All Questions
21
questions
1
vote
0
answers
55
views
extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
- 123
0
votes
1
answer
135
views
Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
- 2,712
1
vote
0
answers
243
views
Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle
Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\...
- 1,915
5
votes
2
answers
244
views
Patching up two trivial fibre bundles induces homology equivalence
I was wondering to ask this question may be it's a silly one. I could not prove or disprove it.
Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap ...
- 585
5
votes
1
answer
518
views
When is a diffeomorphism a bundle map?
Let $F\rightarrow E_0 \rightarrow B$ and $F\rightarrow E_1\rightarrow B$ be two smooth fiber bundles. Suppose $E_0$ and $E_1$ are diffeomorphic.
What are the obstructions for $E_0$ and $E_1$ to be ...
- 641
1
vote
0
answers
115
views
Defining the cospecialization in topology
Below is an excerpt from part V of Deligne's Étale cohomology - starting points.
Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...
- 10.4k
7
votes
2
answers
333
views
Foliation of $\mathbb R^n$ by connected compact manifolds
Does there exist a smooth nontrivial fiber bundle $p: F \hookrightarrow \mathbb R^n \to B$ such that $F$ and $B$ are connected manifolds with $F$ compact? "Nontrivial" here means the fiber $F$ is not ...
- 2,525
5
votes
0
answers
108
views
Induced new structures on Poincare dual manifolds
"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
- 10.4k
3
votes
0
answers
169
views
Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$
Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...
- 2,147
11
votes
1
answer
896
views
Equivariant sections of fiber bundles
One of the fundamental facts in fiber bundle theory is the following result for existence and extension of sections (see Thm. 9 in this paper of Palais, and compare with Thm. 12.2 in Steenrod's book):...
- 7,087
4
votes
0
answers
131
views
Equivalence of Flat Fiber Bundles vs Equivalence of Group Actions on the Fiber
Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those ...
- 394
5
votes
2
answers
1k
views
Triviality of the adjoint and endomorphism bundles
Let $P$ be be a principal bundle over a manifold $M$ with structure group $G$, where $G$ is a Lie group. Let $E = P\times_{\rho} \mathbb{R}^{k}$ be a vector bundle associated to $P$ through a faithful ...
- 3,154
24
votes
1
answer
1k
views
All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$?
I ran into the following sanity check. Is the following statement true?
Every smooth fiber bundle (with compact fiber) over $S^2$ can be extended to a smooth fiber bundle over $\mathbb{C}P^\infty$ (...
- 707
5
votes
0
answers
129
views
Smoothing a continuous section in 1-jet bundle
Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle).
I ...
- 51
16
votes
2
answers
1k
views
Classification of $O(2)$-bundles in terms of characteristic classes
I had asked this question in stackexchange but there seems to be no consensus in the answer
It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by ...
- 3,154