Questions tagged [spectral-sequences]
372 questions
43
votes
6 answers
Simple examples for the use of spectral sequences
I'm looking for basic examples that show the usefulness of spectral sequences even in the simplest case of spectral sequence of a filtered complex.
All I know are certain "extreme cases", where the spectral sequences collapses very early to yield…
Hanno
- 2,736
20
votes
3 answers
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). Then it extends to a product
$E^{r+1}_{p_1,q_1}…
Martin Brandenburg
- 61,443
9
votes
0 answers
Mahowald uncertainty outside of homotopy theory
In homotopy theory there is the following informal idea:
The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named using homological algebra will be infinitely far…
user138661
5
votes
0 answers
Reference request: a Künneth spectral sequence map from equivariant K-theory to cohomology
The analogue of the Künneth formula for Borel $G$-equivariant cohomology can be obtained as the Eilenberg–Moore spectral sequence of a pullback
$\require{AMScd}$
\begin{CD}
(X \times Y)_G @>>> Y_G\\
@VVV @VVV\\
X_G @>>>…
jdc
- 2,984
4
votes
1 answer
Collapse of spectral sequence computing Equivariant cohomology
I have already posted this question on math.stackexchanges but I got no answer and I decided to post it here.
Let us consider the fibration
$$
M\hookrightarrow EG\times_{\varphi}M\twoheadrightarrow BG
$$
where $M$ is a $G-$space, $\varphi$ is the…
RiemannGauss
- 43
4
votes
1 answer
Comparing Spectral Sequences
There is a comparison theorem for spectral sequnces in Weibel's book (5.2.12) stating;
Assume $E_{p,q}$ and $\bar E_{p,q}$ converge to $H_* $ $\bar H_*$ respectively. Furthermore we have given a map $h: H_{*} \to \bar H_{*} $ compatible with a…
Grilo
- 235
2
votes
0 answers
Local coefficients system
Let $G$ be a compact group. Then there exists a universal principal $G$-bundle $G\rightarrow E_{G}\rightarrow B_{G}$. Let $X$ be a paracompact $G$-space and suppose that $X\rightarrow X_{G}\rightarrow B_{G}$ is fiber
bundle (called as the Borel…
Mehmet Onat
- 1,161
1
vote
1 answer
A Question on McCleary's book on Spectral Sequences
I am reading John McCleary's A User's Guide to Spectral Sequence and was quite confused about one result: On page 15 of the version I was reading, it says that if $E^{\star,\star}_2$ is the bigraded vector space in Example 1.E, then…
Zuriel
- 1,108
1
vote
0 answers
Spectral sequence with a column isomorphic to its homology
I have a first-quadrant spectral sequence $E^r_{p, q}$ of abelian groups of finite rank converging to $E^{\infty}_{p, q}$. We have $E^{\infty}_{p, q}=E^{\infty}_{r, s}$ if $p+q=r+s$. We also have $E^2_{p, 0}$ abstractly isomorphic to $E^{\infty}_{p,…
misha
- 287
1
vote
0 answers
Spectral sequence and HOM functor
I work with the category $A-{\rm Mod}$ of left modules over a unital ring $A$, but I could ask the same question for any abelian category with enough projectives. Let $M$ and $N$ be two $A$-modules and take projective resolutions $P_\star…
Paul Broussous
- 6,176