Questions tagged [spectral-sequences]
Spectral sequences compute homology groups by taking a sequence of approximations, and generalise exact sequences. They find application in algebraic topology.
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Collapsing of Kunneth formula for equivariant K-theory of homogeneous spaces.
Minami in "K-groups of symmetric spaces" (equations 1.1, 1.2) states the following, originally due to Hodgkins:
Suppose that $G$ is a compact connected Lie group such that $\pi_1(G)$ is ...
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Computing a Massey product.
Here is the question I am trying to solve:
Can anyone help me in showing me how to compute this Massey Product?
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Do we need $M$ simply connected for $H^*(E)=H^*(M)\otimes H^*(F)$ for $E$ a fibre bundle?
I'm following notations from sec. 14 Bott and Tu here:
Suppose $\pi :E\rightarrow M$ be a fibre bundle on a manifold with fibre $F$ and $\mathcal{U}=\{U_\alpha\}_\alpha$ be a good cover of $M$. Then ...
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Homology of the diagonal sequence of 3x3 commutative diagram of modules
Suppose we have modules $M_{i,j}$ over a commutative ring $R$ (or members of some abelian category, like quasi-coherent sheaves of modules),
and suppose that we have a 3x3 commutative diagram, where ...
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What is "spectral development" of a linear operator?
I want to find the "spectral development" (or spectral series, spectral decomposition, not sure how to translate to english) of the linear operator.
What is that? I can't even google it. I ...
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Finding Cohomology of a Fiber Bundle from Serre Spectral Sequence
I have a fiber bundle $F\rightarrow E\rightarrow \mathbb{RP}^3$ and I know the cohomology groups of $F$ and $\mathbb{RP}^3,$ $H^0(F,\mathbb Z)=H^3(F,\mathbb Z)=\mathbb Z$, $H^2(F,\mathbb Z)=(\mathbb Z/...
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Degeneration of a spectral sequence
I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrechts. On page 140, he is written that due to some results the spectral sequence
$$E^{p,q}_2=H^p(X\times X,\...
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Simplicial resolutions and the homotopy fixed points spectral sequence
According to this set of notes, which says (paraphrasing):
"To construct the homotopy fixed points spectral sequence, we use the fact that the bar construction gives a simplicial resolution of $(...
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If $E_r^{p,q}\cong {E_r'}^{p,q}$ for $r=\{r_0,\infty\}$ and all $p,q$, then $E_r\cong E_r'$
Let $\{E_r\}$ and $\{E_r'\}$ be two (cohomological) spectral sequences of vector spaces (to avoid extension problems).
Suppose that, for certain $r>0$,
$$
E_r^{p,q} \cong {E_r'}^{p,q} \qquad \...
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Proving Hochschild-Serre spectral sequence (of group (co)homology) via Grothendieck spectral sequence
Let $G$ be a group and $N$ be its normal subgroup. Let $A$ be a $G$-module, then we have the Hochschild-Serre spectral sequence
$$
E_2^{p,q} = H^p(G/N, H^q(N,A)) \Rightarrow H^{p+q}(G,A).
$$
and
$$
E_{...
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Tensor product commutes with homology for a "flat chain complex" -- A step for proving the universal coefficient theorem
I'm trying to prove the universal coefficient theorem (UCT) for homology and cohomology using spectral sequences following Section 5.6 of Weibel's book Introduction to homological algebra and coming ...
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Frolicher spectral sequence of a surface
Let $ S $ be a compact complex surface. Can anyone provide a proof (or a reference to a proof) of the fact that the Frolicher spectral sequence of $ S $ degenerates at $ E_1 $?
Note, no assumptions on ...
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How can a decomposable element be transgressive without violating Leibniz rule?
Consider the fibration $K(\Bbb Z_2,1)\to \ast\to K(\Bbb Z_2,2)$. As we know from Serre (see Hatcher's SSAT), $H^*(K(\Bbb Z_2,n);\Bbb Z_2)$ is the polynomial ring on generators $\operatorname{Sq}^I(\...
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What is the spectral sequence associated to this filtration on the de Rham complex?
I am trying to calculate some relative de Rham cohomology, but I am not too skilled with hypercohomology or spectral sequences, and the situation becomes more complicated because (1) the base is not ...
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Edge maps in Grothendieck spectral sequence
Let $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ be abelian categories such that $\mathcal{A},\mathcal{B}$ have enough injectives (we can WLOG assume they're categories of modules for the following) ...
