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Results tagged with spectral-sequences
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user 9684
17
votes
Differentials in the Adams Spectral Sequence for spheres at the prime p=2
With the aid of machine computations, you can readily determine the Adams differentials up to $t-s=30$ using the multiplicative structure, the relation between Steenrod operations in $\text{Ext}_A$ an …
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14
votes
What are some good examples of spectral sequences which degenerate after the first nontrivia...
Some examples with one nonzero family of differentials:
The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p= …
10
votes
Multiplicative structure on spectral sequence
As far as I know that 1954 paper of Massey is faulty, and you cannot get multiplicative spectral sequences just from such stucture on an exact couple. The best I know that you can do is to use Cartan- …
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8
votes
Accepted
Third differential in the homology AHSS
The realification map $BU \to BSO$ lifts through $BSpin^c$, because the first Chern class provides an integral lift of the second Stiefel-Whitney class. Calculating with the Serre spectral sequence f …
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8
votes
Accepted
$BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$
For $3$-primary homotopy of $S$ there is early work by
Nakamura, Osamu
Some differentials in the mod 3 Adams spectral sequence.
Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. No. 19 (1975), 1– …
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8
votes
How to compute the cohomology of the general linear group with integral entries
There are homological stability results (due to Ruth Charney and Hendrik Maazen around 1979, if I recall correctly) saying that $H_*(GL_n(Z); Z) \to H_*(GL_{n+1}(Z); Z)$ is about $n/2$-connected. So …
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8
votes
Spectral sequences and short exact sequences
If you instead work with a cofiber sequence of filtered spectra, then I gave a sufficient condition in Proposition 5.4 of
https://www.mn.uio.no/math/personer/vit/rognes/papers/highfix.pdf
(JPAA, 1999) …
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6
votes
Accepted
Conditionally convergent spectral sequences with exiting and entering differentials
After a long life in preprint form, Boardman's paper was published in the conference proceedings celebrating his 60th birthday:
\bib{MR1718076}{article}{
author={Boardman, J. Michael},
…
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6
votes
Relation between cohomology operations and the Adams spectral sequence
Maybe these references can help:
Mauder, C. R. F.
On the differentials in the Adams spectral sequence.
Proc. Cambridge Philos. Soc. 60 (1964), 409–420.
Mosher, Robert E.; Tangora, Martin C.
Cohomolo …
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6
votes
Accepted
Loop-space functor on cohomology
If you are willing to read a little French, look at page 434 of Serre's "Homologie singuliere des espaces fibres" (1951).
For an exercise in English, with a hint, try Exercise 2 on page 155 in Mosher …
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4
votes
Accepted
Strong convergence of whole-plane spectral sequences
The spectral sequences $E^\alpha$ and $E^\beta$ arise from two different filtrations of $C$, which have different completions, say $\widehat C^\alpha$ and $\widehat C^\beta$. The first converges stro …
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4
votes
Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
This is Theorem 4.8 in Chapter XIII of
Whitehead, George W.
Elements of homotopy theory.
Graduate Texts in Mathematics, 61.
Springer-Verlag, New York-Berlin, 1978.
xxi+744 pp. ISBN: 0-387-90336-4
an …
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3
votes
Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum
(A comment to Tyler's answer.) Strictifying pairings from the stable homotopy category to spectra can be tricky. To even get started with an inductive approach let me assume $E$ is connective, so th …
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3
votes
Accepted
The Hochschild–Serre spectral sequence and cup products
The original reference is
Hochschild, G.; Serre, J.-P.
Cohomology of group extensions.
Trans. Amer. Math. Soc. 74 (1953), 110–134.
On page 118 they introduce a filtration $\{A_j\}_j$ of the cochain c …
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2
votes
Accepted
Does a filtered A_N algebra give rise to a multiplicative spectral sequence?
For the $d_r$-differentials to be derivations, i.e., to satisfy the Leibniz rule $d_r(x \cdot y) = d_r(x) \cdot y \pm x \cdot d_r(y)$ with $x \cdot y = m_2(x \otimes y)$, it is enough to have a filter …
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