Questions tagged [spectral-sequences]
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388 questions
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Trivial action in the Hochschild-Serre spectral sequence
I probably don't understand something very basic about Hochschild-Serre spectral sequence.
Let $G$ be a group with normal subgroup $N$ and $M$ a $G$-module with trivial action. Then as far as I ...
3
votes
1
answer
428
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Leray spectral sequence from hypercohomology
Context: Deligne, Theorie de Hodge II, section 1.4.8.
Let $f:X\rightarrow Y$ be a map between spaces; $\mathcal{F}$ a sheaf of abelian groups on $X$, and $\mathcal{F} \rightarrow \mathcal{F}^{\...
- 35.2k
5
votes
1
answer
549
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Show that if $p\neq 2$, then $\mathbb{Z}_p$ cannot act freely on $\mathbb{C}P^n$
If $p\neq 2$, then the cyclic group $\mathbb{Z}_p$ has no free continuous action on $\mathbb{C}P^n$. My question is how to prove the above fact using Leray-Serre spectral sequence associated to the ...
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0
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448
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Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,713
5
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0
answers
102
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Group cohomology of "twisted" projective SU(N) with various coefficients
Given a group
$$
G= PSU(N) \rtimes \mathbb{Z}_2,
$$
where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then
$$
c a c= a^*,
$$
which $c$ flips $a$ to its ...
- 10.5k
7
votes
1
answer
347
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Invariants in relative cohomology and compact support cohomology of the quotient
Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence ...
- 1,424
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0
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300
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Degeneracy of the Serre Spectral Sequence
I am learning the Serre spectral sequence and I am intrigued about the degeneracy of such at the $E_2$-page. Assuming field coefficients in cohomology for simplicity.
In fact, for a Serre fibration $...
- 81
1
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0
answers
43
views
Spectral sequence with a column isomorphic to its homology
I have a first-quadrant spectral sequence $E^r_{p, q}$ of abelian groups of finite rank converging to $E^{\infty}_{p, q}$. We have $E^{\infty}_{p, q}=E^{\infty}_{r, s}$ if $p+q=r+s$. We also have $E^...
- 287
4
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1
answer
332
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Kernels and cokernels of multicomplex homomorphisms
Let $\mathcal A$ be a (complete and cocomplete) Abelian category.
A multicomplex in $\mathcal A$ is a bigraded object $X^{(\bullet,\bullet)}$ with differentials
$$
d^{(i,j)}_r\colon X^{(i,j)}\to X^{(...
- 25.2k
7
votes
1
answer
372
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Serre spectral sequence degeneration in homology vs cohomology
Let $\pi\colon E \rightarrow B$ be a fiber bundle with fiber $F$. I am not assuming that $B$ is simply-connected. We then have Serre spectral sequences in both rational homology and rational ...
- 35.9k
9
votes
0
answers
421
views
Hochschild-Serre spectral sequence via explicit filtration
Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
- 163
3
votes
0
answers
71
views
Deformations of nilpotent parts of superalgebras
I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598
After ...
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answers
174
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Induced Homomorphism on Cohomology of Symmetric Group 3
For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
- 63.9k
8
votes
0
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views
Relating bordism generators in d and d+2 dimensions --- an explicit example
This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
CommunityBot
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4
votes
0
answers
71
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Relating bordism invairants in $d$ and $d+2$ dimensions
Are there some relationship between mapping the bordism invairants of eq.1 and eq.2?
$$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$
and
$$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \...
- 2,147
2
votes
1
answer
142
views
Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group
Let $M^d$ be a nontrivial bordism generator for the bordism group
$$
\Omega_d^G= \mathbb{Z}_n,
$$
suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...
- 6,881
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votes
1
answer
475
views
Spin cobordism v.s. KO theory in low or in any dimensions
It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...
- 3,051
3
votes
0
answers
166
views
Edge map in derived categories
Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...
- 5,670
3
votes
0
answers
226
views
Does Leray Spectral sequence degenerates at $E_2$ over product of curves
Let $C$ be a smooth, projective curve (can assume to be rational) and $X:=C \times C$. Denote by $p:X \to C$ one of the two natural projections. Let $E$ be a vector bundle on $X$. Is it true that,
$$...
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votes
1
answer
413
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Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$
The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...
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5
votes
1
answer
516
views
Computation of $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$ and Smith isomorphism
question: I am looking for the literature with the result or the computation of Pin- bordism group: $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$. Can someone point out some useful ways to do this or any helpful ...
- 6,881
1
vote
0
answers
112
views
Reference request for Leibniz rule and spectral sequences
Suppose $A_*$,$B_*$, and $C_*$ are chain complexes equipped with filtrations and a map $m:A_* \otimes B_* \to C_*$ respecting these filtrations. I am looking for a reference for the fact that the map $...
- 41
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votes
2
answers
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views
Sphere spectrum, Character dual and Anderson dual
The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
What is the significance of ...
