# Questions tagged [spectral-sequences]

The spectral-sequences tag has no usage guidance.

**7**

votes

**1**answer

135 views

### Invariants in relative cohomology and compact support cohomology of the quotient

Let $\cal H$ be the Poincare upper half-plane and $\overline {\cal H}$ the union of $\cal H$ with the set of cusps $\bf P^1 (\bf Q)$, provided with its usual topology. Let $\Gamma$ a congruence ...

**6**

votes

**0**answers

120 views

### Degeneracy of the Serre Spectral Sequence

I am learning the Serre spectral sequence and I am intrigued about the degeneracy of such at the $E_2$-page. Assuming field coefficients in cohomology for simplicity.
In fact, for a Serre fibration $...

**1**

vote

**0**answers

31 views

### Spectral sequence with a column isomorphic to its homology

I have a first-quadrant spectral sequence $E^r_{p, q}$ of abelian groups of finite rank converging to $E^{\infty}_{p, q}$. We have $E^{\infty}_{p, q}=E^{\infty}_{r, s}$ if $p+q=r+s$. We also have $E^...

**4**

votes

**1**answer

255 views

### Kernels and cokernels of multicomplex homomorphisms

Let $\mathcal A$ be a (complete and cocomplete) Abelian category.
A multicomplex in $\mathcal A$ is a bigraded object $X^{(\bullet,\bullet)}$ with differentials
$$
d^{(i,j)}_r\colon X^{(i,j)}\to X^{(...

**5**

votes

**1**answer

358 views

### Using spectral sequence show that if $p\neq 2$, then $\mathbb{Z}_p$ cannot act freely on $\mathbb{C}P^n$

If $p\neq 2$, then the cyclic group $\mathbb{Z}_p$ has no free continuous action on $\mathbb{C}P^n$. My question is how to prove the above fact using Leray-Serre spectral sequence associated to the ...

**9**

votes

**0**answers

132 views

### Hochschild-Serre spectral sequence via explicit filtration

Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...

**6**

votes

**1**answer

172 views

### Serre spectral sequence degeneration in homology vs cohomology

Let $\pi\colon E \rightarrow B$ be a fiber bundle with fiber $F$. I am not assuming that $B$ is simply-connected. We then have Serre spectral sequences in both rational homology and rational ...

**3**

votes

**0**answers

63 views

### Deformations of nilpotent parts of superalgebras

I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598
After ...

**3**

votes

**0**answers

110 views

### Induced Homomorphism on Cohomology of Symmetric Group 3

For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...

**8**

votes

**0**answers

101 views

### Relating bordism generators in d and d+2 dimensions — an explicit example

This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...

**2**

votes

**1**answer

124 views

### Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group

Let $M^d$ be a nontrivial bordism generator for the bordism group
$$
\Omega_d^G= \mathbb{Z}_n,
$$
suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...

**4**

votes

**0**answers

64 views

### Relating bordism invairants in $d$ and $d+2$ dimensions

Are there some relationship between mapping the bordism invairants of eq.1 and eq.2?
$$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$
and
$$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \...

**2**

votes

**0**answers

116 views

### Does Leray Spectral sequence degenerates at $E_2$ over product of curves

Let $C$ be a smooth, projective curve (can assume to be rational) and $X:=C \times C$. Denote by $p:X \to C$ one of the two natural projections. Let $E$ be a vector bundle on $X$. Is it true that,
$$...

**6**

votes

**1**answer

264 views

### 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

60 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 $...

**3**

votes

**0**answers

107 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 ...

**8**

votes

**2**answers

454 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 ...

**3**

votes

**0**answers

132 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, ...

**3**

votes

**0**answers

80 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}},...

**8**

votes

**1**answer

260 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**

votes

**0**answers

124 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 ...

**9**

votes

**0**answers

99 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 ...

**6**

votes

**0**answers

99 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**

votes

**0**answers

104 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 ...

**5**

votes

**0**answers

122 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

121 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

135 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}, \...

**1**

vote

**0**answers

50 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

159 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

51 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

86 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

108 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

144 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

201 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

285 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 ...

**3**

votes

**0**answers

98 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 ...

**8**

votes

**1**answer

138 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

263 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

361 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

216 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

212 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 $...

**1**

vote

**0**answers

55 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 ...

**5**

votes

**0**answers

92 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 $[...

**2**

votes

**0**answers

99 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

148 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

342 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 ...

**16**

votes

**1**answer

452 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

110 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

224 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

91 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 ...