Questions tagged [fibre-bundles]
for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.
338
questions
4
votes
1
answer
326
views
Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$
Define the map $$P:TS^{n}\to S^{n} \;\;\;\text{by}\;\; P((x,v))=\frac{x+v}{\parallel x+v \parallel}$$ where $$TS^{n}=\{(x,v)\in S^{n} \
\times \mathbb{R}^{n+1}\mid v \perp x \}$$
This map is ...
- 312
5
votes
2
answers
510
views
A $\mathbb{R}^{n}$ -fiber bundle which do not admit a n-dimensional vector bundle structure
Is there a fiber bundle $(E,B, \mathbb{R}^{n})$, with typical fiber $\mathbb{R}^{n}$, such that there is no any $n$-dimensional vector bundle structure on the pair $(E,B)$? That is there ...
- 312
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
2
votes
2
answers
325
views
Constructing jet bundles from a cocycle of smooth transition functions
Suppose we are given an open cover $\mathcal{U}=(U_{i})_{i \in I}$ of a smooth manifold $M$, a cocycle of smooth transition functions $g_{ij}: U_{ij} \to G$ where $G$ is a Lie group, and a (not ...
- 5,933
3
votes
1
answer
316
views
Transformation between two conventions of Hitchin equation
Recall that for a given Riemann surface $\Sigma$ Hitchin's self-duality equation consists of a complex rank $r$ vector bundle $E$ (with degree 0 for simplicity), a connection $d_A: \Omega^k(\Sigma, E) ...
- 367
10
votes
3
answers
622
views
Spin 4-manifold bounded by a mapping torus of tori
Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...
- 1,582
5
votes
2
answers
384
views
Examples and references for Kan-like extensions?
Left and right Kan extensions are both "push-forwards" that share a certain property. I'd like to hear other, non-Kan, examples of such push-forwards, as well as perhaps a better way to think about ...
- 8,559
1
vote
1
answer
504
views
group extensions and principal bundles
Given an extension of groups, say
$$0 \to H \stackrel{i}{\to} G \stackrel{q}{\to} G/H \to 0$$
and a $H$-principal fiber bundle $P \to X$, one can use induction to obtain bundles with fibers $G$ and $...
- 154
2
votes
2
answers
967
views
Is there always a universal bundle over a classifying space?
Consider some kind of bundles, for instance vector bundles or fibre bundles with a certain structure group, such that there are bundle morphisms and pull-backs.
Let then $F(X)$ denote the ...
- 1,062
1
vote
0
answers
46
views
How to find $\beta^\prime_e(t)$ where $\beta_e(t)=\textrm{Hol}^\sigma_{\gamma_{1, t}}(e)$?
Let $p:E\longrightarrow B$ be a surjective submersion and $\sigma: p^*(TB)\longrightarrow TE$ a complete connection. Given a path $\gamma: [a, b]\longrightarrow B$ and $s, t\in [a, b]$ such that $s<...
- 383
2
votes
1
answer
566
views
What is the structure group of the Hopf fibration $S_1\rightarrow S_3 \stackrel{p}\rightarrow S_2$?
I am studying fiber bundles and have thoroughly reviewed the famous example of the Möbius strip. In that example, I learned how to discover that the structure group of the Möbius strip fiber bundle ...
18
votes
2
answers
2k
views
Serre fibration vs Hurewicz fibration
What are the simplest/ typical examples of a Serre fibration which is not a Hurewicz fibration? Is it something pathological?
Sorry if the question is too elementary for MO.
- 21.1k
5
votes
1
answer
392
views
Categorification of covering morphisms
Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct ...
- 10.3k
2
votes
1
answer
485
views
Normal bundle to fibers of a rational morphism
Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...
- 165
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 ...
- 3,064
4
votes
1
answer
2k
views
The space of homotopy classes of maps of products of spheres
Proposition 17.6.1 of "Differential form in Algebraic Topology" by Bott and Tu proves the following beautiful result:
$[S^{q}, X]\simeq \frac{\pi_{q}(X,x)}{\pi_{1}(X,x)}$
where $S^{q}$ is the $q$-...
- 3,064
4
votes
1
answer
719
views
When is the semidirect product of principal fiber bundles a fiber bundle
Let $P_{H}$ be a principal bundle over a manifold $M$ with fiber the Lie group $H$ and let $P_{G}$ be a principal bundle with fiber the Lie group $G$ over the same manifold $M$. Let $h_{ab}\colon U_{...
- 3,064
5
votes
0
answers
257
views
How to visualize the dual objects of jets of functions?
I work with a smooth $f: M \to \Bbb C$ and I would like to have an object mimicking the concept of "$k$-th order differential" from multivariate calculus. For various reasons that are not important ...
- 5,282
-1
votes
1
answer
229
views
Construction of fibration over Riemannian Manifold
Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$.
More precisely:
I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some $C^{\infty}...
