All Questions
35 questions
20
votes
5
answers
3k
views
Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?
One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
- 25.2k
20
votes
1
answer
2k
views
Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence?
Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex:
We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by ...
- 10.8k
17
votes
2
answers
2k
views
Are the homology and cohomology Serre spectral sequences dual to each other?
If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...
- 8,167
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
10
votes
2
answers
1k
views
cup product and Steenrod operations in Serre spectral sequence
Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
- 2,223
9
votes
0
answers
516
views
extension problem for the Atiyah-Hirzebruch spectral sequence
For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
- 1,219
8
votes
1
answer
503
views
Cohomology ring of a fiberwise join
I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...
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
7
votes
1
answer
2k
views
Cohomology groups of quotient by finite group
I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question.
Let $G$ be a finite group acting on a manifold $M$ without fixed ...
- 71
7
votes
1
answer
347
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 ...
- 26.1k
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
7
votes
0
answers
149
views
Cohomology of Lie group $E_8$, e.g. $H^d(E_8,\mathbb{R}/\mathbb{Z})$
What is the $d$-th cohomology of a Lie group $E_8$, say $H^d(E_8,\mathbb{R}/\mathbb{Z})$ with $\mathbb{R}/\mathbb{Z}$ coefficient?
I suppose that there are many nontrivial groups of $H^d(E_8,\mathbb{...
- 10.5k
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
6
votes
1
answer
2k
views
cohomology version of Cartan-Leray spectral sequence that deduces cup product
On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:
If $X$ is a connected pace on which the group $\...
- 1,990
6
votes
0
answers
237
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 ...
- 863
5
votes
1
answer
407
views
spectral sequence for cobordism without leaving smooth category
In Bott & Tu's marvelous book there is a derivation of the spectral sequence for a (smooth) fiber bundle for deRham cohomology done entirely in the realm of the smooth category. Unfortunately, as ...
- 13.5k
5
votes
1
answer
186
views
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
5
votes
0
answers
290
views
Two natural maps asssociated with the nerve of a cover
Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...
- 3,749
4
votes
2
answers
514
views
stability results for the Atiyah-Hirzebruch spectral sequence
For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}...
- 1,219
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
$$\...
- 663
4
votes
1
answer
315
views
Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request
I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
- 301
4
votes
0
answers
397
views
Eilenberg-Moore spectral Sequence calculation
I want to use the cohomology Eilenberg-Moore spectral sequence to calculate the cohomology of the fibre of the map
$$
S^{n} \to \Omega S^{n+1}.
$$
Question 1: Is anyone aware of any references for ...
- 901
4
votes
0
answers
576
views
generalized Atiyah-Hirzebruch spectral sequence from Postnikov truncation
The Atiyah-Hirzebruch spectral sequence
\begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*}
computes the generalized homology $h$ of a total space $E$ of a Serre fibration ...
- 1,219
3
votes
1
answer
421
views
Spectral sequence in Adams's book, Theorem 8.2
I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
- 224
3
votes
2
answers
319
views
cohomology algebra of braid spaces, configuration spaces
In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for $p$...
- 2,223
3
votes
1
answer
260
views
non-simple local coefficient system on a fibration of classifying spaces
Long story short; I posted in MSE
https://math.stackexchange.com/questions/2500745/local-system-of-coefficients-on-a-fibration-of-classyfing-spaces
It is well known that if $G$ is a lie group ...
- 283
3
votes
1
answer
463
views
cohomology module of unit tangent vector bundles over spheres
Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow \tau(S^m)\...
- 2,223
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
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
3
votes
0
answers
234
views
How can I find the differential in the Serre spectral sequence for this sphere fibration?
Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials
$$
d_4^{p,m}:...
- 1,716
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
2
votes
0
answers
123
views
cohomology ring of cross-section space of one-point compactification of tangent bundle
Let $M$ be an $m$-manifold whose cohomology is known. Let $TM$ be the tangent bundle of $M$ and $\xi$ be the fibre-wise one-point compactification of $TM$. Then $\xi$ is a $m$-sphere bundle over $M$. ...
- 2,223
1
vote
1
answer
293
views
Unordered configuration space of $\mathbb{R}P^1$
In the paper
GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
159-165,
Theorem 2. (b): $TP^n(\mathbb{R}P^1)$ is ...
- 1,990
1
vote
1
answer
376
views
Multiplicative structure in the cohomological Leray-Serre spectral sequence - please elucidate a proof
Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set:
$X_p = \pi^{-1}(B^p)$,
$F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, the ...
- 19