37
votes
Accepted
All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$?
No it isn't, but I had to dig quite deep to get a counterexample. Let us look at smooth $(D^7, \partial D^7)$-bundles over $S^2$, i.e $D^7 \to E \overset{\pi} \to S^2$ with an identification $\partial ...
19
votes
Accepted
Serre fibration vs Hurewicz fibration
A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space ...
19
votes
Anomaly in QFT physics v.s. determinant line bundle
The partition function should assign to each possible field configuration $\Phi$ (or field history) in your quantum field theory a number $Z(\Phi)$. That is, it should be a function on the collection ...
17
votes
Accepted
Foliation of $\mathbb R^n$ by connected compact manifolds
There does not, even if you don’t require the fiber and base to be manifolds (or even connected, just that $F$ is not a single point). See
Borel, Armand; Serre, Jean-Pierre,
Impossibilité de fibrer ...
16
votes
Accepted
Classification of $O(2)$-bundles in terms of characteristic classes
The $O(2)$ bundles $\xi$ over a manifold $M$ are classified by their first Stiefel-Whitney class $w_1(\xi)\in H^1(M;\mathbb{Z}/2)$ and their twisted Euler class $e(\xi)\in H^2(M;\mathbb{Z}_{w_1(\xi)})$...
16
votes
Accepted
Is a topological fiber-bundle, whose total space admits a retraction onto a fiber, trivial?
The statement is not true. Let $\pi:V\to M$ be a vector bundle over a manifold which is non-trivial as a fiber bundle. Let $U$ be an open neighborhood of $M$ over which $V$ is trivial, fix $x\in U$, ...
16
votes
Is the concept of a "numerable" fiber bundle really useful or an empty generalization?
In algebraic topology, it is often more convenient to know that a map is a fibration (has the homotopy lifting property with respect to all spaces) than a fibre bundle, because then calculational ...
14
votes
Serre fibration vs Hurewicz fibration
These show up all the time in various generalized covering space theories, the reason being that you want homotopy lifting with respect to a certain class of spaces that you're interested in (e.g. ...
14
votes
Foliation of $\mathbb R^n$ by connected compact manifolds
On the other hand, if you only mean "foliation" as in your title, and not "fibration", then there is Vogt's foliation of R^3 by circles! (But it is not C^1, only differentiable).
Vogt, Elmar, "A ...
13
votes
Accepted
Surface bundles associated to a short exact sequence of groups
(1) It is not true that these groups are precisely the fundamental groups of $S$-bundles. The correct statement is that these groups are precisely the fundamental groups of $S$-bundles over a base ...
12
votes
Is a topological fiber-bundle, whose total space admits a retraction onto a fiber, trivial?
It was already pointed out that the statement is not true in the point-set sense. It is, however, true up to homotopy. This is a theorem of Dold and follows from his
Partitions of unity in the ...
11
votes
Cohomology ring of mapping torus
Not a full answer, but a very natural computational tool here (only for the additive structure) is the Leray spectral sequence
$$E_2^{p,q} = H^p\big(S^1,R^qf_*\underline{\mathbf Z}\big) \Rightarrow H^{...
10
votes
Classification of $O(2)$-bundles in terms of characteristic classes
To complement Mark Grant's excellent answer, I'll say something more about the general case. This topic goes under the name of obstruction theory.
The first observation is that a $G$-bundle on $X$ is ...
10
votes
Accepted
A $\mathbb{R}^{n}$ -fiber bundle which do not admit a n-dimensional vector bundle structure
Such a fibre bundle does not exists if you suppose that it is endowed with a differentiable structure. Stewart has shown that the group of diffeomorphisms of $R^n$ retract to $O(n)$. So every $Diff(R^...
10
votes
Classification of bundles, Postnikov towers, obstruction theory, local coefficients
I'll try to answer question 1. Unfortunately, I know of no especially convenient reference for the case of understanding general Postnikov towers; however, everything I say below is "well known".
...
10
votes
Circle bundle with homotopically trivial fiber in the total space
Yes.
