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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Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is connected,...
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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 ...
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Vanishing terms in Leray spectral sequence

Let $f:X\to Y$ be a continuous map of complex manifolds. Consider $H^p(Y;R^qf_*V)$, where $V$ is a vector bundle on $X$, and $R^qf_*V$ are the higher direct images. (This is the $E_2^{p,q}$ term in ...
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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 ...
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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 ...
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How can I determine the Steenrod Square $Sq^2$ for complex projective space?

I am trying to learn about Steenrod Squares for algebraic varieties so that I can compute examples of complex topological K-theory using the Atiyah-Hirzebruch Spectral Sequence (AHSS). One of the key ...
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145 views

What is the integral cohomology of $\mathrm{PGL}_n(\mathbb{C})$ as a space?

The computation of the cohomology of $\mathrm{GL}_n(\mathbb{C})$ is one of the basic applications of the Serre spectral sequence, using the fiber bundle $\mathrm{GL}_{n-1}(\mathbb{C})\to \mathrm{GL}_n(...
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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: ...
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Spectral sequences: equivalence of exact couples and classic (?) method

By the 'classic' method I mean the construction of the spectral sequence associated to a filtration as found in Weibel's book p. 133-134. There is also the method of construction through exact couples ...
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Surjectivity of Edge morphism in A-H cohomology Spectral Sequence

As the title suggests, I'm interested in proving the following claim: Recall the AH-spectral sequence:$$ E_2^{pq}=H^p(X,\mathcal{H}^q(\ast)) \Longrightarrow \mathcal{H}^{p+q}(X)$$ and since $\...
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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 $K(ku)$ ...
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Kernel of Hurewicz map using the spectral sequence of the universal cover

In Davis & Kirk's Lecture Notes in Algebraic Topology, Exercise 159 reads: Use the spectral sequence of the universal cover to show that for a path-connected space $X$ the sequence $$\pi_2(X)\...
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Confused about Hypercohomology terminology and meaning

check this: Given a sheaf complex $F^\bullet$, let's say I want to compute the hypercohomology of this complex, if we consider the bicomplex of sheaves $C^\bullet(F^\bullet) = (C^p(F^q))\quad (p,q\in\...
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An Exercise in Allen Hatcher's book on Spectral Sequences

anyone knows how to solve Exercise 3 of Chapter 1 of Allen Hatcher's book on Spectral Sequences? The question is as follows: For a fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$ associated to ...
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Isomorphism between Dolbeault Cohomology groups

Let $M^n$ be a compact Kahler manifold and consider the product complex manifold $M \times \mathbb{C}^m$. By the Leray spectral sequence associated to the projection $\pi_{\mathbb{C}^m} : \mathbb{C}^m ...
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Serre Spectral sequence [Hatcher]

I'm having trouble understanding Serre Spectral sequence given in Hatcher. https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf In the beginning of the section, (Page 526 of the book, or rather, Pg 9 of ...
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Differentials on the second page of the spectral sequence of a first quadrant double complex

Suppose we have some (homological) double complex $\{E_{pq}\}$ with $p$ labelling the row and $q$ the column (is this standard or not?). Taking the homology of the vertical maps, it's easy enough to ...
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Showing when the spectral sequence associated to a filtered complex $C$ collapses using a similar complex $C\otimes\mathbb{F}[t]$

Let $\mathbb{F}$ be a field, and let $C=\bigoplus_{i,j\in\mathbb{Z}} C^{i,j}$ be a bigraded $\mathbb{F}$-vector space of finite total dimension. Suppose there are two differentials $d$ and $\widetilde{...
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What can we say about the category of spectral sequences?

For simplify, let's just consider the following category first: object: "spectral sequences", that is a sequence of pages $(E_r)_{r\ge0}$, where each page $E_r$ is a differential object in an abelian ...
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Edge homomorphisms for Atiyah-Hirzebruch spectral sequence for a spectrum $X$

I'm interested in some nice identifications (together with explanations) about the edge homomorphisms in the AHSS for a spectrum $X$. $$\times \times \times $$ Let $X$ be a connective spectrum, as ...
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Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth's theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,...
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When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
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Proving that Tor is a balanced functor using the derived category

At this end of this expository article on derived categories, R.P. Thomas says the following. There are two main advantages of this approach. Firstly that we have managed to make the complex $RF(...
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$h^{0, 1}$ of K3 surface (a priori non-Kahler)

I am trying to understand paper by Siu "Every K3 surface is Kahler". Let $M$ be a K3 surface. Siu wrote $H^1 ( M , \mathscr{O}_M ) =0$ without any references. It is written on fifth page. Maybe I ...
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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: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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A form of Künneth formula?

Problem (Exercise 15.12 in Bott & Tu's Differential Forms in Algebraic Topology) If $X$ is a space having a good cover, e.g., a triangularizable space, and $Y$ is any topological space, prove ...
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Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
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Examples of exact couples of abelian groups

Exact couples are really important when defining spectral sequences. However, I have never really seen a simple non-trivial example of two exact couples of abelian group with a morphism between them. ...
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Spectral sequences on their own right

I have begun to read a bit about Spectral Sequences. I have been studying some topics of algebraic topology and this "tool" came up from time to time. I didn't need to go deep, but I felt I had to ...
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Homology of $K(\mathbb{Z}/n,1)$ via Fibrations.