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Functoriality of filtration's spectral sequence
I read in McCleary's "User's Guide to Spectral Sequences" that a morphism between filtered chain complexes (say $f:X\to Y$, and say that both are as nice as needed, even first quadrant would ...
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Spectral Sequence induced by filtered dg modules
I am a bit confused about Theorem 2.6 in McCleary's "A USer Guide to Spectral Sequence". The theorem says that
Each filtered dg module $(A,d,F^*)$ (with decreasing filtration and $\deg(d)=1$...
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Homology groups of unit tangent bundle
Let $M$ be the closed connected orientable $n$-dimensional manifold and let $S(M)$ be the unit tangent bundle i.e. a set $\lbrace (x,v)| x\in M,\ v\in \mathrm{T}M,\ |v|=1\rbrace$. I want to compute ...
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spectral sequence of filtration for homotopy groups
I was listening to a course on algebraic topology. Our professor constructed a spectral sequence for a filtered space which computes homology. The construction only used relative homology groups and ...
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Is $B^{\infty}E_r^{p,q}$ always a subobject of $Z^{\infty}E_r^{p,q}$?
I'm reading about spectral sequences from various places (e.g. "The heart of cohomology" by Goro Kato, the Stack Project, Wikipedia), and I have a doubt.
We consider a bigraded cohomological ...
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Spectral Sequences - John McCleary Example 1.A (First Quadrant Topological Spectral Sequence)
On John McCleary's book "A user's guide to sepctral sequences", on page 6 he gives the following example:
Example 1.A. Suppose that there is a first quadrant spectral sequence of ...
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A mistake in Grothendieck's Tôhoku paper? Theorem 5.2.1
I was reading the ``Sur quelques points d'alegbre homologique'' English translation when I came across the spectral sequences for equivariant cohomology shown in Theorem 5.2.1 (in the original French ...
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Serre spectral sequence, question on Hatcher's proof
I'm trying to study the proof of Serre spectral sequence in Hatcher's pdf on spectral sequences. On page 530 he says if we have a Hurewicz fibration $\pi:X\rightarrow B$ with $B$ filtered by skeletons ...
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Relationship between mapping cones and spectral sequences
I'm learning homological algebra using Weibel's book. And I have trouble in exercise 5.4.4:
Let $f : B \to C$ be a map of filtered chain complexes. For each $r \geq 0$, define a filtration on the ...
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Equivalent definitions of n-acyclic morphisms / Problem with spectral sequence
I'm trying to understand section (VI.4) on smooth base change in Milne's Étale Cohomology.
He defines a morphism $g \colon Y \longrightarrow X$ to be $n$-acyclic (I only care about the case $n \geq 0$)...
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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 ...
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Step in building up spectral sequences
I'm currently reading Timothy Chow's article defining spectral sequences, and I'm stuck on one step. Here is a summary:
We have a finite-dimensional chain complex $... \overset{\partial}{\...
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Explicit Description of the map from Adam's $E_2$ term to Continuous Group Cohomology
I am currently working through the paper ''The Homotopy of $L_2V(1)$ for the Prime $3$" by Goerss, Henn, Mahowald which can be found in the book Categorical Decomposition Technique in Algebraic ...
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Confusions about the statement of Leray's theorem for spectral sequences
I am trying to understand more precisely some aspects of the statement of Leray's theorem on spectral sequences, as stated in Loring Tu's "Introduction to Equivariant Cohomology". The ...
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Compute $\pi_{3+k}(S^3)$ with $k\le 3$ using Steenrod squares
As an application of the theorem
(Serre) The cohomology ring $\mathcal{H}^*(K(\mathbb{Z}_2,n);\mathbb{Z}_2)$ is isomorphic to $\mathbb{Z}_2[Sq^I(\iota_n)]$ where $\iota_n$ is a fondamental class of $\...
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Sheaf cohomology of a cover (please find my mistake)
This question will be a bit long because its subject is quite technical.
First some preliminaries: If you have a cover of a space that is open or locally finite and closed you can try to compute sheaf ...
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Proof of Kunneth formula over a commutative coefficient ring by spectral sequences
In "Differential forms in algebraic topology" by Bott & Tu, I am asked to prove the Kunneth Formula for singular cohomology by spectral sequence:
Exercise 15.12 (Kunneth Formula for ...
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Atiyah-Hirzebruch spectral sequence for twisted K-theory for infinite-dimensional CW-complexes
I'm writing a thesis that runs an AHSS for twisted K-theory over an Eilenberg-MacLane space, namely an infinite-dimensional CW-complex.
In Atiyahs works Twisted K-Theory and Twisted K-Theory and ...
