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.

Filter by
Sorted by
Tagged with
2
votes
0answers
58 views

Spectral Sequence associated to a filtration abuts because we can find closed representatives

Let $(K,D)$ be a differential complex of abelian groups, and $K = K_0 \supset K_1 \supset K_2 \supset \cdots \supset K_{p+1} = 0$ a filtration of $K$ by sub-complexes. Let $(E^{r},d^r)_{r\ge 1}$ be ...
7
votes
0answers
104 views

Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy answer,...
3
votes
0answers
35 views

How networks with high largest eigenvalues are more robust?

In the literature, it sometimes indicates that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust. Robustness here is relevant to link/...
8
votes
1answer
1k views

Calculate the cohomology group of $U(n)$ by spectral sequence.

Here $U(n)$ is the unitary group, consisting of all matrix $A \in M_n (\mathbb{C})$ such that $AA^*=I$ Problem How to calculate the integer cohomology group $H^*(U(n))$ of $U(n)$? What if $O(n)$ ...
2
votes
1answer
243 views

Tensor product of homology equivalences

Let $f : C \to C'$ and $g : D \to D'$ be chain maps of non-negative chain complexes of $R$-modules, where $R$ is any commutative ring. Assume that $f$ and $g$ are homology equivalences. Is the same ...
3
votes
1answer
104 views

Computation of Ext as a cohomologies of certain complex

Let $R$ be a ring and $K^\bullet$ be a complex of $R$-modules such that $K^\bullet$ has only one nontrivial cohomology $H^0(K^\bullet)=M$. Suppose that $R$-module $N$ is such that $\text{Ext}^i_R(K_j,...
1
vote
1answer
167 views

Multiplicative spectral sequence

I have a simple question regarding the definition of a multiplicative spectral sequence, which I couldn't answer myself by looking at the definitions in various texts: Is the product assumed to be ...
15
votes
1answer
674 views

Group cohomology of dihedral groups

If $m$ is odd, the group cohomology of the dihedral group $D_m$ of order $2m$ is given by $$H^n(D_m;\mathbb{Z}) = \begin{cases} \mathbb{Z} & n = 0 \\ \mathbb{Z}/(2m) & n \equiv 0 \bmod 4, ~ n &...
4
votes
0answers
232 views

Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
4
votes
1answer
133 views

How to define the natural map on the second page of a spectral sequence?

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
13
votes
1answer
442 views

Vector space identity from Chow's “You Could Have Invented Spectral Sequences”

In Chow's You Could Have Invented Spectral Sequences (3rd page, left column) appears the following isomorphism of vector spaces: $$\frac{Z_d}{B_d}\cong \frac{Z_d+C_{d,1}}{B_d+C_{d,1}}\oplus \frac{Z_d\...
5
votes
0answers
153 views

Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title $E_2^{p,q}=\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{...
2
votes
1answer
71 views

The spectral sequence of the path fibration of $S^2$

Bott and Tu use the fibration $\Omega S^2 \to PS^2 \to S^2$ to compute the cohomology of $\Omega S^2$. They do this by looking at the (Serre?) spectral sequence. Then $E_2^{p,q} = H^p(S^2,H^q(\Omega ...
2
votes
1answer
410 views

Spectral sequence page isomorphism

Suppose we have a map of spectral sequences $\{E_{p,q}^r,d^r\}\to \{{E'}_{p,q}^r,d'^r\}$, both generated from total chain complexes, $C$ and $C'$ respectively, such that for some $r$ the map between ...
5
votes
1answer
208 views

Module structure in the Serre spectral sequence of the Borel construction

Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its ...
2
votes
0answers
130 views

cohomology of orbit space by a free group action

Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$. Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get $H^*(B;...
6
votes
1answer
256 views

Homotopy of double chain complexes

Consider complexes $(A,d_1), (A',d_1)$, $(C,d_2), (C',d_2)$ and morphisms $f_1,f_2: (A,d_1)\to (A',d_1)$ and $g_1,g_2: (C,d_2)\to (C',d_2)$ of degrees $0$. Consider the functor $(-\otimes-)$, then $$...
4
votes
1answer
193 views

Does every elliptic cohomology theory represent a complex-orientable $E_\infty$-ring spectra and vice-versa?

