# 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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### Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is connected,...
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### extension problem of a spectral sequence

From Hatcher's SSAT, If the coefficient group $G$ is a field, then $H_n(X;G)$ is the direct sum $\oplus_p E^\infty_{p,n-p}$ of the terms along the $n^\text{th}$ diagonal of the $E^\infty$ page. For ...
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### Vanishing terms in Leray spectral sequence

Let $f:X\to Y$ be a continuous map of complex manifolds. Consider $H^p(Y;R^qf_*V)$, where $V$ is a vector bundle on $X$, and $R^qf_*V$ are the higher direct images. (This is the $E_2^{p,q}$ term in ...
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### definition of convergence of a spectral sequence

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant and ...
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### How to define the natural map on the second page of a spectral sequence?

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
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### How can I determine the Steenrod Square $Sq^2$ for complex projective space?

I am trying to learn about Steenrod Squares for algebraic varieties so that I can compute examples of complex topological K-theory using the Atiyah-Hirzebruch Spectral Sequence (AHSS). One of the key ...
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### Does every elliptic cohomology theory represent a complex-orientable $E_\infty$-ring spectra and vice-versa?

The last paragraph in Two-Vector Bundles and Forms of Elliptic Cohomology remarks that neither the spectrum $K(ku)$ nor tmf is complex orientable. In the case of $K(ku)$: "...the unit map for $K(ku)$ ...
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### Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence

For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
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### Reference Request: The Atiyah-Hirzebruch Spectral Sequence

I have just finished learning the Serre spectral sequence and I would like to learn about the Atiyah-Hirzebruch spectral sequence. Could someone suggest an accessible reference? Thank you in advance.
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### Spectral sequence of a filtered complex: convergence conditions and abelian categories

There is a theorem that if given a filtered complex and the filtration is bounded then there is a spectral sequence whose 0th and 1st page have specific forms and the sequence converges to (co)...
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### filtered modules (LNAT, Davis & Kirk)

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written: Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a ...
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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 ...
### 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 ...