Questions tagged [fiber-bundles]
For questions about fiber bundle, which is a space that is locally a product space, but globally may have a different topological structure.
2
votes
1answer
56 views
What is Thom Isomorphism?
I am reading the following post on Thom Isomoprhism and I also have the Thom Isomoprhism from Hatcher's, Corollary 4.9,pg441
nlab's: Let $V \rightarrow X$ be a rank $n$ vector bundle over a simply ...
2
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0answers
20 views
Is the map $\bigsqcup_{w } \frac{BwP}{P} = \frac{GL_n(\mathbb{C})}{P} \to GL_n(\mathbb{C})$ via $bwP \mapsto bw$ a morphism of algebraic varieties?
Let $G = GL_n(\mathbb{C})$, $P \subseteq G$ a parabolic subgroup, and consider the decomposition $\frac{G}{P} = \bigsqcup_{w \in W^P} \frac{BwP}{P}$ where $W \cong S_n$ is the Weyl group of $G$, $W_P$ ...
1
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0answers
20 views
Classifying maps and transition functions
Suppose a space $X'$ is obtained from $X$ by attaching an $i$-cell, i.e., $$X' = X \cup_\phi e^i$$ where $\phi: \partial e^i \to X$ is the attaching map. Let $G$ be a structure group, and $P$ be a ...
0
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0answers
16 views
Change of trivialization
Let $\pi:P\to M$ a smooth $G$-principal bundle with action $\beta:G\times P\to P$, $G$ a Lie group and $M$ a diff. manifold. Let also $T^*P$ be its cotangent bundle and $\sigma$ be a section of this ...
1
vote
0answers
15 views
Principal fibre bundles with constant transition functions
I guess this is not limited to principal bundles, however those are my primary interest in asking this question.
Let $(P,\pi,M,G)$ be a principal fibre bundle over $M$ with structure group $G$.
...
4
votes
1answer
49 views
Classification of circle bundles over $\mathbb{RP}^2$
I am trying to understand isomorphism classes of bundles
$$\mathbb{S}^1\hookrightarrow E\to \mathbb{RP}^2.$$
These are classified by homotopy classes $[\mathbb{RP}^2,BAut(\mathbb{S}^1)]$.
ATTEMPT 1.
...
0
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0answers
12 views
If $G$ is a Lie group and $H$ is a closed Lie subgroup, then $G\to G/H$ is a principal- $H$ bundle.
Let $G$ be a Lie group and $H$ be a closed Lie subgroup of $G$. Let $G/H$ has the quotient topology. Then $ p: G\to G/H$ is a principal-$H$ bundle.
I was reading the above theorem from the book ...
1
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0answers
44 views
Fibre bundles over interval are trivial - but in a smooth way?
I'm trying to understand the homotopy invariance theorem for smooth fibre bundles.
A main step is to show that the interval $[0,1]$ only admits trivial fibre bundles.
I found this proof (see Lemma 3) ...
6
votes
0answers
48 views
induced map in homology on a fiber bundle
Let $F \rightarrow E \rightarrow B$ be a fiber bundle of compact connected smooth manifolds and $B$ simply connected. Suppose that there is a map
$f: E \rightarrow E$ that covers a map $g: B \...
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0answers
20 views
Reference Request for $\frac{G}{B} \rightarrow \frac{G}{P}$ being a fiber bundle for $G = GL_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, let $B$ be the Borel subgroup of upper triangular matrices, and let $P$ be a parabolic subgroup. Then we may identify $G/B$ with the complete flag variety and $G/P$ with ...
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0answers
24 views
Proof fiber bundle construction theorem
Do you think this proof is correct?
https://en.wikipedia.org/wiki/Fiber_bundle_construction_theorem
Have you got any further reference?
1
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0answers
23 views
Jet bundle cohomology
Consider a base manifold $\mathcal{M}$ and a smooth bundle $E\to\mathcal{M}$. I am interested in the cohomology groups of the variational bicomplex associated with the jet bundle $J^{\infty}E$. In ...
1
vote
0answers
23 views
Construction of fiber bundles via transition maps: how to find smooth structure.
I'm trying to prove the reconstruction theorem for general smooth fiber bundles, that is, no extra algebraic structure over my bundle, only smoothness.
Let $M$ and $F$ be smooth manifolds of ...
3
votes
2answers
53 views
Monodromy when fiber is connected
When I learned about monodromy, it was in the context of covering spaces. There, if $p$ is the covering map and $l$ is a loop with $\gamma(0)=\gamma(1)=x$, then a $\bar \gamma$ lift of $\gamma$ is ...
0
votes
0answers
22 views
parallel transport is independent of the bases chosen in each of the two tangent spaces
Connections on principal fibre bundles
In the above set of notes on page-3 section 2.2 under the heading Parallel transport there is a statement that equivariance ensures that the parallel transport ...
