All Questions
Tagged with at.algebraic-topology spectral-sequences
218 questions
2
votes
1
answer
148
views
An attempt at an alternative calculation of the rank of $\pi_n(MO)$
$\newcommand{\a}{\mathfrak a}\newcommand{\Z}{\mathbb Z}$Let $MO$ be the Thom Spectrum, then I am trying to come up with an alternative calculation that the rank of $\pi_n(MO)$ as a $\Z_2$ vector space ...
- 391
6
votes
0
answers
141
views
Are the $K(n)$-local $E_n$-Adams spectral sequences isomorphic to the Adams-Novikov spectral sequences?
Let $H$ be a closed subgroup of the Morava stabilizer group $\mathbb G_n$. [Devinatz-Hopkins, Prop. 6.7] identifies the $K(n)$-local $E_n$-Adams spectral sequence for $E_n^{hH}$ as the homotopy fixed ...
- 155
2
votes
1
answer
201
views
Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\...
- 508
3
votes
0
answers
90
views
Topological groups satisfying the Borel transgression theorem
I am using the Borel transgression theorem as given in Mimura and Toda's "Topology of Lie groups I and II", page 378, Theorem 2.7. I know that it applies when the fiber has the homotopy type ...
- 61
4
votes
1
answer
514
views
A question about spectral sequences
In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
- 1,367
4
votes
2
answers
290
views
Loop-space functor on cohomology
For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$.
More concretely, $\omega$ is given by the Puppe sequence
$$\...
- 663
1
vote
0
answers
78
views
A question about the localization theorem of Borel-Hsiang and spectral sequence
Suppose that $T$ is a torus acting on a topological space $X
$. Let $T\longrightarrow E_{T}\longrightarrow B_{T}$ be the universal $T$-bundle. Let $X\longrightarrow X_{T}\longrightarrow B_{T}$ be the ...
- 1,367
13
votes
5
answers
2k
views
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live ...
5
votes
1
answer
472
views
Two spectral sequences arising from a simplicial spectrum
Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization.
Let's assume each $X_n$ is connective.
From this situation, we can form two filtrations on $X$: the ...
- 339
2
votes
0
answers
179
views
Differentials in the Atiyah-Hirzebruch spectral sequence for a bounded generalized cohomology theory
$\newcommand{\res}{\mathrm{res}}$Let $X$ be a connected finite CW complex, and let $E$ be a bounded spectrum. For simplicity, let me assume that it has homotopy groups concentrated in degrees 0,1,2.
...
- 1,001
11
votes
2
answers
858
views
Spectral sequences and short exact sequences
Suppose I take a short exact sequence of filtered chain complexes:
$$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$
We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
- 721
6
votes
1
answer
375
views
Different flavours of Vassiliev Conjecture
There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
- 2,152
3
votes
1
answer
421
views
Spectral sequence in Adams's book, Theorem 8.2
I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
- 224
2
votes
0
answers
284
views
Notation for spectral sequences [closed]
Every single spectral sequence I have seen in my life was denoted by $E$. Even when there is more than one spectral sequence, people tend to use the same letter with some workaround (e.g. a fourth ...
- 2,152
4
votes
2
answers
409
views
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
- 441
16
votes
1
answer
776
views
The second stable homotopy group
I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\...
- 663
0
votes
0
answers
185
views
Interpreting the edges in the Serre spectral sequence
Let $F \hookrightarrow E \stackrel{\pi}{\rightarrow} B$ be a fiber bundle. For simplicity, assume that $F$ is connected, that $B$ is $1$-connected, and that $B$ is a CW complex. Consider the Serre ...
1
vote
0
answers
167
views
Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\begin{matrix}
\ F & \to & E &\to & B \\\
\downarrow & & \downarrow & & \downarrow \\\
\ F'...
- 103
4
votes
0
answers
170
views
infinite families in stable homotopy groups
The question is about infinite families in stable homotopy groups. Yes, there are some Q&A about the topic.
