Questions tagged [spectral-sequences]
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388 questions
3
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Strong convergence of whole-plane spectral sequences
I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-...
- 205
4
votes
0
answers
419
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Cohomology of double complex with exact rows
Let $(C^{p,q},d_h,d_v)$ be an unbounded double complex of modules of some algebra $A$ in an abelien category. Let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that
(1) ...
- 205
5
votes
0
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544
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Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple explanation?)
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than ...
- 1,229
3
votes
1
answer
260
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non-simple local coefficient system on a fibration of classifying spaces
Long story short; I posted in MSE
https://math.stackexchange.com/questions/2500745/local-system-of-coefficients-on-a-fibration-of-classyfing-spaces
It is well known that if $G$ is a lie group ...
- 283
4
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1
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195
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Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?
I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.
How about for multiple summands in the ...
- 1,229
15
votes
2
answers
1k
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Pullback and homology
Suppose I have two maps of topological spaces, $f:X\rightarrow B$ and $g:Y\rightarrow B$, such that $f$ induces a homology isomorphism and $g$ is a fibration and $B$ is connected. Is it true that the ...
- 1,629
2
votes
1
answer
712
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Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$
Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω_{n,O}(pt)$ and $Ω_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$?
Here are some ...
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15
votes
1
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988
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Why is it difficult to obtain the next differential in a spectral sequence?
I am following Hatcher's notes on spectral sequences. On page 522 an exact couple $(A, E, i, j, k)$ is defined. You can construct a derived exact couple easily from this to get your desired $E'$ ...
- 151
10
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1
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719
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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 $M$...
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168
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Adams spectral sequence for loop spaces
Let $X = \Omega_0^3S^3$ a connected component of $\Omega^3S^3$. I am interested in explicit construction of spectral sequence converging to odd prime torsion in homotopy groups of $X$.
There is a ...
- 1,129
8
votes
1
answer
615
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Functorial construction of ("pre"-)spectral sequences? (Or - what is the "higher structure" underlying spectral sequences?)
Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the ...
- 7,799
6
votes
1
answer
242
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For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My ...
- 16.9k
3
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0
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234
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How can I find the differential in the Serre spectral sequence for this sphere fibration?
Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials
$$
d_4^{p,m}:...
- 1,716
2
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0
answers
77
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hirzebruch spectral sequence for a cohomology theory on a subcategory of TOP
The answer to my question is probably going to be 'yes, sometimes'. So I'll give my motivation first.
I am trying to give a short argument that for a fibration $F \hookrightarrow E \to B$ and a $\...
- 123
6
votes
1
answer
399
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Leray-Serre spectral sequence for algebraic groups
Let $G$ be a semisimple, simply-connected, complex algebraic group. Fix a Borel subgroup $B$ and let $P$ be a parabolic subgroup properly containing $B$. If $M$ is a $B$-module, then we have the Leray-...
- 422
2
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0
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151
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Monodromy and simple system of local coefficients
I was interested in the following question: if one has a fibration
$F\to E\to B$
there is associated a monodromy map, that is basically an action of the fundamental group $\pi_1(B)$ on the ...
- 41
8
votes
1
answer
363
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Adams spectral sequence and short exact sequences. Some clarifications
as the title suggests I'm looking for some clarifications in the computations of the ext charts of some $A(1)$-modules arising as extensions of other modules. In particular, I've the following example ...
- 503
4
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0
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343
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Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
- 1,716
6
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167
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A spectral sequence problem in Alejandro Adem's Paper
I am reading Adem's paper Periodic Complexes and Group actions. But I can't give an argument about a statement on spectral sequences.
Suppose you have an orientable fibration of CW-complexes like ...
- 61
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0
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163
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Question about the precise statement of Leray spectral sequences and a simple example
On Bott's paper "Homogeneous vector bundles" there is the following statement of Leray spectral sequence:
Let $X$ and $Y$ be paracompact and locally compact spaces and $f : Y \to X$ be a proper map....
7
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149
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Cohomology of Lie group $E_8$, e.g. $H^d(E_8,\mathbb{R}/\mathbb{Z})$
What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient?
I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{...
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1
answer
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Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space
Let $G$ be a compact, connected Lie group. There is an Atiyah–Hirzebruch spectral sequence
$$H^*(BG;K^*) \implies K^*(BG)$$
connecting $H^*BG$, which generally contains torsion, with $K^*BG \cong \...
- 2,995
2
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0
answers
71
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Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
- 2,147
5
votes
0
answers
521
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Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
- 16.9k
3
votes
0
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310
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Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
1
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1
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379
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Spectral sequence of Galois cohomology over local fields
On page 530 in his paper,
Notes on etale cohomology of number fields, Ann. Scient. ENS (1973),
Mazur insisted that
$$
\text{Ext}^q_{G_K} (M,~\bar{K}^*) \...
- 367
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0
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108
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Relating inflation maps from spectral sequences in lower and higher dimensions
The spectral sequence has some nice property.
Consider $ N \to G \overset{R}{\to} Q $ and $G/N=Q$. There is a spectral sequence $\{E^{p,q}_n, d_n\}$ with: (i) The differential is defined as a map $...
- 10.5k
9
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1
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455
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Is this sequence of Lie algebra cohomology a part of spectral sequence?
There is an exact sequence
$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \...
- 1,980
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0
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109
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Empty regions on the second list of unstable Adams spectral sequence
Define $\phi(n) = 4n - 2$. Is there a proof that on the second list of unstable Adams spectral sequence (for all spheres) there are no elements in squares $(n, m)$ such that $m < \phi(n)$. The ...
