
...
- 30.4k
5
votes
1
answer
407
views
spectral sequence for cobordism without leaving smooth category
In Bott & Tu's marvelous book there is a derivation of the spectral sequence for a (smooth) fiber bundle for deRham cohomology done entirely in the realm of the smooth category. Unfortunately, as ...
- 9,870
2
votes
1
answer
186
views
Can I bound the degree of a contracting homotopy in an exact filtered complex?
Suppose that I have given you a bigraded vector space $V = \bigoplus_{i,j} V_{i,j}$. The first grading is a "homological" $\mathbb Z$-grading, and the second is an independent $\mathbb Z$-grading. ...
- 55.9k
6
votes
1
answer
858
views
What kind of spectral sequences come from double complexes?
Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex.
My ...
- 20.9k
2
votes
1
answer
615
views
Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?
For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...
- 16.9k
6
votes
1
answer
1k
views
Unbounded complexes, resolutions and computation of derived functors
Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...
CommunityBot
- 1
9
votes
0
answers
680
views
Can the Bockstein spectral sequence be used to compute cohomology rings ?
If $G$ is a finite group then there is the so-called Bockstein spectral sequence $$E_2^n = H^n(G,\mathbb{F}_p) \Rightarrow \begin{cases} \mathbb{F}_p & n =0 \newline 0 & n>0\end{cases}$$ ...
- 2,160
8
votes
1
answer
377
views
Elementary computation of direct image sheaves.
I am a physicist and would like to understand the section 1 of
this math paper, which explains how the SYZ conjecture implies topological mirror symmetry. I have some technical problem and would ...
- 13.2k
7
votes
1
answer
2k
views
Cohomology groups of quotient by finite group
I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question.
Let $G$ be a finite group acting on a manifold $M$ without fixed ...
- 149k
4
votes
1
answer
2k
views
Comparing Spectral Sequences
There is a comparison theorem for spectral sequnces in Weibel's book (5.2.12) stating;
Assume $E_{p,q}$ and $\bar E_{p,q}$ converge to $H_* $ $\bar H_*$ respectively. Furthermore we have given a map ...
- 3,504