But I wonder if the order of Mahowald's elements is known?
in Green Book it mentioned ...
- 171
4
votes
0
answers
102
views
"Standard computations" with stable Hopf invariants
I am struggling in understanding the proof of Lemma 10.6 of the paper "Mapping class groups and function spaces" by Bodigheimer, Cohen and Peim http://www.math.uni-bonn.de/people/cfb/...
- 641
7
votes
1
answer
261
views
Relation between cohomology operations and the Adams spectral sequence
$\newcommand{\Z}{\mathbb Z} \DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Cone}{Cone}$
I'm trying to understand how higher order cohomology operations are related to the Adams spectral ...
- 71
3
votes
0
answers
264
views
Explicit description of the Leray spectral sequence with compact supports for a fibration
Consider a locally trivial fibration $f: E \to B$ with fiber $F = \mathbb{C}^n$. The Leray spectral sequence with compact supports is
$$ E_2: H^p_c(B, \underline{H^q_c(F)}) \implies H^{p+q}_c(E). $$
...
3
votes
0
answers
249
views
Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
2
votes
0
answers
98
views
Name for the "other term" in a derived exact couple
I'm building a spectral sequence using an exact couple $D^1 \to D^1 \to E^1 \to D^1$, with $k$th derived exact couple $D^k \to D^k \to E^k \to D^k$. In this case, I happen to have more information ...
- 455
4
votes
1
answer
320
views
Higher order differentials of Bockstein spectral sequence
The Bockstein SS is obtained from the exact sequence
$$0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$$
with $E_1^p=H^p(X,\mathbb{Z}/2)$ and the differential $d_1=Sq^1$.
How to identify ...
- 918
3
votes
0
answers
118
views
The $\pi_1(BM)$ action on $H^*(BH,R)$ in a Serre fibration $BH\to BG\to BM$
$R$ is a ring. Applying the Serre spectral sequence to a fibration $F\to E\to B$, to avoid local coefficients, we need to require that $\pi_1(B)$ acts on $H^*(F, R)$ trivially. Consider a fibration of ...
- 663
1
vote
0
answers
93
views
Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
- 1,805
4
votes
1
answer
448
views
The Hochschild–Serre spectral sequence and cup products
Let $X$ be a variety over a field $k$ with separable closure $k_s$. Let $A$, $B$ be étale sheaves on $X$. Consider now the Hochschild–Serre spectral sequences.
\begin{align*}
E_2^{pq}: H^p(k, H^q(X_{...
- 388
3
votes
0
answers
240
views
Does Kudo's transgression theorem still hold when the coefficient is ℤ/pⁿℤ?
I am trying to compute a Lyndon–Hochschild–Serre spectral sequence with coefficient $\mathbb{Z}/p^2\mathbb{Z}$, where $p$ is an odd prime, and Kudo's transgression theorem looks like a good tool for ...
- 31
3
votes
0
answers
224
views
Cohomology ring $H^*(BG,\mathbb{Z}_2)$ for $G=\mathbb{Z}_2\ltimes B^2\mathbb{Z}^2$
$$
\newcommand{\Z}{\mathbb{Z}}
$$
Consider the group $G=\Z_2\ltimes B^2\Z^2$ where $\Z_2$ acts on $\Z^2$ by interchanging the two factors of $\Z^2$, hence $\Z_2$ also acts on the Eilenberg-MacLane ...
- 10.5k
1
vote
0
answers
222
views
Cohomology spectral sequence of a CW complex filtered by its skeletons
Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$
is a filtration of $X$ by its skeletons $X^i$. Now ...
- 191
5
votes
0
answers
290
views
Thom spectrum of $(\mathrm{Spin}\times_{Z_2} \mathrm{SO}(d))$
$\DeclareMathOperator\MSO{MSO}\DeclareMathOperator\MSpin{MSpin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}\DeclareMathOperator\BG{BG}\DeclareMathOperator\BO{BO}\DeclareMathOperator\MG{MG}\...