- 5,770
3
votes
0
answers
122
views
Twisted spin cobordism v.s. KO theory in low dimensions
Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like
$${\Omega_d^{(\mathrm{spin} \times G)/N}},...
3
votes
0
answers
165
views
Reference for specific detail on Serre spectral sequence
In "A primer on spectral sequences" http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf (by J.P.May apparently, although no name is given in the pdf) I found a very detailed version of the ...
- 404
9
votes
0
answers
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views
Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...
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6
votes
0
answers
122
views
Bordism groups and a short exact sequence
Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...
- 10.5k
5
votes
1
answer
311
views
Transgression image and Serre spectral sequence for tori
Let $\mathbb{K} \subset \mathbb{T}$ be two tori acting on a topological space $X$ (with all the properties you want). We use the notations $$X_{\mathbb{T}} := (X \times E \mathbb{T}) / \mathbb{T}, \...
CommunityBot
- 1
6
votes
0
answers
562
views
The $E_2$-page of the May spectral sequence
I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS.
At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...
- 4,189
6
votes
0
answers
237
views
A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?
This question is a follow-up to my previous question:
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
In that question, I discussed two different spectral sequences for ...
- 18.2k
5
votes
0
answers
721
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 ... \...
- 1,227
1
vote
0
answers
91
views
Continuous maps vs filtrations construction of the Leray spectral sequence
The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more ...
- 1,227
4
votes
1
answer
227
views
How are p-primary parts determined for odd p?
When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away.
How are odd primary part calculations done in relation ...
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2
votes
0
answers
69
views
Condition for a map to carry over to Leray spectral sequences
I am trying to understand the conditions for two Leray spectral sequences to be related by a map.
Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps of topological spaces (with ...
- 1,227
2
votes
1
answer
532
views
Construction of differentials in the spectral sequence for double complexes
I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (...
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2
votes
1
answer
275
views
Why only consider decreasing filtrations on cochain complexes?
When reading various literature on spectral sequences one always comes across two setups:
A chain complex with an increasing filtration
A cochain complex with a decreasing filtration
My question is ...
- 52.7k
4
votes
1
answer
598
views
Thom space, homotopy group and cohomology group
In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
- 3,051
3
votes
0
answers
506
views
Degeneration of relative Hodge-de Rham spectral sequence
$$\require{AMScd}$$
$$\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\Hdr}{H_{\mathrm{dRh}}}
\newcommand{\tensor}{\otimes}
\newcommand{\Ohol}{\mathcal{O}}$$
Please excuse that ...
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votes
1
answer
456
views
Does the Serre spectral sequence of the Fadell-Neuwirth fibration collapse if there is a cross-section?
I had asked a vague question in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by Mike Miller in the comments, but received no answers. Here I will specialize to an interesting ...
- 35.9k
3
votes
0
answers
149
views
Gel'fand's and Fuks' calculation of cohomology of formal vector fields - isomorphic spectral sequences yield isomorphic cohomology?
In this book (and, in what seems to be an equivalent fashion, in this article), Gel'fand and Fuks calculate the Lie algebra cohomology of the formal vector fields $W_n$ on $\mathbb{R}^n$ with trivial ...
CommunityBot
- 1
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votes
1
answer
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views
The converse of Vietoris-Begle theorem
It is well known the following result:
Lemma: Let $F\rightarrow E\rightarrow B$ be a fibration with $B$ connected and
simply connected. Suppose that $F$ is $n$-acyclic, i.e. $H^{p}\left( F;%
%...
CommunityBot
- 1
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votes
2
answers
408
views
Homology spectral sequence for function space
The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\...
- 10.9k
3
votes
1
answer
465
views
What bigrading is used in this spectral sequence?
I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ...
- 1,554
6
votes
1
answer
244
views
to compare cohomologies of fibers of two fiber bundles
Consider the following commutative diagram of the fiber bundles $%
F\rightarrow E\rightarrow B$ and $F^{\prime }\rightarrow E^{\prime
}\rightarrow B^{\prime }$ where $B^{\prime }$ is simply connected ...
- 56.9k
2
votes
2
answers
323
views
homology of a base space of a a fiber sequence
Suppose we have a fiber sequence of connected spaces $A\rightarrow B\rightarrow C$ and suppose we know the homology of A and B, is there a homological spectral sequence converging to the homology of $...
- 56.9k
1
vote
0
answers
122
views
maps between two Leray spectral sequences based on maps on cochain complexes
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 ...
- 1,227
6
votes
0
answers
163
views
Spectral Sequence for Twisted K-theory
Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[...
- 61
2
votes
0
answers
143
views
Maps between 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. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...
- 1,227
8
votes
2
answers
500
views
To compare the total, base and fiber spaces of two fiber bundles
Consider the following commutative diagram of the fiber bundles $%
F\rightarrow E\rightarrow B$ and $F^{\prime }\rightarrow E^{\prime
}\rightarrow B^{\prime }$ where $B^{\prime }$ is simply connected ...
- 5,770
16
votes
1
answer
808
views
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...
- 35.9k