27
votes
Accepted
Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space
Of course, in any spectral sequence $E_{r+1}$ is a subquotient of $E_r$ (the kernel of $d_r$ divided by the image of $d_r$). And in general new torsion can appear in the sense of torsion elements in $...
- 54.4k
20
votes
Accepted
Pullback and homology
This is not necessarily true. For example, there is a space $X$ constructed by attaching a 3-dimensional cell to $S^1 \vee S^2$, which serves as a standard counterexample to several questions. The map ...
- 51.5k
20
votes
Accepted
Calculation of $H^{10}(K(\mathbb{Z}, 3); \mathbb{Z})$
I do not like naming a cohomology class $n$ because that deserves to be the name of an integer. I will use the name Hatcher does and call the generator of $H^2(K(\Bbb Z, 2); \Bbb Z)$ by the name "$a$"....
- 9,388
19
votes
Accepted
Relating two different approaches to the Atiyah-Hirzebruch Spectral Sequence
For cohomology, this is theorem 3.3 in
Maunder, C.R.F., The spectral sequence of an extraordinary cohomology theory, Proc. Camb. Philos. Soc. 59, 567-574 (1963). ZBL0116.14603.
Theorem 3.3 If $...
- 16.2k
17
votes
Accepted
Sphere spectrum, Character dual and Anderson dual
The Anderson dualizing spectrum $I_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded ...
- 5,550
15
votes
Accepted
What is the relationship between spectral sequences and obstruction theory?
This is a partial answer, but every obstruction theory (in some precise sense) provides you with a spectral sequence (in fact several). Let me clarify what do I mean with obstruction theory. All this ...
- 16.2k
14
votes
Accepted
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
Good question. I think the answer is yes.
The unnamed spectral sequence is usually referred to as the isotropy spectral sequence. For a group $G$ acting on $X$ and an abelian group $A$ of ...
- 35k
14
votes
To compare the total, base and fiber spaces of two fiber bundles
No. Consider the map from the fibre bundle
$$B\mathbb{Z} \to BD_\infty \to B\mathbb{Z}/2$$
to $* \to * \to *$. Here $D_\infty = \mathbb{Z} \rtimes \mathbb{Z}/2$ is the infinite dihedral group.
You ...
- 17.7k
13
votes
Multiplicative structure on spectral sequence
This is an expansion of John Rognes' answer. I have filled in a few details in Douady's seminare notes and noticed that one gets away with slightly weaker axioms. If there is already a reliable ...
- 6,666
13
votes
Why is it difficult to obtain the next differential in a spectral sequence?
Expanding on Tyler Lawson's comment, the point of a spectral sequence is often that we know what $E$ is concretely, and we want to use this to compute $A$. The issue is that if we want to explicitly ...
- 513
13
votes
Accepted
Multiplicativity of the homology Atiyah-Hirzebruch spectral sequence for a ring spectrum
You can give a proof of multiplicativity by using that the smash product preserves connectivity. Here is a sketch proof.
(EDIT: Denis Nardin pointed me towards this reference by Dugger. This ...
- 51.5k
13
votes
Accepted
Zero differential in Serre spectral sequence for configuration spaces
I'll write $C_n$ for the configuration space, and $X_n$ for $\mathbb{R}^2$ with $n$ points removed. You are presumably thinking about the spectral sequence
$$ E_2^{pq} = H^p(C_{n-1};H^q(X_{n-1})) \...
- 55k
12
votes
Accepted
Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$
Unoriented cobordism: can be read off from the structure of the unoriented cobordism ring (calculated in Thom's thesis): $\Omega_6^O = (\mathbb Z/2)^3$, $\Omega_7^O = \mathbb Z/2$, $\Omega_8^O = (\...
- 6,766
12
votes
Accepted
Hodge Numbers and Leray Spectral Sequence
I don't think I defined the Hodge numbers in this way. Rather, the argument in Section 1 shows that the Hodge numbers agree with the dimensions of
the terms in the $E_2$ page of the Leray spectral ...
- 985
12
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
Some examples with one nonzero family of differentials:
The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p=...