- 4,969
2
votes
0
answers
97
views
Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold
The symplectic blow-up of a compact symplectic manifold $(X,ω)$ along a compact symplectically embedded submanifold $(M,σ)$ results in another compact manifold $(\tilde{X},\tilde{ω})$ given by
$$\...
- 191
0
votes
1
answer
322
views
group actions on fibre bundles
Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram
If $\xi$ is a trivial bundle, i.e....
- 1,970
8
votes
2
answers
444
views
The action of $GL_{\infty}$ on the infinite wedge space
This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...
- 671
2
votes
1
answer
106
views
A question on 2-bundles
In this paper, the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a ...
- 733
2
votes
2
answers
171
views
Is every closed Sasakian 3-manifold a circle bundle on a Riemann surface?
It suffices to say that all circle bundles on compact Riemann surfaces admit the structure of a closed Sasakian 3-manifold. The question is, the converse of this statement and/or what are the ...
- 133
2
votes
0
answers
603
views
Some examples of non trivial principal bundles
1.Is there a nontrivial pricipal bundle $P(M,G)$, with $G$ connected, such that the total space $P$ admit a foliation such that each leaf is diffeomorphic to $M$(Not necessarily via projection $\pi$...
- 312
3
votes
1
answer
329
views
"Ambient homotopy" between preimages under a fiber bundle?
Choose a notion of an "ambient homotopy" between maps of topological spaces. For example, say that two embeddings $Y \rightarrow X$ are ambiently homotopic if there is a path between them in the space ...
- 2,302
1
vote
1
answer
277
views
triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle
Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle
$$
\xi: \...
- 2,213
5
votes
2
answers
781
views
triviality of Whitney sums of a vector bundle
Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid\sum_{i=1}^4x_i=0\}.$$ The alternating group $A_4$ acts on $V$ by
$$\sigma(x_1,x_2,x_3,x_4)=(x_{\...
- 2,213
7
votes
1
answer
515
views
vector bundles associated to a covering space
Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S_n$ on $M$. Then we have a $n!$-sheeted ...
- 2,213
11
votes
1
answer
2k
views
Does pullback in the category of smooth manifolds always exists?
I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist.
Remarks:
1) A pullback in a certain category is defined as ...
- 6,621
5
votes
0
answers
238
views
characteristic classes of a covering space with symmetric group action
Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...
- 2,213
11
votes
2
answers
960
views
first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles
Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(...
- 2,213
4
votes
2
answers
1k
views
Cup product of cohomology in a Serre spectral sequence
How to use Serre spectral sequence to compute cup product structures?
Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...
- 2,213
12
votes
1
answer
789
views
Stiefel-Whitney class of fibre bundles
Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...
- 2,213
2
votes
0
answers
41
views
Why generalized vectors can be written locally as sum of vectors and 1-forms?
I would like to understand better this point.
In generalized complex geometry the generalized bundle $E$ is defined as a non-trivial fibration of the cotangent bundle $T^*M$ over the tangent bundle $...
- 159
6
votes
0
answers
424
views
When is a circle fibration a circle bundle?
Let $\pi : E \to B$ be a Serre fibration over a CW complex, with circle fibers.
In the orientable case, it is easy to see that $\pi$ is fiber homotopy equivalent to a principal $SO(2)$--bundle.
...
- 589
2
votes
0
answers
121
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$. ...
- 2,213
2
votes
0
answers
217
views
cross-sections of a sphere bundle
Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...
- 2,213
1
vote
0
answers
123
views
Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$
I ask this question in MSE and I received no answer, so I repeat it here:
Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...
- 312
13
votes
2
answers
2k
views
Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
I previously asked this on Math.SE but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\...
- 655
3
votes
0
answers
199
views
Fiber Bundle with a perfect bilinear map
Let $\pi:X\rightarrow Y$ a double cover of Riemann surfaces, and consider a $SL_n-$bundle $E$ with a perfect $\mathcal O_Y-$bilinear pairing $$\psi:\pi_*E\times\pi_*E\rightarrow\pi_*\mathcal O_X$$
...
- 1,871
7
votes
2
answers
945
views
Explicitly describing the region of the plane "outward of" a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$
Some Context:
I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...
- 279
18
votes
3
answers
906
views
When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$
For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case ...
- 1,873
2
votes
0
answers
235
views
Almost complex structure gluing
Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ ...
6
votes
1
answer
414
views
When are principal bundles preserved by colimits?
Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant ...
- 489
5
votes
1
answer
436
views
Frame-bundle reduction from spinor-bundle reduction
Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...
- 3,064
9
votes
3
answers
512
views
A conjecture about parallelizable generalized spheres
Let $S^{d}$ denote the standard $d$-dimensional sphere. I heard from a physicist that from physical arguments they have been able to show that the vector bundle:
$E_{d} = TS^{d}\oplus \Lambda ^{d-2}...
- 3,064
2
votes
1
answer
160
views
Space of invariant sections
I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...
- 209
8
votes
1
answer
1k
views
Curvature of a principal bundle and the exterior covariant derivative
I am sorry if this is too elementary; I had posted it on math.stack but no one answered.
Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...
- 3,064