Questions tagged [spectral-sequences]
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388 questions
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Has anyone seen this generalization of the snake lemma? Is it useful?
I originally posted this question on MSE (link), but was suggested to post here instead.
While learning about spectral sequences a friend of mine found a proof of the snake lemma using spectral ...
- 497
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0
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328
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Bousfield-Kan and Generalized Eilenberg-Moore spectral sequences
Building on the work of Anderson and Rector, Bousfield's paper "On the homology spectral sequence of a cosimplicial space" constructs a spectral sequence which takes in a cosimplicial space (here ...
- 131
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0
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Pro-trivial cosimplicial tower of spaces
Let $\{X^\bullet_s\}$ be a cosimplicial tower of spaces. In other words, for each fixed "tower degree" n, we have a (let's assume Reedy fibrant) cosimplicial space $X^\bullet_n$, and for each fixed ...
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When the local system of coefficients are simple in the Leray-Serre spectral sequence
Let $F\to E\to B$ a fibration and $\{E_{r}^{\ast,\ast},d_{r}\}$ the Leray-Serre Spectral sequence converging to $H^{\ast}(E;R),$ such that
$$E_{2}^{p,q}=H^{p}(B;\mathcal{H}^{q}(F;R))$$
is the ...
- 29
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1
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Spectral sequence associated with a Postnikov tower (Solved by myself)
Suppose $E$ is an $S^1$-spectra of simplicial Nisnevich sheaves. For any $r\in\mathbb{Z}$, we have a distinguished triangle
$$E_{\geq r+1}\longrightarrow E_{\geq r}\longrightarrow F_r\longrightarrow ...
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0
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$\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence
I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the ...
- 521
2
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1
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Poset filtrations
Consider a chain complex $C$ and a poset $P$ so that there is a filtration by subcomplexes $C^p$ of $C$ where $p\in P$ in such a way that $p<q$ implies $C^p \leqslant C^q$.
As a second option, ...
- 1,554
6
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3
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460
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multiplicative structure of Ext
Basically, I am trying to compute something with the Adams spectral sequence (as a toy example). The $E^2$ page reduced to computing $Ext^{s,t}_{\Gamma} (\mathbb{F}_2, \mathbb{F}_2)$, where $\Gamma = \...
- 225
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0
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Computations of Bredon homology of $S(1+\sigma)$ with Universal Coefficient S.S
What I am trying to do is to compute $\mathbb{Z}$-graded Bredon homology of $S(1+\sigma)$ over $Q\times\Sigma_2$, where
$Q$ is a cyclic group of order 2
$\sigma$ is its real sign representation
$\...
- 1,759
14
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0
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Computations using "Stover's spectral sequence"
In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.
The second ...
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1
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Calculation of $H^{10}(K(\mathbb{Z}, 3); \mathbb{Z})$
I was trying to calculate $H^q(K(\mathbb{Z}, 3); \mathbb{Z})$ for some $q$ with the Serre spectral sequence associated to the fibration $K(\mathbb{Z}, 2) \to PK(\mathbb{Z}, 3) \simeq * \to K(\mathbb{Z}...
- 1,263
3
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0
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Characterization of degeneracy of spectral sequence of a fiber bundle at the second term
Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients ...
- 21.8k
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multiplication in spectral sequence
I am trying to understand this paper. Let $M$ be a compact Kaehler manifold of dimension $n$, $X$ is a holomorphic vector field, $i_X$ the contraction operator, i.e. for $\alpha$ a $p$-form, then $i_X(...
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Computation of mod p homology of $MSU$
I am trying to proof Novikov theorem
\begin{equation}
MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i.
\end{equation}
This can be proved by using ...
user131113
5
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1
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Functoriality of filtered spectral sequences
What is the appropriate functoriality statement of a filtered chain map between filtered spectral sequences?
Suppose that we have two filtered chain complexes $C,C'$ and a filtered chain map $f\colon ...
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1
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Zero differential in Serre spectral sequence for configuration spaces
I moved this question from Math StackExchange.
I am trying to compute homology of $Conf(n, \mathbb{R}^2)$ - ordered configurations of $n$ points on the plane - using Serre spectral sequence. I know ...
- 1,759
1
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0
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174
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Characterization of weakly convergence of spectral sequences
Let $C$ be a chain complex (in any abelian category) and let $\{F_p\}$ be a decreasing filtration of $C$. It induces a filtration on the homologies of $C$, given by $$F_pH=im(H(F_p)\rightarrow H(C)),$$...
- 183
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A diagram in the proof of Theorem 2.5.5 of 'Cohomology of Number Fields' and the Tate Spectral Sequence
I've been reading the book 'Cohomology of Number Fields' for years.
But I couldn't check the commutativity of the diagram
on page 126 until now. So I ask for help.