I would like to calculate the $\mathbb{Z}$-Homology of $K(\mathbb{Z}/n,1)$ via the Fibration $$K(\mathbb{Z},1)\hookrightarrow K(\mathbb{Z},1)\twoheadrightarrow K(\mathbb{Z}/n,1)$$ and the Serre-...
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Product structure in Cohomology Spectral sequence

In the Serre Spectral sequence, we know, the cup product structure induces a canonical product in all $E_{r}$ pages which is compatible with respect to the differential. I am trying to understand ...
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166 views

Morphisms in the derived category vs. morphisms on cohomology

Let $A$ and $B$ be complexes $R$-modules. Assume that $A^\bullet$ is bounded above and $B^\bullet$ is bounded below. Then there is a convergent spectral sequence $$\prod_{r\in \mathbb{Z}} \...
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Cohomology of Symmetric Group 3 using Lyndon-Hochschild-Serre spectral sequence

For the symmetric group $S_{3}$ we have the short exact sequence $$0\rightarrow C_{3}\rightarrow S_{3}\rightarrow C_{2}\rightarrow 0,$$ where $C_{n}$ is the cyclic group of order $n$. Using the Lyndon-...
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Construction of a sequence

Can I find a sequence $(f_j)_{j\in\Bbb{N}}\in C^{\infty}(\Bbb{R^+})$ such that : $$ \lim_{j\to\infty}\int^\infty_0 \big(\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\big)^2 rdr=0$$ and $$ \...
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Explicit formula for Higher Bockstein

The formula for the Bockstein $\beta:H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p\mathbb{Z})$ is $$\beta[c\otimes 1]=[\frac{1}{p}\partial c\otimes 1]$$ (McCleary page 456) How about for ...
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Questions regarding Bockstein Spectral Sequence (McCleary's book)

I have some questions regarding Bockstein homomorphism in John McCleary's book (pg 455-456). Q1) Is there a typo, is it supposed to be $\bar{u}\in H_n(X;\mathbb{Z}/r\mathbb{Z})$? Q2) How do we see ...
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Reference for Bockstein Homomorphisms / Spectral Sequence for Homology

Is there any book for Bockstein Homomorphisms specialising in the case of homology? So far the books I read (Hatcher, Munkres) discuss it for cohomology. I am aware that it is said to be similar for ...
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Showing that filtered complexes induce a spectral sequence

Let $(F_p C_\bullet)_p$ be a filtered complex of $R$-modules, with chain maps $$\cdots \xrightarrow{}C_n\xrightarrow{d_n}C_{n-1}\xrightarrow{}\cdots$$ such that $$\cdots \hookrightarrow F_pC_n\...
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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,...
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differential in AHSS for spin cobordism

According to these solutions, the differential $d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})$ is the dual of $Sq^2$. Why? This MO post asks a similar question (but about $d_3$ in ...
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Example 1.K in A User's Guide to Spectral Sequences

I'm having trouble with Example 1.K, p.25, of John McCleary's book A User's Guide to Spectral Sequences. Specifically, I don't understand how he defines the "obvious map" in the second paragraph : ...
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Serre Spectral Sequence I

I am trying to follow the proof in Kochman's Introduction to Stable Homotopy Theory, page 59. This will be first part of a series of post. (Serre Spectral Sequence) Let $R$ be a commutative ring ...
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118 views

Homology of a trivial fibre bundle via a spectral sequence

We have a fibration $F \rightarrow X \rightarrow B$. If $X = B \times F$, then combining Künneth formulas and universal coefficient theorem gives an isomorphism $H_n(X;G) \simeq \oplus H_p (B;H_{n-p}(...
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Reference Request: The Atiyah-Hirzebruch Spectral Sequence

I have just finished learning the Serre spectral sequence and I would like to learn about the Atiyah-Hirzebruch spectral sequence. Could someone suggest an accessible reference? Thank you in advance.
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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)...
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filtered modules (LNAT, Davis & Kirk)

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written: Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a ...
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Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence

Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$. In the beginning he states that there exisits ...
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Spectral sequence $B^1_{p,q}= B^\infty_{p,q}$ definition, nlab

In an introductory notes to spectral sequences, nlab, Definition 1.26 we define $$B^r_{p,q} = \partial(F_{p+r-1} C_{p+q+1})$$ and $$B^\infty_{p,q}= \partial(F_{p}C_{p+q+1})$$ So what is the ...
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Structure of module over Eilenberg MacLane spectrum

Let $HR$ be the Eilenberg-Maclane spectrum for a commutative ring $R$ and $M$ be a module over $HR.$ Then I want to prove that $M$ is a product of Eilenberg-Mac Lane spectra. Construction: Let $\...
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A question related to group cohomology and spectral sequences

It is actually a follow-up question of a mathoverflow question. I don't quite understand the answer there. I tried to compute the group cohomology of $H^n(\mathbb{Z}_4,\mathbb{Z})$ via the Lyndon-...