All Questions
19 questions with no upvoted or accepted answers
10
votes
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813
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On functoriality of the Leray spectral sequence
The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
- 40.3k
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 ...
- 965
7
votes
0
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541
views
Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
- 1,325
7
votes
0
answers
436
views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
- 353
6
votes
0
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237
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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 ...
- 863
6
votes
0
answers
723
views
On the multiplicative structure in spectral sequences.
Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice
topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow
F_i \rightarrow F_{i+1}\to \dots)$ be a bounded ...
- 21.8k
5
votes
0
answers
544
views
Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple explanation?)
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than ...
- 1,229
4
votes
0
answers
397
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}.
$$
Question 1: Is anyone aware of any references for ...
- 901
3
votes
0
answers
249
views
Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
3
votes
0
answers
101
views
Geometric filtration for Eilenberg-Moore spectral sequence
I'm reading the paper by Eilenberg-Moore (https://link.springer.com/content/pdf/10.1007/BF02564371.pdf) about the Eilenberg-Moore spectral sequence.
In section 11, they introduce the notion of ...
- 449
3
votes
0
answers
310
views
Functoriality of Leray homology spectral sequences of fibrations
Let $p\colon E\to B$ and $p'\colon E'\to B'$ be two fibrations. Assume for simplicity that $B,B'$ are simply connected. Let we have morphism of these fibrations, i.e. two continuous maps
$$f\colon E\...
- 21.8k
2
votes
0
answers
269
views
Dress' construction and Serre spectral sequence
Currently, I am reading Serre spectral sequence, given below, using Dress' construction.
Let $f:E\to B$ be a Serre fibration. Then, there is a first quadrant
spectral sequence $\big\{E^r,d^r\}_{...
- 632
2
votes
0
answers
486
views
An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology
I am currently reading Künneth spectral sequence, which is given below.
Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
- 632
2
votes
0
answers
108
views
Relating inflation maps from spectral sequences in lower and higher dimensions
The spectral sequence has some nice property.
Consider $ N \to G \overset{R}{\to} Q $ and $G/N=Q$. There is a spectral sequence $\{E^{p,q}_n, d_n\}$ with: (i) The differential is defined as a map $...
- 10.5k
2
votes
0
answers
216
views
completion and convergence of spectral sequence
I would like to understand the connection between $p$-adic completion and the strong convergence of a spectral sequence. Precisely, suppose $E^2_{s,t}\implies G_{s+t}$ is a first quadrant strongly ...
- 21
1
vote
0
answers
167
views
Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\begin{matrix}
\ F & \to & E &\to & B \\\
\downarrow & & \downarrow & & \downarrow \\\
\ F'...
- 103
1
vote
0
answers
93
views
Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
- 1,805
1
vote
0
answers
222
views
Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
- 191
1
vote
0
answers
184
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A naturality question concerning the universal coefficient spectral sequence
I am reading Hillman's book "algebraic invariants of links" and on page 20 he mentions the following universal coefficient spectral sequence.
Let X be a connected finite CW complex.Let $H$ be a ...
- 1,033