11
votes
Accepted
Cohomology ring of a fiberwise join
What you wish to prove is not true. Namely, if $I=\operatorname{ker} f^*$ then $I^2\subseteq \operatorname{ker}(f\ast f)^*$ but the inclusion may be strict.
The fibred join construction comes up in ...
- 35k
11
votes
Pullback and homology
Here is a positive answer to a slightly different question.
Call a map $X\to B$ "acyclic" if it induces an isomorphism in homology for every coefficient system on $B$. (If $B$ is simply connected ...
- 54.4k
11
votes
to compare cohomologies of fibers of two fiber bundles
No. Let $B'$ be any space, and take $E'=PB'$ and $F'=\Omega B$. The Kan-Thurston theorem gives a map $f\colon B\to B'$ such that $H^*(f;\mathbb{Q})$ is an isomorphism but $\Omega B$ is discrete, so $...
- 55k
11
votes
Accepted
The second stable homotopy group
I love this question! I've enjoyed thinking of it. Below, I show why the sequence splits always.
Let $X\to Y=K(H_1(X,\mathbb{Z}/2),1)$ be the map inducing the identity in $H_1(-,\mathbb{Z}/2)$. By ...
- 14.9k
10
votes
Multiplicative structure on spectral sequence
As far as I know that 1954 paper of Massey is faulty, and you cannot get multiplicative spectral sequences just from such stucture on an exact couple. The best I know that you can do is to use Cartan-...
- 8,942
10
votes
Where does the primary obstruction of a fibration show up in its spectral sequence?
In the general case of integral coefficients and possibly non-trivial local coefficient system, let $\pi=\pi_1(B)$.
A cocycle for the obstruction class is an element $o\in Hom_{\mathbb Z\pi}(C_{k+1}\...
- 451
10
votes
Accepted
Reference request: cohomology of Eilenberg Maclane spaces with $p$-local groups
Here is a sketch proof.
Step 1: For sensible spaces or spectra (connected, finite type) $X$, $H^*(X;\tau) $ will have exponent $p$ for all the coefficient groups $\tau$ you list exactly when the ...
- 10.6k
10
votes
Accepted
Torsion in the integral cohomology of $BPU_{n}$
You may want to have a look at this paper:
X. Gu. On the cohomology of classifying spaces of projective unitary groups. arXiv:1612.00506, (link to arXiv)
The spectral sequence involving $BSU_n$ ...
- 17.3k
10
votes
In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?
Not in general, no. The problem is that animated functors play well with colimits, and homotopy groups play better with limits. However, if your functors $\mathcal{A}\xrightarrow{F}\mathcal{B}\...
- 8,779
10
votes
Accepted
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
You may want to look at the classical paper of Jack Milnor, "On the homology of Lie groups made discrete." The Friedlander-Milnor conjecture states that the map $BG^\delta \to BG$ (where $G$ ...
- 11.8k
10
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The Serre spectral sequence for the path-loop fibration for an $n$-sphere is a positive answer to question 1, a negative answer to question 2. More generally, a fibration in which either the base or ...
9
votes
Sphere spectrum, Character dual and Anderson dual
One way to think of these spectra is in terms of the cohomology theories they define. In other words, if
$E$ is a spectrum, what is $[E, I\mathbb Z]$? This is less of a description of what they are ...
- 6,766
9
votes
Accepted
Show that if $p\neq 2$, then $\mathbb{Z}_p$ cannot act freely on $\mathbb{C}P^n$
Consider the cohomology with $\mathbb{Z}$ coefficients (and reduce the 0-th term modulo $p$ to get uniform description of it). Then we have a spectral sequence starting from $\mathbb{F}_p[x,y]/x^{n+1}$...
- 4,064
9
votes
Accepted
Has anyone seen this generalization of the snake lemma? Is it useful?
Spectral sequences are obtained by applying the snake lemma ad nauseam, essentially. You are doing that (finitely many times) in your spectral sequence argument.
Let me illustrate. Consider the case ...
- 1,534
9
votes
Accepted
Two spectral sequences arising from a simplicial spectrum
The precise relation between the skeleton filtration and the levelwise Postnikov filtration is that the décalage of the first is isomorphic to the second. This is explained in [Ariotta: Coherent ...
- 2,003
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