All Questions
Tagged with spectral-sequences fibre-bundles
9 questions
2
votes
1
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191
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Cohomology class of fiber bundle
Suppose I have a fiber bundle $\pi: E\rightarrow B$, with fiber $F$, such that the Serre spectral sequence on cohomology is immediately degenerate. In other words, $H^*(E)=H^*(B)\otimes H^*(F)$.
I ...
- 39.3k
5
votes
1
answer
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
8
votes
3
answers
914
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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 ...
- 3,173
8
votes
2
answers
492
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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 $p$-...
- 35.9k
3
votes
1
answer
260
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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 ...
- 11.2k
3
votes
0
answers
234
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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}:...
- 366
4
votes
2
answers
1k
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Cup product of cohomology in a Serre spectral sequence
How to use Serre spectral sequence to compute cup product structures?
Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...
- 20k
2
votes
0
answers
123
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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
10
votes
3
answers
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Serre Spectral Sequence of Representations
Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...
- 6,210