The diagram is induced by taking ...
- 1,013
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160
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Splitting of Atiyah-Hirzebruch Spectral Sequence
Suppse E is a cohomology theory which has Kunneth Formula, i.e $ E(A \wedge B)= E(A) \otimes_{E(pt)} E(B) $. What happens to the Atiyah Hirzebruch Spectral sequence while we compute $ E(A \wedge B) $?...
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When is the cohomology of a fiber bundle a tensor product?
Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...
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Conditionally convergent spectral sequences with exiting and entering differentials
I have to deal with unbounded filtrations and want to use the conditional convergence of spectral sequences and the results from
[1]: J. Michael Boardman, Conditionally Convergent Spectral Sequences,...
- 466
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0
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240
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Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence
Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$.
In the beginning he states that there exisits ...
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Kuenneth short exact sequence for K-homology
Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one?
Using general spectra stuff, one gets a ...
- 2,998
11
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3
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846
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Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum
Let $E$ be a ring spectrum and $F$ a connective spectrum. Then we have a convergent Atiyah-Hirzebruch spectral sequence $H_s(F,E_t) \Rightarrow E_{s+t}(F)$. Suppose now that $F$ is also a ring ...
- 64k
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286
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Mahowald uncertainty outside of homotopy theory
In homotopy theory there is the following informal idea:
The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named ...
user138661
8
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1
answer
474
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Third differential in the homology AHSS
I need some guidance in identifying the third differential in the homology AHSS for $\Omega_{\ast}^{\text{Spin}^c}(X)$ in degrees $\leq 4$.
Remember that $\pi_0(M\text{Spin}^c)=\Bbb Z$, $\pi_2(M\...
- 2,018
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Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in the $n$-line of the $E_2$ of the ASS of the sphere?
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element $...
- 7,799
5
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1
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Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra
Consider the extension
$$1\to SU(2)\to X\to O\to1,$$
there are 4 possibilities for $X$:
$X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\...
- 2,147
11
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1
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861
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What is the relationship between spectral sequences and obstruction theory?
Let $X,Y$ be objects of some category $\mathcal C$, and suppose I want to study homotopy classes of maps from $X$ to $Y$ (almost everything one does in algebraic topology can be viewed this way). It ...
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3
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1
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Question about spectral sequences associated to filtered complexes with unbounded filtrations
All references below are from McCleary's book, second edition.
Suppose that we have a filtered complex where the filtration is unbounded. Suppose that the associated spectral sequence is weakly ...
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Grading in Eilenberg-Moore spectral sequence
I am puzzled over something I read in Quillen's On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field.
On page 557, when computing the $E_2$ page of a case of the Eilenberg-...
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0
answers
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Eilenberg-Moore spectral Sequence calculation
I want to use the cohomology Eilenberg-Moore spectral sequence to calculate the cohomology of the fibre of the map
$$
S^{n} \to \Omega S^{n+1}.
$$
Question 1: Is anyone aware of any references for ...
- 901
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Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper
In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
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0
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non zero differential in a spectral sequence
This is the situation:
Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...
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1
answer
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Leray-Serre spectral sequence for projective bundles
Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that ...
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The computation of $d_2$ in the Hochschild-Serre spectral sequence
I'm trying to understand the Hochschild-Serre spectral sequence by an example. Consider the short exact sequence of groups:
$1\to N\to G\to G/N\to 1$
where $G\cong \mathbb{Z}_4$, $N\cong\mathbb{Z}_2$.
...
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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
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1
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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}^{\...
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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
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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^*,
$$
which $c$ flips $a$ to its ...
- 10.5k
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Invariants in relative cohomology and compact support cohomology of the quotient
Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence ...
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Degeneracy of the Serre Spectral Sequence
I am learning the Serre spectral sequence and I am intrigued about the degeneracy of such at the $E_2$-page. Assuming field coefficients in cohomology for simplicity.
In fact, for a Serre fibration $...
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Spectral sequence with a column isomorphic to its homology
I have a first-quadrant spectral sequence $E^r_{p, q}$ of abelian groups of finite rank converging to $E^{\infty}_{p, q}$. We have $E^{\infty}_{p, q}=E^{\infty}_{r, s}$ if $p+q=r+s$. We also have $E^...
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Kernels and cokernels of multicomplex homomorphisms
Let $\mathcal A$ be a (complete and cocomplete) Abelian category.
A multicomplex in $\mathcal A$ is a bigraded object $X^{(\bullet,\bullet)}$ with differentials
$$
d^{(i,j)}_r\colon X^{(i,j)}\to X^{(...
- 2,452
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1
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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 ...
- 395
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0
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421
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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
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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 ...
- 163
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0
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71
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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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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}(\...
- 39
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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 ...
- 2,147