By the homotopy long exact sequence
$$\dots \to \pi_2(B) \to \pi_1(S^1) \to \pi_1(E) \to \dots$$ and the fact that the generator of $\pi_1(S^1)$ is homotopically trivial in $B$, we see that $\...
10
votes
Cohomology ring of mapping torus
This is also an incomplete answer but from a different angle than the previous ones. Let $A=\begin{pmatrix} a&b\\c&d\end{pmatrix}$.
EDIT: the answer below only deals with the orientable case $\...
9
votes
Accepted
Are topological fiber bundles on the same base with homeomorphic fibers isomorphic?
If $F$ is locally connected, locally compact and Hausdorff (or alternatively compact Hausdorff), then the inverse function from $Homeo(F, F)$ to itself is continuous. Moreover, such a space $F$ is ...
9
votes
Classifying space of semidirect product of groups
I am adding my comment as an answer.
Every extension of groups $1 \to H \to G \to K \to 1$ corresponds to a fibration $$BH \to BG \to BK,$$ or a little more precisely at the space level $$EG/H \to (...
9
votes
Accepted
Intersection form of surface bundle over surface
Yes, such a thing exists, but I don't know an explicit example.
To see that it exists, it is clearest to me to consider the universal situation. For any $k \in \mathbb{Z}$ there is a space $\mathcal{S}...
9
votes
Cohomology ring of mapping torus
I just want to point out that one can go quite far using algebraic topology (in this special case), although computing the cohomology ring completely may require some other ingredient.
Lets call the ...
9
votes
Accepted
A question regarding isomorphism in cohomology for moduli space of stable bundles over a compact Riemann surface
Things are actually simpler. View $\Gamma _n=H^1(X,\mathbb{Z}/n)$ as the group of line bundles $L\in \operatorname{Pic}^{0}(X) $ with $L^{{\tiny \otimes }n}=\mathscr{O}_X$. The
map $N_0(n,k) \times \...
9
votes
Accepted
Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?
I think the answer is yes.
Since $G$ is compact, there is a $G$-invariant Riemannian metric on $M$ (by averaging any metric). The orthogonal distribution to the fiber for this metric is a $G$-...
8
votes
Accepted
Surface bundles over surfaces with(out) flat structure
It's still open for all values of $g$ and $h$. One reference for it is
M. Bestvina, T. Church, and J. Souto,
Some groups of mapping classes not realized by diffeomorphisms,
Comment. Math. Helv. 88 (...
8
votes
Action of the spin covariant derivative on gamma matrices?
The question is not clearly stated, but if appropriately interpreted it does make sense.
Let $S$ be a spinor bundle over a pseudo-Riemannian manifold $(M,g)$, whose bundle of Clifford algebras is ...
8
votes
Signature of the manifold of the multiple fibrations over spheres
The total space of a fiber bundle over a sphere that isn't a circle has zero signature by a result of
Chern, Hirzebruch, and Serre that the signature is multiplicative when the fundamental group of ...
8
votes
Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$
Firstly, yes, your examples are all correct. However, in example~(B) we just have $O(3)=\{\pm I\}\times SO(3)$ as groups, so your fibration is just the product of
$$BSO(3)\xrightarrow{1}BSO(3)\to 1$$...
8
votes
Accepted
The existence of the extension of a non-trivial line bundle
This is a bordism problem, and as such can be answered using algebraic topology. I'll answer in the unoriented setting, then indicate how to modify things if $M$ and $W$ are required to be oriented.
...
8
votes
Accepted
Smooth structure on the space of sections of a fiber bundle and gauge group
Your intuition is right. To endow the space of sections of a fiber bundle $F$ with a manifold structure at $\phi \in \Gamma^\infty(F)$ you consider a tubular neighborhood (respecting the fiber ...
8
votes
Is the concept of a "numerable" fiber bundle really useful or an empty generalization?
Dold's condition in on the bundle, while paracompactness is a condition on the base space. These are two very different sorts of concepts, and the former is more categorical. Note that the property ...
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