# Questions tagged [spectral-sequences]

Spectral sequences compute homology groups by taking a sequence of approximations, and generalise exact sequences. They find application in algebraic topology.

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### Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
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### Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
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### Module structure in the Serre spectral sequence of the Borel construction

Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its ...
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### What can we say about the category of spectral sequences?

For simplify, let's just consider the following category first: object: "spectral sequences", that is a sequence of pages $(E_r)_{r\ge0}$, where each page $E_r$ is a differential object in an abelian ...
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### Edge homomorphisms for Atiyah-Hirzebruch spectral sequence for a spectrum $X$

I'm interested in some nice identifications (together with explanations) about the edge homomorphisms in the AHSS for a spectrum $X$. $$\times \times \times$$ Let $X$ be a connective spectrum, as ...
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### $h^{0, 1}$ of K3 surface (a priori non-Kahler)

I am trying to understand paper by Siu "Every K3 surface is Kahler". Let $M$ be a K3 surface. Siu wrote $H^1 ( M , \mathscr{O}_M ) =0$ without any references. It is written on fifth page. Maybe I ...
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### Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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### A form of Künneth formula?

Problem (Exercise 15.12 in Bott & Tu's Differential Forms in Algebraic Topology) If $X$ is a space having a good cover, e.g., a triangularizable space, and $Y$ is any topological space, prove ...
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### Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
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### Examples of exact couples of abelian groups

Exact couples are really important when defining spectral sequences. However, I have never really seen a simple non-trivial example of two exact couples of abelian group with a morphism between them. ...
In Lecture Notes in Algebraic Topology by Davis and Kirk it seems that given a (convergent, bigraded) spectral sequence $(E^r, d^r)$ one defines the $E^{\infty}$-page as $$E^r_{p, q} = \operatorname{... 0answers 173 views ### Spectral sequence B^1_{p,q}= B^\infty_{p,q} definition, nlab In an introductory notes to spectral sequences, nlab, Definition 1.26 we define$$B^r_{p,q} = \partial(F_{p+r-1} C_{p+q+1})$$and$$B^\infty_{p,q}= \partial(F_{p}C_{p+q+1})So what is the ... 0answers 89 views ### Structure of module over Eilenberg MacLane spectrum Let HR be the Eilenberg-Maclane spectrum for a commutative ring R and M be a module over HR. Then I want to prove that M is a product of Eilenberg-Mac Lane spectra. Construction: Let \... 0answers 73 views ### A question related to group cohomology and spectral sequences It is actually a follow-up question of a mathoverflow question. I don't quite understand the answer there. I tried to compute the group cohomology of H^n(\mathbb{Z}_4,\mathbb{Z}) via the Lyndon-... 0answers 172 views ### Cohomology of double complex with exact rows Let (C^{p,q},d_h,d_v) be a double complex of modules and let d_h,d_v be the horizontal and vertical differential respectively. Suppose that (1) the horizontal rows are exact, (2) the columns ... 0answers 154 views ### Local-to-Global Spectral Sequence for locally free resolution Consider two coherent sheaves \mathcal{E},\mathcal{F}, with locally free resolutions \begin{align} &0 \longrightarrow \mathcal{E}_1 \longrightarrow \cdots \longrightarrow \mathcal{E}_n \... 0answers 56 views ### Spectral sequence of non-cohomologically trivial fibration. It is a well known that in case when we have \xi: E\to B - a Serre fibration, with the trivial action of \pi_1(B) on a fiber F, we have standart Atiyah - Hirzebruch spectral sequence. What we ... 0answers 172 views ### Equivariant cohomology with respect to Lie group and its maximal torus Let G be a compact connected Lie group, T \subset G its maximal torus and W=N(T)/T its Weyl group. The formula 2.11 of Atiyah, Bott The Moment Map and Equivariant Cohomology states that for any ... 0answers 83 views ### Calculating torsion in \pi_i(S^{2n}) Let p be an odd prime and n>1. I want to prove that \pi_i(S^n) has no p-torsion for i<n+2p-3. For odd n this is Proposition 6.26 in McCleary's book (p.206). He mentions afterward ... 0answers 64 views ### computing \pi_1 S^1 from a spectral sequence In most of my calculations that I have done based off of mosher and tangora, calculations have proceeded by knowing for example that the fiber of some fibration is say a 1-sphere. From this we are ... 0answers 75 views ### Comparison Criterion for Atiyah - Hirzebruch Spectral Sequence Let us denote with E(X) the A-H spectral sequence associated to a CW complex X and homology theory h_*: E(X)_{pq}^2 = H_p(X, h_q(\ast))\Longrightarrow h_{p+q}(X)$$and with E(Y) the one ... 0answers 130 views ### Reference on Weibel's Homological Algebra: “G/H acts by conjugation in LHS-spectral sequence” I'm studying the Lyndon-Hochschild-Serre spectral sequence for H\triangleleft G:$$ H_p(G/H;H_q(H;A))\Rightarrow H_{p+q}(G;A) $$where A is a G-module. I was told (w/o giving a proof) that ... 0answers 149 views ### Mapping Lemma for Spectral Sequences in Weibel's book - Help with the proof We start by recalling the definition of a morphism in the category of Spectral Sequences: a morphism f \colon A \to E is a family of maps f^r_{pq}\colon A^r_{pq}\to E^r_{pq} in the abelian ... 0answers 206 views ### Basic questions about convergence of spectral sequences. I have a fairly rudimentary understanding of spectral sequences. I have a couple questions though. In Algebra, Lang states what seems to me, to be two slightly different notion of convergence of a ... 0answers 35 views ### How networks with high largest eigenvalues are more robust? In the literature, it sometimes indicates that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust. Robustness here is relevant to link/... 0answers 372 views ### Gysin sequence and Serre spectral sequence Given an oriented S^k bundle E over a compact manifold M we get the Gysin Sequence (I am interested in the DeRham cohomology). We can obtain this sequence from the Serre Spectral Sequence if the ... 0answers 76 views ### Generalisation of Adams spectral sequence to triangulated categories We have the Adams SS with$$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$where E is the Eilenberg-Maclane Spectrum yielding \mathbb{Z}/p coefficients. I was wondering if there is a SS for ... 0answers 77 views ### Reference request: where can I find illustrative, concrete examples of the use of the Eilenberg–Moore spectral sequence? Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle S^3 \to S^2... 0answers 117 views ### Serre spectral sequence and locally constant coefficients I have a brief question - In the Serre Spectral sequence for a fibration$$F \rightarrow E \rightarrow B$$one can require, to avoid using local system of coefficients, that the action \pi_1(B) on ... 0answers 187 views ### Spectral Sequence and Stiefel Manifold Let Spin(3) be embedded in Spin(5) by the spin embedding then we have a fibration:$$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3) Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel ...
Let $E$ be a spectrum and let $X$ be a space or connective spectrum. Then the cohomological Atiyah-Hirzebruch spectral sequence is of the form: $H^s(X,E^t) \Rightarrow E^{s+t}(X)$ This is a half-...