# Tagged Questions

The spectral-sequences tag has no usage guidance.

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votes

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+50

### Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...

**1**

vote

**0**answers

54 views

### Reference request for Leibniz rule and spectral sequences

Suppose $A_*$,$B_*$, and $C_*$ are chain complexes equipped with filtrations and a map $m:A_* \otimes B_* \to C_*$ respecting these filtrations. I am looking for a reference for the fact that the map $...

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**0**answers

86 views

### Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum

This is a following up question of Sphere spectrum, Character dual and Anderson dual.
What are the differences and the significances of the following:
(1). Homotopy classes of maps from a Thom ...

**6**

votes

**2**answers

390 views

### Sphere spectrum, Character dual and Anderson dual

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
What is the significance of ...

**2**

votes

**0**answers

92 views

### Edge map in derived categories

Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...

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74 views

### Twisted spin cobordism v.s. KO theory in low dimensions

Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like
$${\Omega_d^{(\mathrm{spin} \times G)/N}},...

**5**

votes

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114 views

### Spin cobordism v.s. KO theory in low or in any dimensions

It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...

**3**

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**0**answers

121 views

### Reference for specific detail on Serre spectral sequence

In "A primer on spectral sequences" http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf (by J.P.May apparently, although no name is given in the pdf) I found a very detailed version of the ...

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88 views

### Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$

I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...

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93 views

### Bordism groups and a short exact sequence

Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...

**5**

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**0**answers

99 views

### The $E_2$-page of the May spectral sequence

I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS.
At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on ...

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votes

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113 views

### 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 ...

**3**

votes

**0**answers

106 views

### What is the filtration in Leray's spectral sequence?

Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration
$$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...

**4**

votes

**1**answer

129 views

### Transgression image and Serre spectral sequence for tori

Let $\mathbb{K} \subset \mathbb{T}$ be two tori acting on a topological space $X$ (with all the properties you want). We use the notations $$X_{\mathbb{T}} := (X \times E \mathbb{T}) / \mathbb{T}, \...

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vote

**0**answers

46 views

### Continuous maps vs filtrations construction of the Leray spectral sequence

The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more ...

**4**

votes

**1**answer

158 views

### How are p-primary parts determined for odd p?

When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away.
How are odd primary part calculations done in relation ...

**2**

votes

**0**answers

47 views

### Condition for a map to carry over to Leray spectral sequences

I am trying to understand the conditions for two Leray spectral sequences to be related by a map.
Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps of topological spaces (with ...

**5**

votes

**0**answers

84 views

### Does Serre spectral sequence degenerate for fiber bundles that are trivial restricted to the boundary?

Let $W$ be an $n$-dimensional manifold with boundary. Assume $W$ admits an exhausting Morse function with indexes smaller or equal to $k$. Let $\pi: E\to W$ be fiber bundle with compact fiber $F$ and $...

**2**

votes

**1**answer

103 views

### Construction of differentials in the spectral sequence for double complexes

I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (...

**2**

votes

**1**answer

141 views

### Why only consider decreasing filtrations on cochain complexes?

When reading various literature on spectral sequences one always comes across two setups:
A chain complex with an increasing filtration
A cochain complex with a decreasing filtration
My question is ...

**2**

votes

**0**answers

189 views

### Degeneration of relative Hodge-de Rham spectral sequence

$$\require{AMScd}$$
$$\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\Hdr}{H_{\mathrm{dRh}}}
\newcommand{\tensor}{\otimes}
\newcommand{\Ohol}{\mathcal{O}}$$
Please excuse that ...

**3**

votes

**1**answer

277 views

### Thom space, homotopy group and cohomology group

In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...

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95 views

### Gel'fand's and Fuks' calculation of cohomology of formal vector fields - isomorphic spectral sequences yield isomorphic cohomology?

In this book (and, in what seems to be an equivalent fashion, in this article), Gel'fand and Fuks calculate the Lie algebra cohomology of the formal vector fields $W_n$ on $\mathbb{R}^n$ with trivial ...

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votes

**1**answer

127 views

### Does the Serre spectral sequence of the Fadell-Neuwirth fibration collapse if there is a cross-section?

I had asked a vague question in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by Mike Miller in the comments, but received no answers. Here I will specialize to an interesting ...

**6**

votes

**2**answers

261 views

### Homology spectral sequence for function space

The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\...

**3**

votes

**1**answer

355 views

### What bigrading is used in this spectral sequence?

I am reading this paper of Positselski and Vishik, in particular the main theorem: the cohomology algebra of a conilpotent algebra (i.e. the cohomology of its cobar construction) is Koszul if it ...

**6**

votes

**1**answer

209 views

### to compare cohomologies of fibers of two fiber bundles

Consider the following commutative diagram of the fiber bundles $%
F\rightarrow E\rightarrow B$ and $F^{\prime }\rightarrow E^{\prime
}\rightarrow B^{\prime }$ where $B^{\prime }$ is simply connected ...

**2**

votes

**2**answers

209 views

### homology of a base space of a a fiber sequence

Suppose we have a fiber sequence of connected spaces $A\rightarrow B\rightarrow C$ and suppose we know the homology of A and B, is there a homological spectral sequence converging to the homology of $...

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vote

**0**answers

52 views

### maps between two Leray spectral sequences based on maps on cochain complexes

Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...

