Questions tagged [spectral-sequences]

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3
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0answers
136 views

Cobordism theory of some weird space

Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$. The $W$ is a homogeneous space (also a quotient space), but not a group. Previously, I am aware of the ...
2
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0answers
132 views

A Thom isomorphism for sheaves

Let say $\mathcal{F}$ is a locally free sheaf of abelian groups over $X$, where $X$ is an algebraic variety over $\mathbb{C}$ (or a field $k$) with analytic (or étale) topology and $Z$ is a closed ...
5
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0answers
134 views

Convergence of a spectral sequence of a double complex

In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
14
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1answer
911 views

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 ...
4
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0answers
162 views

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 ...
2
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0answers
68 views

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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100 views

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 ...
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1answer
163 views

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 ...
5
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131 views

$\mathbb{Z}/2\mathbb{Z}$ 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 ...
2
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1answer
122 views

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, ...
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3answers
303 views

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 = \...
3
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0answers
109 views

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 $\...
11
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0answers
239 views

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 ...
13
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1answer
564 views

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}...
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56 views

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 ...
5
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121 views

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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82 views

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 ...
5
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1answer
251 views

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 ...
6
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1answer
239 views

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 ...
13
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243 views

Gel'fand and Fuks' “Globalizing” of cohomology of formal vector fields

I apologize for the length of this question. If anybody already spent some time with cohomology of (formal) vector fields and the results of Gel'fand and Fuks, I imagine a lot can be skipped. I do ...
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46 views

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 $$F_pH=im(H(F_p)\rightarrow H(C)),$$...
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231 views

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 ...
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0answers
122 views

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) $?...
2
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1answer
240 views

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{...
5
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1answer
168 views

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,...
4
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0answers
210 views

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 ...
4
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1answer
98 views

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 ...
10
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3answers
581 views

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 ...
7
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0answers
106 views

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 ...
8
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1answer
302 views

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\...
5
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0answers
200 views

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 $...
5
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1answer
130 views

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_{\...
11
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1answer
485 views

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 ...
3
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1answer
148 views

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 ...
3
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1answer
182 views

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-...
4
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0answers
211 views

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 ...
11
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0answers
214 views

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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0answers
65 views

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 ...
2
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1answer
176 views

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 ...
4
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1answer
181 views

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 $G\cong \mathbb{Z}_4$, $N\cong\mathbb{Z}_2$. ...
3
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0answers
128 views

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 ...
1
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1answer
218 views

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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0answers
229 views

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 ...
5
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0answers
89 views

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 ...
7
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1answer
216 views

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 ...
6
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0answers
163 views

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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0answers
33 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^...
4
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1answer
301 views

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^{(...
5
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1answer
465 views

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 ...
9
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0answers
211 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 ...

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