- 487
5
votes
1
answer
425
views
Can we construct a filtered chain complex from a spectral sequence?
Suppose $\{(E_r,d_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E_r,d_r)$ is a chain complex over $\mathbb{F}$ and $E_{r+1}=H(E_r,d_r)$. For simplicity, ...
- 673
6
votes
1
answer
411
views
Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity
Suppose $G$ is a Lie group, with $\pi_0(G)$ not necessarily finite, but might as well assume $G_0$, the connected component of the identity, is compact.
In the case that $\pi_0(G)$ is finite, then we ...
- 35.5k
5
votes
0
answers
714
views
Spectral sequence from a stratification by closed subvarieties
I am looking for a reference for the following result: If $X$ is an algebraic variety and
$$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$
is a stratification (edit: filtration) ...
10
votes
0
answers
325
views
Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$
First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
- 101
3
votes
0
answers
101
views
Geometric filtration for Eilenberg-Moore spectral sequence
I'm reading the paper by Eilenberg-Moore (https://link.springer.com/content/pdf/10.1007/BF02564371.pdf) about the Eilenberg-Moore spectral sequence.
In section 11, they introduce the notion of ...
- 449
27
votes
0
answers
1k
views
Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived ...
- 64k
8
votes
3
answers
914
views
Spectral sequences in algebraic topology [duplicate]
What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...
8
votes
2
answers
492
views
Conditions under which the preimage of a submanifold in nontrivial in homology
Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-...
- 2,095
7
votes
0
answers
270
views
Differentials in spectral sequences and Massey products
Consider a multiplicative spectral sequence such as the cohomological Serre spectral. It is known that differentials will satisfy a Leibniz rule. Is there a clean statement involving differentials and ...
- 965
10
votes
1
answer
321
views
$\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}_{p^{r+1}}$ summand
I am reading Neisendorfer's paper Samelson products and exponents of homotopy groups and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}_{p^{r+1}}$ ...
- 1,263
3
votes
1
answer
205
views
Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$
It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds:
$$
\Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = ...
- 2,147
4
votes
1
answer
315
views
Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request
I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
- 301
4
votes
0
answers
201
views
Using the Serre spectral sequence - moving between $\mathbb{Z}/2$ and $\mathbb{Z}$ information
I am trying to understand the computation of $\pi_5(S^3)$ and $\pi_6(S^3)$ using the Serre spectral sequence. I know already that $\pi_5(S^3)$ is only 2-torsion and $\pi_6(S^3)$ is 2-torsion together ...
- 5,349
2
votes
0
answers
269
views
Dress' construction and Serre spectral sequence
Currently, I am reading Serre spectral sequence, given below, using Dress' construction.
Let $f:E\to B$ be a Serre fibration. Then, there is a first quadrant
spectral sequence $\big\{E^r,d^r\}_{...
- 632
2
votes
0
answers
486
views
An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology
I am currently reading Künneth spectral sequence, which is given below.
Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
- 632
3
votes
0
answers
186
views
Cobordism theory of some weird space
Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the ...
- 10.5k
7
votes
0
answers
541
views
Convergence of a spectral sequence of a double complex
In Weibel's book, a spectral sequence $E^r_{p,q}$ is said to weakly converge to a graded object $H_{\ast}$ if for every $n$ there exists a filtration $\dots \subset F_{r}H_{n} \subset F_{r-1}H_{\ast} \...
- 1,325
5
votes
0
answers
328
views
Bousfield-Kan and Generalized Eilenberg-Moore spectral sequences
Building on the work of Anderson and Rector, Bousfield's paper "On the homology spectral sequence of a cosimplicial space" constructs a spectral sequence which takes in a cosimplicial space (here ...
- 131