Questions tagged [spectral-sequences]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
116 views

Galois-action on spectral sequence

Let $X_\bullet\to S$ be a proper surjective hypercover of a $k$-scheme by smooth proper $k$-schemes. This gives a proper surjective hypercover $X'_\bullet\to S_{\bar{k}}$ where $X'_n:=X_n\times_k \bar{...
5
votes
1answer
168 views

Can we construct a filtered chain complex from a spectral sequence?

Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...
6
votes
1answer
214 views

Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity

Suppose $G$ is a Lie group, with $\pi_0(G)$ not necessarily finite, but might as well assume $G_0$, the connected component of the identity, is compact. In the case that $\pi_0(G)$ is finite, then we ...
4
votes
0answers
212 views

Spectral sequence from a stratification by closed subvarieties

I am looking for a reference for the following result: If $X$ is an algebraic variety and $$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$ is a stratification (edit: filtration) ...
9
votes
0answers
200 views

Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$

First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
3
votes
0answers
91 views

Group cohomology with coefficients in a graded module

I am working in a problem group cohomology and nailed it down to compute a cohomology using an spectral sequence argument. The situation is as follows: Let $G = C_4 = \langle \sigma \rangle$ be the ...
3
votes
0answers
70 views

Geometric filtration for Eilenberg-Moore spectral sequence

I'm reading the paper by Eilenberg-Moore (https://link.springer.com/content/pdf/10.1007/BF02564371.pdf) about the Eilenberg-Moore spectral sequence. In section 11, they introduce the notion of ...
24
votes
0answers
871 views

Spectral sequences as deformation theory

I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
8
votes
3answers
577 views

Spectral sequences in algebraic topology [duplicate]

What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...
8
votes
2answers
357 views

Conditions under which the preimage of a submanifold in nontrivial in homology

Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-...
5
votes
1answer
382 views

About an argument in Olsson's book

The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson. I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end. It seems that there might ...
5
votes
1answer
135 views

Cohomology of doubly pinched torus via spectral sequences

Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
5
votes
0answers
175 views

Differentials in spectral sequences and Massey products

Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
9
votes
1answer
270 views

$\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}_{p^{r+1}}$ summand

I am reading Neisendorfer's paper Samelson products and exponents of homotopy groups and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}_{p^{r+1}}$ ...
3
votes
1answer
147 views

Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$

It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds: $$ \Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = ...
6
votes
1answer
191 views

Weibel's H-book, Milnor's exact sequence for spectral sequence of filtered complex, Theorem 5.5.5

This is a question which I asked on StackExchange first, but might be more suited here. I got stuck on the proof of Theorem 5.5.5 in Weibel's book. Not only that, but I also could not even find the ...
4
votes
1answer
239 views

Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request

I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
1
vote
0answers
106 views

On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)

I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
3
votes
1answer
258 views

Weak Lefschetz theorem for Lef line bundles

I'm studying M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. The premises are the following....
4
votes
0answers
143 views

Using the Serre spectral sequence - moving between $\mathbb{Z}/2$ and $\mathbb{Z}$ information

I am trying to understand the computation of $\pi_5(S^3)$ and $\pi_6(S^3)$ using the Serre spectral sequence. I know already that $\pi_5(S^3)$ is only 2-torsion and $\pi_6(S^3)$ is 2-torsion together ...
2
votes
0answers
149 views

Dress' construction and Serre spectral sequence

Currently, I am reading Serre spectral sequence, given below, using Dress' construction. Let $f:E\to B$ be a Serre fibration. Then, there is a first quadrant spectral sequence $\big\{E^r,d^r\}_{...
2
votes
0answers
186 views

An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology

I am currently reading Künneth spectral sequence, which is given below. Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
5
votes
1answer
383 views

Cohomology of derived tensor product of complexes and Künneth spectral sequence

Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
3
votes
0answers
154 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
votes
0answers
144 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 ...
7
votes
0answers
206 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
votes
1answer
1k 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 ...
5
votes
0answers
195 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
votes
0answers
69 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 ...
1
vote
0answers
132 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 ...
1
vote
1answer
190 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 ...
6
votes
0answers
146 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
votes
1answer
148 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, ...
6
votes
3answers
330 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
votes
0answers
122 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
votes
0answers
254 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 ...
15
votes
1answer
640 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}...
3
votes
0answers
62 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
votes
0answers
127 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(...
2
votes
0answers
89 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
votes
1answer
322 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
votes
1answer
366 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 ...
14
votes
0answers
261 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 ...
1
vote
0answers
75 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 $$F_pH=im(H(F_p)\rightarrow H(C)),$$...
7
votes
0answers
236 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 ...
0
votes
0answers
128 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
votes
1answer
269 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{...
6
votes
1answer
226 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
votes
0answers
213 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
votes
1answer
102 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 ...

1
2 3 4 5
7