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votes

**0**answers

90 views

### Spectral Sequence for Twisted K-theory

Atiyah and Segal wrote in their Twisted K-theory that one can compute twisted K-theory using a spectral sequence similar to an Atiyah-Hirzebruch spectral sequence. They claimed that for any twisting $[...

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vote

**0**answers

92 views

### Maps between Leray spectral sequences

Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...

**4**

votes

**1**answer

144 views

### The converse of Vietoris-Begle theorem

It is well known the following result:
Lemma: Let $F\rightarrow E\rightarrow B$ be a fibration with $B$ connected and
simply connected. Suppose that $F$ is $n$-acyclic, i.e. $H^{p}\left( F;%
%...

**8**

votes

**2**answers

334 views

### To compare the total, base and fiber spaces of two fiber bundles

Consider the following commutative diagram of the fiber bundles $%
F\rightarrow E\rightarrow B$ and $F^{\prime }\rightarrow E^{\prime
}\rightarrow B^{\prime }$ where $B^{\prime }$ is simply connected ...

**15**

votes

**1**answer

420 views

### “Rotated” version of the Atiyah-Hirzebruch spectral sequence

Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...

**2**

votes

**1**answer

107 views

### Leray spectral sequence for continuous functions on pairs of topological spaces

Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$.
The Leray spectral sequence (with complex ...

**5**

votes

**1**answer

214 views

### On the Leray spectral sequence and sheaf cohomology

I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...

**2**

votes

**0**answers

87 views

### 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 ...

**5**

votes

**1**answer

128 views

### Torsion in the integral cohomology of $BPU_{n}$

I would like to prove that the integral cohomology of $BPU_{n}$ the classifying space of the projective unitary group of order $n$ has $n-$primary torsion.
We have a fiber sequence of the form $BSU_{...

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votes

**0**answers

139 views

### Is there a spectral sequence of Atiyah's topological KR-theory that can be used to compute basic examples?

For Segal's complex $G$-equivariant $K$-theory, it is well-known that there is an Atiyah-Hirzebruch spectral sequence. If say $G$ a finite group and $X$ a finite CW-complex, the second page of this ...

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votes

**1**answer

145 views

### Reference request: cohomology of Eilenberg Maclane spaces with $p$-local groups

In Rudyak's 'On Thom Spectra, Orientability, and Cobordism', the following fact is used:
Let $\mathbb{Z}_{(p)}$ be $\mathbb{Z}$ localized at the ideal $(p)$. Let $\pi,\tau$ be two cyclic $\mathbb{...

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votes

**1**answer

304 views

### Relating two different approaches to the Atiyah-Hirzebruch Spectral Sequence

Given a (for simplicity connective) spectrum $E$ and a pointed CW-space $X$ there is "the" (homological) Atiyah-Hirzebruch spectral sequence
$$E_{pq}^2 = \tilde{H}_p( X, \pi_q(E)) \Rightarrow \pi_{p+...

**2**

votes

**1**answer

184 views

### Étale cohomology of tensor product

Let $X$ be a smooth projective variety over a field $k$.
Suppose we have étale abelian sheaves $A, B$ on $X_{\rm ét}$ such that
$$H^j(X_{\rm ét}, A),\ H^j(X_{\rm ét}, B)$$
are finitely generated ...

**8**

votes

**1**answer

293 views

### fibrations of classifying spaces - Leray Hirsch Theorem converse

Let $G$ be a topological group and let $H$ be a closed subgroup. Assume that $G \rightarrow G/H$ is a principal $H$-bundle. We have a fibration of classyifing spaces
$$G/H \rightarrow BH \rightarrow ...

**5**

votes

**0**answers

151 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 ...

**2**

votes

**1**answer

163 views

### Hochschild-Serre filtration and etale cohomology

I encountered the Hochschild-Serre spectral sequence in étale cohomology
$$H^i(\text{Gal}(\overline{k}/k), H^j_{et}(X_{\overline{k}}, F))\Rightarrow H^{i+j}_{et}(X_{{k}}, F)$$
How is the filtration ...

**6**

votes

**1**answer

212 views

### Calculating topological index

Consider the space $X=BSL(8,\mathbb{C})/(\mathbb{Z}/2)$. The topological Brauer group of $X$ is given by $Br_{top}(X)=Tor(H^{3}(X;\mathbb{Z}))=\mathbb{Z}/2$. I'm studying concepts of topological ...

**6**

votes

**1**answer

458 views

### Hodge Numbers and Leray Spectral Sequence

Mark Gross' notes survey of SYZ fibrations and toric degenerations begin by explaining why dual torus fibrations interchange Hodge numbers. But he defined the Hodge numbers in an unusual way
$$h^{p,q}(...

**7**

votes

**1**answer

183 views

### Does a filtered A_N algebra give rise to a multiplicative spectral sequence?

The question is pretty much in the title. It is a classical fact that a filtered dga gives rise to a multiplicative spectral sequence. It is claimed in Remark 4.1 of https://arxiv.org/pdf/1410.6728....

**3**

votes

**1**answer

160 views

### Strong convergence of whole-plane spectral sequences

I am trying to understand strong convergence for whole-plane spectral sequences in the paper by J.Boardman:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-...

**4**

votes

**0**answers

169 views

### Cohomology of double complex with exact rows

Let $(C^{p,q},d_h,d_v)$ be an unbounded double complex of modules of some algebra $A$ in an abelien category. Let $d_h,d_v$ be the horizontal and vertical differential respectively. Suppose that
(1) ...