The last paragraph in Two-Vector Bundles and Forms of Elliptic Cohomology remarks that neither the spectrum $K(ku)$ nor tmf is complex orientable. In the case of $K(ku)$: "...the unit map for $K(ku)$ ...
4
votes
1answer
168 views

Examples spectral sequence

I have to make a talk about spectral sequences, so I'd like to present some concrete examples of computation, after the general definition. I'd like to present three examples of spectral sequences: ...
13
votes
0answers
460 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
1
vote
1answer
326 views

turning a map into a fibration

In Allen Hatcher's book Spectral Sequence page 29 Example 1.18, What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a ...
3
votes
1answer
359 views

Spectral sequence of a filtered complex: convergence conditions and abelian categories

There is a theorem that if given a filtered complex and the filtration is bounded then there is a spectral sequence whose 0th and 1st page have specific forms and the sequence converges to (co)...
6
votes
1answer
303 views

Derived functor vs. spectral sequence

I heard many times that because of introducing derived category, we can avoid cumbersome spectral sequence. However, I don't quite understand its meaning. Here is a precise example people talking ...
0
votes
1answer
62 views

Question about spectrum versus spectral sequences

What is the difference between spectrum sequences and spectral sequences? Are they considered to be the same? I know that the spectrum sequence of a real number $\alpha$ is the sequence that has $[n\...
3
votes
0answers
369 views

Gysin sequence and Serre spectral sequence

Given an oriented $S^k$ bundle $E$ over a compact manifold $M$ we get the Gysin Sequence (I am interested in the DeRham cohomology). We can obtain this sequence from the Serre Spectral Sequence if the ...
6
votes
0answers
619 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if $\check{H}^q(...
3
votes
0answers
185 views

Spectral Sequence and Stiefel Manifold

Let $Spin(3)$ be embedded in $Spin(5)$ by the spin embedding then we have a fibration: $$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3)$$ Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel ...
20
votes
4answers
1k views

Useful fibrations

What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to S^{...
3
votes
0answers
75 views

Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
4
votes
1answer
445 views

extension problem of a spectral sequence

From Hatcher's SSAT, If the coefficient group $G$ is a field, then $H_n(X;G)$ is the direct sum $\oplus_p E^\infty_{p,n-p}$ of the terms along the $n^\text{th}$ diagonal of the $E^\infty$ page. For ...
1
vote
0answers
41 views

The product of $E_2$-degenerate spectral sequences also $E_2$-degenerates?

Assume the Leray spectral sequence of a map $f_i:X_i\rightarrow B_i$ $E_2$-degenerates for $i=1,2$. Is it true that the Leray spectral sequence of the map $f_1\times f_2:X_1 \times X_2 \rightarrow B_1 ...
2
votes
0answers
36 views

what is the natural map from BP to $HF_p$?

what is the natural map from BP to $HF_p$? BP is the Brown-Peterson spectrum and $HF_p$ is Eilenberg Maclane spectrum. I am trying to learn the connections between ASS and ANSS. This map should ...
7
votes
0answers
250 views

Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
4
votes
1answer
586 views

why $HF_p$(Eilenberg Mac Lane spectrum) smash X (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$?

why $HF_p$(Eilenberg Mac Lane spectrum) smash $X$ (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$? This is Prop 2.1.2 (g) in Ravenel's green book. Can ...
1
vote
1answer
42 views

What is the topology of $E^*(E)$ where $E$ is a ring spectrum?