2
votes
1answer
53 views
Consequences from Bott Periodicity
I have some questions about some arguments used in the discussion about consequnces of Bott periodity in in A Concise Course in Algebraic Topology by P. May at page 207. Her the excerpt:
Following ...
5
votes
0answers
164 views
Are most physics books wrong about the covariant derivative and connection?
I have always read in many physics books that a valid way of intuitively introducing the covariant derivative and the connection was the following: (example in GR but same thing for gauge theories)
...
1
vote
2answers
27 views
G-structure on fiber bundles
I was studying about fiber bundles with a $G$-structure, and I arrive to the definition (below all the spaces are smooth manifolds):
Given a topological group $G$ a $G$-structure on fiber bundle $(E,...
4
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0answers
65 views
Problem with equivalent definition of a integrable $G$-structure
I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 2. It this proposition:
My problem is that I don't understand the converse of this ...
0
votes
1answer
53 views
Section and pull-back bundle
I find in general that the pull-back of a section of a vector bundle is a section of the pull-back bundle, but this seems to be false for the cotangent bundle.
Let $\phi:M\to N$ be a smooth map, $\...
3
votes
0answers
51 views
Pulling back $\mathfrak g $-valued 1-forms
Let $\omega$ be a vector-valued 1-form where the vector space is the Lie algebra $\mathfrak g$ and let $\mathcal P$ be a trivial principal bundle over a manifold $M$. Thus
$$
\omega\in\Omega^1(\...
1
vote
1answer
60 views
Tangent bundle of a trivial bundle
I was asking myself if the tangent bundle of a trivial bundle $\mathcal{P}=M\times V$ with fiber $\pi: \mathcal{P}\to M$ (actually it would be a principal trivial bundle, but I think it doesn't matter ...
2
votes
1answer
54 views
Torus bundle over a torus
Let $G$ be a nilpotent Lie group and $H<J$ be two closed subgroups such that
$$G/H\to G/J$$ is a fiber bundle with fibers $J/H$.
Suppose that $G/J=(S^1)^m$ and $J/H=(S^1)^m$. Is the bundle trivial?...
2
votes
0answers
38 views
Contractible fibers implies $H^*(E) \cong H^*(B)$ for $p:E \rightarrow B$?
I believe the following is true:
Let $F\rightarrow E \rightarrow B$ be a fiber bundle. If $F$ is contractible, then $H^*(E;R) \cong H^*(B;R)$ for a commutative ring $R$ with identity.
I will ...
0
votes
1answer
43 views
Question about group of automorphism of some $G$-structure.
I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 15.
I don't understand why $U$ consists of transformation $a$ of $M$ that leave each ...
3
votes
1answer
37 views
Can we detect non-trivial bundles using characteristic classes for a custom structure group?
Suppose $E\to X$ is a fibre bundle where say $X$ is a CW complex, with structure group $G$, i.e. it is classified up to bundle isomorphism by a homotopy class of maps $c\colon X\to BG$. It's not ...
0
votes
1answer
28 views
Help find mistake in conclusion about vector fields on principal bundle
Let $P$ be a principal fiber bundle with structure group $G$ acting freely on the right. Let $T_uP = G_u + Q_u$ be a connection on $P$, where $u\in P$ and $G_u$ is the vertical space consisting of ...
0
votes
1answer
69 views
Do the isomorphism classes of fiber bundles constitute a set?
Let $ M $ and $ F $ be smooth manifolds. Is the collection of
isomorphism classes of fiber bundles of fiber type $ F $ over $ M $ a set or not and why?
2
votes
1answer
51 views
Fiber Bundles over spheres
Let $F$ be any topological space.
In many books, for example in http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html, it is said that a continuous map (characteristic map) (where $Homeo(F)$ has the ...
5
votes
1answer
106 views
Is the Hairy Ball Theorem equivalent to saying that the Hopf Fibration has no global sections?
The Hairy Ball Theorem states that $S^2$ has no nonvanishing tangent vector fields. But if we did have such a field then we could normalise each vector so that it lay on the unit circle of the tangent ...
7
votes
1answer
102 views
Local triviality for the fiber bundles
The notations are as follows:
\begin{align*}
& \operatorname{Diff}^+(\mathbb{D}^2):= \{ f:\mathbb{D}^2\to \mathbb{D}^2\ |\ f \text{ is an orientation preserving diffeomorphism} \},\\
& \...
0
votes
1answer
25 views
The orbit space GL(n,R)/O(n)
If $G= GL(n,\mathbb{R})$ and $H= O(n)$ then why the orbit space
$G/H$ is homeomorphic to the space of all upper triangular
matrices with positive diagonal entries?(Here action of $H$ on $G$ is the ...
0
votes
0answers
25 views
Natural bijection between equivariant maps and sections, Principal bundles
This is in page 11, Prop 6.1 of Mitchell's notes on Fibre bundles. I am quoting the proposition below.
Let $\pi:P \rightarrow B$ be a principal $G$ bundle, $X$ a right $G$-space. $Hom_G(P,X)$ ...