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Cohomology of projective space using a spectral sequence
This is exercise 18.3.D of Vakil's Foundations of Algebraic Geometry.
It's to prove that $H^i(\mathbb{P}^n_A, O(m))$ are free, and to compute the dimensions, where $A$ is any commutative ring.
The ...
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Cohomology ring of space localization at a prime
For $X$ a path-connected abelian CW complex with finitely many cells in each dimension, and $X_{(p)}$ be the localization of $X$ at prime $p$ (so that $\tilde{H}_*(X_{(p)}) = \tilde{H}_*(X)\otimes \...
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About $E^\infty$ spectral sequence.
While learning spectral sequence, I have trouble realizing where the terms $E^k$ get stationary
$E^\infty$.
For bounded chain complexes, it happens since the filtration is finite. But, for example, ...
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Spectral Sequences from Derived Categories
I'm trying to understand how derived categories "replace" spectral sequences. More specifically the derived category statement of Grothendieck Spectral Sequence vs the normal version. I been ...
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Cech Spectral Sequence for Cohomology with Compact Support
Using the Cech spectral sequence, we can compute the cohomology of a topological space via the cohomology of open subsets. Is there a version of this where we use cohomology with compact support? Let ...
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Differentials in the spectral sequence associated to a double complex
Edit: I would very much like someone to comment on this. I have worked through Vakil's notes Spectral sequences: friend or foe? and I believe my description is correct. But there are so many indices ...
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Serre spectral sequence and universal coefficient theorem
Let $F \rightarrow E \rightarrow B$ be a Serre fibration. Assume that the cohomological spectral sequence (with integer coefficients) $E_2^{p,q} = H^p(B; H^q(F)) \Rightarrow H^*(E)$ degenerates so ...
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Cohomology of $SU(n)$
Context:
I am trying to show that $H^{\ast}(SU(n))\cong\Lambda[x_{3},\dots x_{2n-1}]$.
My approach was to use induction- $SU(2)\cong S^3$ so $H^{\ast}(SU(2))\cong H^{\ast}(S^3)\cong\Lambda[x_{3}]$ so ...
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Proposition 5.7.6 Weibel Homological Algebra
Let $F:\mathcal{A} \rightarrow \mathcal{B}$ be a right exact functor and suppose $\mathcal{A}$ has enough projectives.
Recall that the left hyper-derived functors of $F$ are defined as
$\mathbb{L}_{i}...
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Spectral sequence which does not converge weakly
I am looking for a simple (if possible) example of a spectral sequence of a filtered complex which does not converge weakly. I don't have great intuition for this, so I'm looking for a simple example. ...
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Final step in Hatcher's construction of the Serre spectral sequence for homology
$\require{AMScd}$
I'm reading Hatcher's chapter/former book on spectral sequences. I'm currently struggling to understand the last substantive step of the proof of Theorem 5.3 (the Serre spectral ...
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On the Leibniz rule in the Serre spectral sequence
Consider the $\mathbb{Z}/2$-cohomological Serre spectral sequence associated to the path fibration
$$ K(\mathbb{Z}/2, 1) \simeq \Omega K(\mathbb{Z}/2, 2) \longrightarrow PK(\mathbb{Z}/2, 2) \...
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Action of fundamental group of the base space on the homology of the fiber
The following was extracted from Hatcher’s book of algebraic topology:
Could you explain me with more detail what is this action of the fundamental group of the base space of a fibration $\pi: E \to ...
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Some confuse in spectral sequence and its calculate
We have the Leray's theorem:
Let $\pi:E\longrightarrow B$ be a fiber bundle with fiber $F$ over a simply connected base space $B$. Assume that in every dimension $n$, $H^{\ast}(F)$ is of finite rank ...
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homomorphism of $G/N$-modules
Let $G$ be a group with $N$ a normal subgroup (not necessarily of finite index). Let $Q$ and $A$ be $G$-modules and $P$ be a $G/N$-module.
I want to make sense of the term ${\mathrm{Hom}}_{G/N}(P, {\...
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Interpreting some cohomology groups
Let $C$ be a smooth geometrically integral curve over a number field $k$, we do not assume $C$ to be proper, i.e., $C$ is not projective.
Under the spectral sequence $H^p(k,H^q_{\mathrm{et}}(C_{\bar{k}...
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Computing cohomology of Cech-De Rahm Complex
Bott & Tu use what they call the "Cech-de Rahm complex" a lot, which is a double complex that uses the Cech differential horizontally and the de rahm differential vertically, with ...
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