In Adams' lectures on generalised cohomology Page 51, it states $E^*(E)$ is a topology ring. I do not know the topology on it. the reference that Adams offered there is by Novikov in Russian. Can ...
2
votes
1answer
86 views

Example leading to spectral sequence

I am reading A User's Guide To Spectral Sequences and I don't understand the example in the informal introduction chapter: We want to compute $H^*$, where $H^*$ is a graded $R$-module or a graded $...
2
votes
2answers
193 views

What is the difference between multiplication and direct sum on homotopy groups of spheres?

In Allen Hatcher's book Algebraic topology he states that factoring out 2-torsion $\pi_{i}(S^{2n})\cong\pi_{i-1}(S^{2n-1})\times\pi_{i}(S^{4n-1})\:\forall n$ but in his book Spectral Sequences in ...
2
votes
0answers
66 views

The differentials of a spectral sequence

Suppose we are on the $E_r$ page and the lattice either consists of 0 or $\mathbb{Q}[x,y]$ in each entry. Suppose in particular that the points $(p,q)$ and $(r, s)$ (and "their codomains") are equal ...
3
votes
0answers
77 views

Reference request: where can I find illustrative, concrete examples of the use of the Eilenberg–Moore spectral sequence?

Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle $S^3 \to S^2$...
1
vote
0answers
88 views

The Poincare series for a bigraded vector space

I don't understand this computation (this is from McCleary's book on spectral sequences, p.15): The Poincare series of a (locally finite) bigraded vector space $E^{\ast,\ast}$ is defined as $P(E^{\...
2
votes
1answer
52 views

What does the $\Omega$ represent in $\Omega S^{n}$?

To put my question in context, I'm reading Hatcher's book on Spectral sequences is which is say " The suspension homomorphism $E$ is the map on $pi_{i}$ induced by the natural inclusion map $S^{n}\...
12
votes
1answer
896 views

Are there spectral sequences for calculating homology or cohomology of homotopy (co)limits?

Suppose my nice topological space $X$ is the homotopy colimit $$\operatorname{hocolim}D\cong X$$ of a diagram $D\colon I\to \mathbf{Top}$ and the homotopy limit $$\operatorname{holim}E\cong X$$ of a ...
1
vote
0answers
79 views

Spectral sequences to involve together two ideals of a ring

I'm looking for spectral sequences to involve together two ideals of a ring. For instance, let $I,J$ be two ideals of Noetherian ring $R$ and $M$ be a finite $R$-module then we have the following ...
10
votes
0answers
307 views

Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...
4
votes
2answers
598 views

definition of convergence of a spectral sequence

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant and ...
2
votes
1answer
286 views

An exact homology sequence associated with a principal SO(n) bundle

Suppose $P$ is a principal $SO(n)$ bundle, X is its base space. Why is there an exact sequence in homology groups $$ 0 \to H^1(X;\mathbb{Z}_2) \to H^1(P;\mathbb{Z}_2) \to H^1(SO(n);\mathbb{Z}_2)\to H^...
3
votes
0answers
117 views

Serre spectral sequence and locally constant coefficients

I have a brief question - In the Serre Spectral sequence for a fibration $$F \rightarrow E \rightarrow B$$ one can require, to avoid using local system of coefficients, that the action $\pi_1(B)$ on $...
1
vote
0answers
152 views

Question About Transgression

I have been working on this question here. Here is the setup: First, all cohomology groups are assume to be with $\mathbb{Q}$ coefficients. We assume that $H^*(K(\mathbb{Q},n))=\mathbb{Q}[x]$, with ...
12
votes
2answers
2k views

Serre Spectral Sequence and Fundamental Group Action on Homology

I am looking at my algebraic topology notes right now, and I am looking at our definition for the Serre Spectral Sequence and it requires that the action of the fundamental group of the base space of ...
2
votes
0answers
380 views

filtration on the cohomology of a complex

Let $K^\bullet$ be a complex and let $F_I$ and $F_{II}$ be two filtrations on it. suppose $F_I^i K^n$ intersects $F_{II}^i K^n$ trivially. It then follows that in the induced filtration on the ...