2
votes
2answers
147 views
Doubt in the definition of principal bundle.
I am following Kobayashi and Nomizu( Foundations of differential geometry) Volume 1.
In page number 50 while defining principle $G$-bundle $P(M,G)$ they said that the action of the Lie group $G$ ...
0
votes
1answer
40 views
Question about a proposition in Kobayashi book about $G$-structures.
I'm reading Kobayashi's book, Transformation Groups in Differential Geometry and at the page 3 is this proposition:
the definition of $K$ is given here:
My question is why this proposition is ...
1
vote
1answer
35 views
Extending a monomorphism of bundles, Lemma 7.3, Atiyah, Shapiro
This is from Lemma 7.2, pg17
Let $E,F$ be vector bundles on $X$ and $f:E \rightarrow F$ a monomorphism on $Y$. Then if $\dim F > \dim E+ \dim X$, $f$ can be extended to a monomorphismon on $X$ ...
1
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0answers
28 views
Definition of fibration from a compact Kahler manifold to a compact complex curve
Let $X$ be a compact Kähler manifold and let f be a holomorphic map from $X$ to a compact complex curve $C$. We say that f is a fibration if it is surjective and if its generic fiber is connected.
I ...
0
votes
0answers
75 views
Must every manifold bundle be a fiber bundle?
I have been learning differential geometry from this lecture series by Frederic Schuller's. Here he defines a bundle (of topological manifolds) as
A triple, $(E, \pi, M),$ where $E$ and $M$ are ...
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0answers
27 views
$H$ is admissible, then $P\rightarrow P/H$ is a principle $H$ -bundle.
Proposition 3.5, page 5: Suppose $P \rightarrow B$ is a principal $G$-bundle, and let $H$ be an admissible subgroup of $G$. Then the quotient map $P \rightarrow P/H$ is a principal $H$-bundle.
...
1
vote
1answer
48 views
Surjective homotopy equivalence which is not a fibration?
This is probably obvious to topologists so I'll just come right out with the question:
What is an example of a surjective homotopy equivalence $E \to B$ of path-connected CW complexes which is not a ...
0
votes
1answer
25 views
Are there topologically trivial bundles with a nonzero curvature?
A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder.
Is it possible to have a ...
1
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0answers
25 views
A theorem on fibre bundles, $H^k(E) \cong H^{k+n}(E,E_0)$.
This is a theorem about fibre bundles, stated in page 21, Theorem 3.6
Let $\xi= (E,B,p)$ be an $n$-vector bundle. Let $F_0$ be the fibre $F$ without its nonzero element, and $E_0$ be the total ...
0
votes
0answers
34 views
differential of integration over fibers
Is there a way to express the differential of a fiber integral that is similar to Reynolds transport problem or Leibniz rule? Here is the following setting:
Let $\pi: X\mapsto Y$ be a fiber bundle on ...
2
votes
1answer
37 views
Integral curves are horizontal
Suppose we're given an Ehresmann connection on a submersion (or a fiber bundle, but I don't think it's needed) $\begin{smallmatrix}X\\
\downarrow\\
Y
\end{smallmatrix}$. Given a curve $\gamma$ in the ...
0
votes
0answers
59 views
Descent data and trivialization of bundles via coherent isomorphisms of fibers
In this MO question I tried to understand how a trivialization of a bundle (continuous map) $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ is related to a coherent family of isomorphisms ...
0
votes
0answers
9 views
Infinitely many fixed-rank bundles on projective space?
I was asked as a question whether a topological space can have infinitely many non-isomorphic bundles (with fixed rank). As a hint we were recommended to consider projective space. I don't really know ...
0
votes
1answer
45 views
Nontriviality of the Hopf Fibration
A simple question how to understand why even though locally $S^3$ is homeomorphic to $S^2\times S^1$, how do you see that globally this is not true?
2
votes
1answer
37 views
Is there a group whose manifold is a fiber bundle with base is $S_1$ and fiber $\mathbb{Z_2}?$
Let's consider a fiber bundle with base $S_1$ and fiber $\mathbb{Z}_2$. I want this manifold to be topologically non-trivial, the edge of the Möbius strip.
How do I know if is it possible to ...
0
votes
0answers
9 views
Universal standard principle bundle of the Gauge group
Let $p:P\rightarrow B$ be a principal $G$-bundle, and let $E_G\rightarrow B_G$ be the universal bundle for $G$.
Let $Aut_B(P)$ be the subspace of $Map_G(P,P)$ of maps over $B$. Let $Map_P(B,B_G)$ be ...
1
vote
0answers
13 views
Is a union of compatible fiber bundles a fiber bundle?
Let $\eta_i:E_i\rightarrow B_i$ be fiber bundles, and let $f_i:\eta_i\rightarrow\eta_{i+1}$ be maps of fiber bundles.
Assuming that the bundles maps $\eta_i$ are maps of smooth compact manifolds, and ...
