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510 views

Why should we study the total complex?

Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
mrtaurho's user avatar
  • 165
7 votes
0 answers
270 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 ...
qqqqqqw's user avatar
  • 965
5 votes
1 answer
185 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_{\...
annie marie cœur's user avatar
4 votes
0 answers
396 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 ...
Niall Taggart's user avatar
1 vote
0 answers
90 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 ...
Vitolo's user avatar
  • 81
6 votes
0 answers
237 views

A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?

This question is a follow-up to my previous question: "Rotated" version of the Atiyah-Hirzebruch spectral sequence In that question, I discussed two different spectral sequences for ...
Dominic Else's user avatar
16 votes
1 answer
808 views

"Rotated" version of the Atiyah-Hirzebruch spectral sequence

Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...
Dominic Else's user avatar
6 votes
1 answer
2k views

cohomology version of Cartan-Leray spectral sequence that deduces cup product

On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology: If $X$ is a connected pace on which the group $\...
Shiquan Ren's user avatar
  • 1,990
6 votes
2 answers
502 views

Interpretations of differentials in hypercohomology spectral sequences as Yoneda products

I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups. More ...
Christopher Drupieski's user avatar
5 votes
1 answer
3k views

Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves

I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...
user5395's user avatar
  • 545