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.
807
questions
2
votes
3answers
39 views
Difference between a function and a section of a fibre bundle
Suppose $E \rightarrow B$ with projection map $\pi$ and fibre F is a fibre bundle, with a section $\sigma$. How is $\sigma$ different from a function $f:B \rightarrow F$? The standard answer I find ...
0
votes
0answers
7 views
Examples of points in the total space of a fiber bundle
What are some examples of notation for points in the total space of a fiber bundle?
I think I have found one variant for vector bundles in this question about trivializations (something like $v^i \...
0
votes
1answer
51 views
Fiber from $S^1$ to $\mathbb{R} P^1$ that does not admit a section.
Consider the map $q:S^1 \to \mathbb{R} P^1$ given by $v \mapsto \mathbb{R} v$. I want to show that this map does not admit sections. I've previously shown that $S^1$ is diffeomorphic to $\mathbb{R} P^...
0
votes
0answers
20 views
Fiber bundle whose fibers are fiber bundles
Suppose $F \rightarrow E \rightarrow M$ is a fiber bundle. I've been considering situations where $F$ can be identified as a product space, but started thinking that it's not necessary for the ...
0
votes
0answers
30 views
Regarding singular points of a fiber bundle
Let $X$, $Y$ be two projective varieties over $\mathbb{C}$, where $Y$ is smooth, and let $f:X\rightarrow Y$ be an etale locally trivial fiber bundle, with fiber a variety $F$. Is there a relation ...
0
votes
1answer
37 views
Fiber bundle $S^{3} \to S^{7} \to \mathbb{H}P^{1}$
The map $h: S⁷ \to HP¹ $ given by $(a,b,c,d) \to \frac{a}{c}$ is a fiber bundle with fiber $S^3$ over basepoint $\infty=\frac{a}{0}$. Thus, $h$ induced a sequece
$$S^3 \cdots S^7 \to HP¹$$
Even more, ...
3
votes
1answer
38 views
Why doesn't the Homotopy group satisfy excision?
I'm studying higher homotopy groups from the book Algebraic Topology by author Allen Hatcher, there he says that the sequence $A \to X \to X/A$ does not induce an exact sequence of homotopy groups. ...
0
votes
0answers
33 views
subbundles & slice charts
Let $F\hookrightarrow E \overset{\pi}{\longrightarrow} M$, $F'\hookrightarrow E' \overset{\pi'}{\longrightarrow} M'$ two smooth fiber bundles.
If $F'\subset F$, $M'\subset M$, $E'\subset E$ are ...
6
votes
0answers
115 views
About theorem 16.2 in Switzer's Algebraic Topology
I have some difficulties understanding a very precise point of Switzer's proof of the existence and unicity of chern classes, which is Theorem 16.2 in his book. Unfortunatly there are many notations ...
3
votes
0answers
28 views
Lens space bundles over a circle must come from sphere bundle over a circle?
The question I would like to answer is the following.
From the classification of sphere bundles we know that the only orientable $S^3$ bundle over $S^1$ is $S^3 \times S^1$.
So suppose we have lens ...
0
votes
0answers
24 views
Principal bundle over another principal bundle again principal?
I want to expand upon an earlier (partially answered) question: Is a principal bundle of a principal bundle still principal?
As in said question, let $M, P_1, P_2$ be manifolds, let $P_1 \overset{\...
6
votes
1answer
56 views
Nontrivial bundle in physics
A student of mine asked me why we are studying bundles in theoretical physics, and why it isn't enough to look at product spaces.
I gave the example of the tangent bundle of the sphere, and of the ...
0
votes
0answers
26 views
connection from covariant derivate
Given a vector bundle $E$ with covariant derivate $\nabla:\mathfrak{X}(M)\times \Gamma E \longrightarrow \Gamma E$, an operator that $\forall g,h \in C^{\infty} M \;\; U,V \in \mathfrak{X}(M) \;\; X,Y ...
2
votes
0answers
33 views
Interesting examples of $Q$-fiber bundles
Let $X$ be a topological space. What are some interesting examples of $\mathbb{Q}$-fiber bundles (fiber bundles with fiber some $\mathbb{Q}$-vector space, though not necesarily vector bundles) on $X$? ...
