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11
votes
3answers
914 views

Multiplicative structure on spectral sequence

Let $E$ be a spectral sequence and assume that there is a product $E^{r}_{p_1,q_1} \times E^r_{p_2,q_2} \to E^r_{p_1+p_2,q_1+q_2}$ which satisfies the Leibniz rule (for all $p_i,q_i$, but $r$ fixed)....
10
votes
1answer
381 views

Multiplicative Structure of the Atiyah-Hirzebruch/Leray-Serre spectral sequence

This is related to this question (edit: now answered, see below). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ ...
83
votes
11answers
6k views

Spectral sequences: opening the black box slowly with an example

My friend and I are attempting to learn about spectral sequences at the moment, and we've noticed a common theme in books about spectral sequences: no one seems to like talking about differentials. ...
42
votes
5answers
5k views

Why are spectral sequences so ubiquitous?

I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with ...
8
votes
2answers
561 views

H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory

Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below $Z$ is the integer $\mathbb{Z}$, and $U(1)$ Abelian group ...
16
votes
2answers
959 views

Are the homology and cohomology Serre spectral sequences dual to each other?

If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...
5
votes
2answers
783 views

On two spectral sequences for the cohomology of a double complex

For a (bounded) double complex (of abelian groups or vector spaces) one can consider two spectral sequences that converge to the cohomology of the totalization: one can first compute either the ...
6
votes
1answer
524 views

What kind of spectral sequences come from double complexes?

Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex. My ...
7
votes
5answers
696 views

Triply graded spectral sequence?

As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply,...
5
votes
2answers
1k views

Construction of Serre Spectral Sequence

I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me. He starts with considering a double complex $C_{\bullet,\bullet}$ with ...
5
votes
1answer
850 views

Natural morphism appearing in Grothendieck spectral sequence

Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now ...
3
votes
1answer
473 views

Cohomology groups of quotient by finite group

I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question. Let $G$ be a finite group acting on a manifold $M$ without fixed ...
3
votes
2answers
259 views

cohomology algebra of braid spaces, configuration spaces

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for $p$...
2
votes
2answers
696 views

Conditions for the restriction $H^i(G,A)\to H^i(H,A)$ being surjective

I was wondering what the condition is for the restriction map (in group cohomology) $H^i(G,A)\to H^i(H,A)$ to be surjective. I am a little confused about when maps between cohomology groups are ...
2
votes
1answer
911 views

Generalized Beilinson spectral sequences

Assume we are workling on $\mathbb{P}^n$ for some $n\geq 1$ and we have a coherent sheaf $F$ on it. Then there are two (well known?) spectral sequences $E_r^{p,q}$ with $E_1$-term: $E_1^{p,q}=H^q(\...