All Questions
Tagged with spectral-sequences at.algebraic-topology
218 questions
1
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331
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When the local system of coefficients are simple in the Leray-Serre spectral sequence
Let $F\to E\to B$ a fibration and $\{E_{r}^{\ast,\ast},d_{r}\}$ the Leray-Serre Spectral sequence converging to $H^{\ast}(E;R),$ such that
$$E_{2}^{p,q}=H^{p}(B;\mathcal{H}^{q}(F;R))$$
is the ...
- 29
6
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0
answers
211
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$\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence
I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the ...
- 521
6
votes
3
answers
460
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multiplicative structure of Ext
Basically, I am trying to compute something with the Adams spectral sequence (as a toy example). The $E^2$ page reduced to computing $Ext^{s,t}_{\Gamma} (\mathbb{F}_2, \mathbb{F}_2)$, where $\Gamma = \...
- 225
3
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0
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180
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Computations of Bredon homology of $S(1+\sigma)$ with Universal Coefficient S.S
What I am trying to do is to compute $\mathbb{Z}$-graded Bredon homology of $S(1+\sigma)$ over $Q\times\Sigma_2$, where
$Q$ is a cyclic group of order 2
$\sigma$ is its real sign representation
$\...
- 1,759
14
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0
answers
404
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Computations using "Stover's spectral sequence"
In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.
The second ...
- 208
15
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1
answer
730
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Calculation of $H^{10}(K(\mathbb{Z}, 3); \mathbb{Z})$
I was trying to calculate $H^q(K(\mathbb{Z}, 3); \mathbb{Z})$ for some $q$ with the Serre spectral sequence associated to the fibration $K(\mathbb{Z}, 2) \to PK(\mathbb{Z}, 3) \simeq * \to K(\mathbb{Z}...
- 1,263
3
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0
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70
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Characterization of degeneracy of spectral sequence of a fiber bundle at the second term
Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients ...
- 21.8k
6
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1
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542
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Zero differential in Serre spectral sequence for configuration spaces
I moved this question from Math StackExchange.
I am trying to compute homology of $Conf(n, \mathbb{R}^2)$ - ordered configurations of $n$ points on the plane - using Serre spectral sequence. I know ...
- 1,759
0
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0
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160
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Splitting of Atiyah-Hirzebruch Spectral Sequence
Suppse E is a cohomology theory which has Kunneth Formula, i.e $ E(A \wedge B)= E(A) \otimes_{E(pt)} E(B) $. What happens to the Atiyah Hirzebruch Spectral sequence while we compute $ E(A \wedge B) $?...
- 341
5
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1
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924
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When is the cohomology of a fiber bundle a tensor product?
Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...
- 7,609
11
votes
3
answers
846
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Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum
Let $E$ be a ring spectrum and $F$ a connective spectrum. Then we have a convergent Atiyah-Hirzebruch spectral sequence $H_s(F,E_t) \Rightarrow E_{s+t}(F)$. Suppose now that $F$ is also a ring ...
- 64k
8
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1
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474
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Third differential in the homology AHSS
I need some guidance in identifying the third differential in the homology AHSS for $\Omega_{\ast}^{\text{Spin}^c}(X)$ in degrees $\leq 4$.
Remember that $\pi_0(M\text{Spin}^c)=\Bbb Z$, $\pi_2(M\...
- 2,018
5
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1
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186
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Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra
Consider the extension
$$1\to SU(2)\to X\to O\to1,$$
there are 4 possibilities for $X$:
$X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\...
- 2,147
11
votes
1
answer
861
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What is the relationship between spectral sequences and obstruction theory?
Let $X,Y$ be objects of some category $\mathcal C$, and suppose I want to study homotopy classes of maps from $X$ to $Y$ (almost everything one does in algebraic topology can be viewed this way). It ...
- 64k
4
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1
answer
290
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Grading in Eilenberg-Moore spectral sequence
I am puzzled over something I read in Quillen's On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field.
On page 557, when computing the $E_2$ page of a case of the Eilenberg-...
4
votes
0
answers
397
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Eilenberg-Moore spectral Sequence calculation
I want to use the cohomology Eilenberg-Moore spectral sequence to calculate the cohomology of the fibre of the map
$$
S^{n} \to \Omega S^{n+1}.
$$
Question 1: Is anyone aware of any references for ...
- 901
11
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0
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266
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Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper
In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
- 1,329
2
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1
answer
355
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Leray-Serre spectral sequence for projective bundles
Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that ...
- 1,325
5
votes
0
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102
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Group cohomology of "twisted" projective SU(N) with various coefficients
Given a group
$$
G= PSU(N) \rtimes \mathbb{Z}_2,
$$
where $PSU(N)$ is a projective special unitary group. Say $a \in PSU(N)$, $c \in \mathbb{Z}_2$, then
$$
c a c= a^*,
$$
which $c$ flips $a$ to its ...
- 10.5k
7
votes
1
answer
347
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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 ...
- 26.1k
6
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0
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300
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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 $...
- 81
5
votes
1
answer
549
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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 ...
- 395
9
votes
0
answers
421
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 ...
- 163
7
votes
1
answer
372
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 ...
- 163
8
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0
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125
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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 ...
- 2,147
2
votes
1
answer
142
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}...
- 2,147
7
votes
1
answer
413
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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,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...
- 2,147
4
votes
1
answer
639
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 ...
- 10.5k
10
votes
2
answers
2k
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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 ...
- 10.5k
8
votes
1
answer
475
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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 $n$) $$\Omega_d(BG)_p.$$
...
- 10.5k
3
votes
0
answers
165
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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 ...
- 404
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0
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131
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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 ...
- 10.5k
6
votes
0
answers
122
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 ...
- 10.5k
6
votes
0
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562
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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 ...
- 4,189
6
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0
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237
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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 ...
- 863
4
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1
answer
227
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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 ...
- 41
4
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1
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598
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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 ...
- 10.5k
9
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1
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456
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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 ...
- 1,886
6
votes
2
answers
408
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_{\...
- 101
6
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1
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244
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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 ...
- 1,367
2
votes
2
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323
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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 $...
- 1,629
6
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0
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163
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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 $[...
- 61
4
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1
answer
182
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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;%
%...
- 1,367
8
votes
2
answers
500
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 ...
- 1,367
16
votes
1
answer
808
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$-...
- 863
8
votes
1
answer
302
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_{...
- 317
8
votes
1
answer
441
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{...
- 235
13
votes
1
answer
957
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+...
- 2,303
8
votes
1
answer
525
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 ...
- 283
7
votes
0
answers
436
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 ...
- 353