3
votes
2answers
70 views
How many $G$ bundles are there over a surface?
Let $G$ be a finite group, $M$ a 2-dimensional manifold. There are
$$ \#\{ (a_1,b_1,a_2,b_2,\cdots,a_g,b_g) \in G^{2g}\,|\, \Pi_i[a_i,b_i]=1\} $$
many $G$ bundles over $M$ up to isomorphism, where $...
2
votes
2answers
39 views
Stiefel manifold as a fibre bundle over another Stiefel manifold.
I want to show $V_n(\mathbb R^k)$ is a fibre bundle over $V_m(\mathbb R^k)$ were $n>m$
Here $V_n(\mathbb R^k)$ is the Stiefel manifold, that is the set of all $n$ orthogonal frames in $\mathbb R^k$...
0
votes
1answer
25 views
Fibre bundle which is not an open map [duplicate]
I am looking for a counterexample to the following claim:
Let $p: E \rightarrow B$ be a fiber bundle, then $p$ is an open map.
This is true when $E \simeq F \times B$ where $F$ is the fiber and $...
2
votes
0answers
23 views
Non-trivial $C^{k_1}$ vector bundle that is trivial as $C^{k_2}$ fibre bundle
For given $\infty \ge k_1 \ge k_2 \ge 0$, are there non-trivial $C^{k_1}$ vector bundles over a "sufficiently nice" space that are trivial as a $C^{k_2}$ fibre bundle? How nice can the space be and ...
0
votes
0answers
13 views
Induced Projective map of a vector $X$
I've been going through connections on fibre bundles in Nakahara's Topology, Geometry and Physics from 2003 and I wondered if someone could answer this question for me (from page 395, exercise 10.1 a.)...
0
votes
1answer
32 views
Yang-Mills Field Strength Compatibility Function
I've been going through Nakahara's Topology, Geometry and Physics from 2003 and I'm struggling to fully understand this derivation from page 410:
Say there exists two different fields, $\mathcal{F}_i ...
8
votes
1answer
161 views
Higher covariant derivatives and the exterior derivative
Let me start with the following tl;dr version of my question
What is a higher-order derivative, in general? How does it relate to the exterior
derivative and to differential forms?
Suppose we ...
2
votes
1answer
31 views
Oriented sphere bundles - Why does pulling back an orientation along a path respect the restriction map?
I am reading Edwin Spanier's book "Algebraic Topology", theorem 5.8.19.
I do not understand the very first sentence of the proof. I hope there is not too much notational confusion here, but the ...
2
votes
1answer
46 views
The relationship between the tubular neighbourhoods of two diffeomorphic manifolds
I'm a beginner of this complex area and want to use the differential geometry as a tool to solve some control problems. So my statement might be a little bit inaccurate...I will try my best.
There ...
1
vote
0answers
43 views
G-connections and holonomy group
$\pi:E \longrightarrow M$ vector bundle with connection $\nabla$ and structure group $G\leq GL(V)$. I read three definition of G-connection:
1) The holonomy group at $p$ are the parallel transports $...
2
votes
1answer
50 views
Tangent and cotangent bundle as an associated bundle
In a book I read the isomorphisms below were mentioned without any explanation.
Is there any intuitive way to see these identities hold?
Let $M$ be a smooth $n$-manifold, $F(M)$ a frame bundle of $...
3
votes
0answers
25 views
Adapted Stochastic Processes and Fiber Bundles
Suppose that we have an adapted stochastic process. In other words fix $T>0$ and suppose that for all $t \in [0,T]$ $X(t)$ is a collection of random variables, $F(t)$ is a $\sigma$-algebra, $F(s) \...
0
votes
0answers
42 views
Proving linearity properties of the curvature in a vector bundle
Let $\mathcal H$ be a connection on a vector bundle $E\overset{\pi}{\longrightarrow} M$. For $X \in \mathfrak X(M)$, let $\bar X \in \Gamma \mathcal H$ his horizontal lift, let $k(v):=\text{pr}_2(Vv):...