- 1,129
10
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1
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370
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Adams Spectral sequence for computing some $B$-bordism groups
As the title suggests, I'm trying to apply the Adams Spectral sequence to get some insights of the bordism group
$$ \Omega_4(\xi)= \pi_4(M\xi)$$
where $\xi \colon BSpin \times K(D_{2n},1) \to BSO$ is ...
- 2,018
7
votes
1
answer
845
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Mayer Vietoris Spectral sequence for topological K theory
In Sheaf theory one can obtain the Mayer Vietoris spectral sequence for cohomology. For $\mathcal{U}$ an open cover of $X$ we get the convergence
$E_2^{pq} = \check H^p(\mathcal{U},H^q(-,F)) \...
- 659
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0
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374
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Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
- 7,377
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257
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Building examples of elements of $\Omega_4(\xi)$ via surgery theory: how to do it?
When computing special bordism groups, I often need to determine existence of (singular) smooth $4$-manifolds with fixed fundamental group and certain properties like the spin behaviour (i.e. being ...
- 2,018
5
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1
answer
516
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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 ...
- 10.5k
5
votes
2
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526
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The inability to continue a fibration sequence even when a delooping exists
$\newcommand{\i}{\iota}$
The general notion that I am trying to disprove is that if we are given a fibration $X \to Y$ with fiber $F$ such that the delooping $BF$ exists, that there is a map $Y \to BF$...
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5
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1
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Spin bordism group of classifying space $BG$ with a finite Abelian $G$
The spin bordism group for the classifying space $BG$ of group $G$ can be denoted as $\Omega^{Spin}_d(BG)$.
For example, $\Omega^{Spin}_d(pt)$ are computed by Anderson-Brown-Peterson (D. W. Anderson, ...
- 755
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2
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651
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Inflate a finite-group cocycle into coboundary in non-Abelian groups
Edit: In case that there is no solution for the original question, I modify to enrich the question.
We like to ask a possible specific inflation a $H^3(Q, \mathbb{R} /\mathbb{Z})$ cocycle with a ...
- 755
0
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1
answer
143
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Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
- 755
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0
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120
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Trivialize a cocycle of a continuous Lie group-cohomology to a coboundary
Someone recently asks a question $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? now inspires me to revisit an earlier general question to ask an example of 3-cocycle
$\omega_3^G$ of a ...
- 10.5k
2
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1
answer
264
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Trivialize a cup-product 3-cocycle of $G$ in a larger group $J$
Inspired by this question, let us take a nontrivial 3-cocycle $\omega_3^G(g_a, g_b, g_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. ...
- 755
4
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1
answer
394
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$SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$?
I was trying to understand this interesting question by example.
Let me follow their previous discussion and ask: Let a generic nontrivial 2-cocycle $\omega_2^G(g_1,g_2) \in H^2(G,\mathbb{R}/\mathbb{...
- 755
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1
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609
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References for properties of Atiyah-Hirzebruch Spectral Sequence for a spectrum $X$ and generalised homology theory $MSpin_*$
Currently I'm working on the following version of the AHSS $$ E^2_{pq}\cong H_p(M\eta; MSpin_q(\ast))\Rightarrow MSpin_{p+q}(M\eta)$$
where $\eta \colon B \to BSO$ is a stable vector bundle, and $M\...
- 503
2
votes
0
answers
206
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Cohomology of fiber bundles with non constant coefficients
Let $E \to B$ be a topological fiber bundle. We know that the Serre spectral sequence allows you to compute the cohomology of $E$ with constant coefficient in terms of the cohomology of $F$ and the ...
- 121
7
votes
1
answer
506
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$G$ cocycle split to a coboundary in $J$, via a group extension
Consider a generic nontrivial $d$-cocycle $\omega_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\...
- 10.5k
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votes
1
answer
495
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Identification of a Serre Spectral Seq. via Thom Isomorphism with the Atiyah-Hirzebruch Spectral Seq
Let $\xi_n$ be an orientable $n$-dimensional vector bundle over a pointed space $B_n$. We can consider the relative Serre Spectral Sequence $$ H_p(B_n; h_q(D(\xi_n|\ast),S(\xi_n|\ast))\Rightarrow h_{p+...
- 2,018
11
votes
2
answers
656
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$G$ cocycle split and trivialized to a coboundary in $J$, given a group homomorphism $J \overset{r}{\rightarrow} G$
Consider a generic nontrivial 3-cocycle $\omega_3^G(g_1,g_2,g_3) \in H^3(G,U(1))$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the 3-cocycle $\...
- 10.5k
7
votes
1
answer
663
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Explicit 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$
I would like to know the explicit expression of 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$ with $U(1)\equiv \mathbb{R}/\mathbb{Z}$ coefficient, or namely $H^2[...
- 10.5k
3
votes
0
answers
274
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Is there a spectral sequence for borel-moore homology associated to a whitney filtration?
Consider a Whitney stratified space
$$
\varnothing = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n
$$
is there a spectral sequence for borel-moore homology which depends on the ...
- 1,716
2
votes
0
answers
326
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A version of Leray Hirsch better for local coefficients
Let $F \hookrightarrow E \to B$ be a fibration, and let everything be over a field $k$.
The leray hirsch theorem in its usual form says that the (homology) cohomology spectral sequence degenerates at (...
- 579
5
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0
answers
368
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Reference request: a Künneth spectral sequence map from equivariant K-theory to cohomology
The analogue of the Künneth formula for Borel $G$-equivariant cohomology can be obtained as the Eilenberg–Moore spectral sequence of a pullback
$\require{AMScd}$
\begin{CD}
(X \times Y)_G @...
- 2,995