1
vote
1answer
58 views
Principal bundle structure of $S^1\to S^1$ given by $z\to z^2$ map
Let $S^1$ be unit circle on $C^\star$(punctured complex plane). Consider $S^1\to S^1$ by $z\to z^2$ map. Identify $Z_2$ action as $e:z\to z$ and $g:z\to -z$ action on $S^1$ of domain. Let $U=S^1-\{1\}$...
3
votes
1answer
63 views
2 dimensional torus bundles over $\mathbb{S}^2$
With my limited knowledge of bundles, it seems that the isomorphism classes of principal torus bundles are in one-to-one correspondence with the homotopy classes of maps $[\mathbb{S}^2,\mathbb{C}P^\...
0
votes
0answers
45 views
The definition of principal G-bundle
I've encountered many different definitons of principal $G$ bundle from Morita's Geometry of differential forms , Hamilton's Mathematical gauge theory , Kobayashi and Nomizu's Foundations of ...
0
votes
0answers
22 views
Do locally trivial and smooth fibrations correspond to subbundles of the tangent bundle
Let $M$ be a manifold of dimension $n$. Let $p:E\rightarrow M$ be a locally trivial smooth fibration. Does this give us a way to construct a subbundle of the tangent bundle $TE$ of the manifold $E$, ...
4
votes
1answer
94 views
Existence of section on fiber bundles with contractible fibers.
I would like a proof (or a sketch of a proof or a reference) for the following fact:
If $\pi \colon P \to M$ is a fiber bundle of manifolds, and its fibers are contractible, then there exist a ...
1
vote
1answer
54 views
Intuition of normal bundle to a manifold
So i recently learned about fibre bundles and tangent bundles in particular. While the definition of tangent bundles seems quite intuitive, i really struggle to understand any definition of the ...
2
votes
0answers
46 views
Fiber bundle maps
I was reading many books about fiber bundles and they don't agree in a general definition of a fiber bundle map.
The categories I was trying to understand were "vector bundles", "principal bundles" ...
1
vote
1answer
97 views
The monodromy representation $\pi_1(X)\to Mod(F)$
Given a fiber bundle $F\to X\to E$, with $F,E$ are Riemann surfaces. I know the monodromy gives a permutation of the fiber. But how to see we have the monodromy representatio: $\pi_1(E)\to Mod(F)$? ...
1
vote
0answers
91 views
understand Torsion using Flat Bundles
My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arised a discussion when the map $H^2(X,\mathbb{...
1
vote
0answers
24 views
May a Trivial Principal Bundle Admit a Non-Trivial Reduction of the Structure Group?
Let $\pi\colon P \to M$ be a principal bundle with structure group $G$ with respect to the Lie group right action $\triangleleft\colon P \times G \to P$.
To my knowledge, a reduction of the structure ...
5
votes
1answer
174 views
$G$-bundles over $S^2$
In light of the fact that the only homogeneous (under a finite--dimensional Lie group action) $S^2$-bundle over $S^2$ is the trivial one. I would like to know if this fact is more general. i.e., is ...
0
votes
0answers
16 views
a little question about conjugate linear map
I'm reading Characteristic Classes written by Milnor and Stasheff,and the problem 13-F on page 153 says:
If $U\subset \mathbb C^n$ is an open set with coordinate functions $z_1,...z_n:U\to \mathbb ...
0
votes
0answers
21 views
Fiber of the Hitchin fibration
In the paper "More On Gauge Theory And Geometric Langlands" by Edward Witten ( https://arxiv.org/abs/1506.04293 ), he writes about the Hitchin fibration
$\pi: \mathcal{M}_H \rightarrow \mathcal{V}, (...
0
votes
1answer
53 views
Homology with local coefficients and flat bundles
I have a problem in understanding Prop.1.10 in
https://pages.uoregon.edu/sadofsky/691/sseq-local-coefficients.pdf
My interpretation is as follows:
Let $X$ be a smooth n-manifold and $U\to X$ be ...
2
votes
1answer
82 views
A small question about fiber bundle [duplicate]
I'm recently studying fiber bundles,and I find many examples of them are motivated from maps like $p:E\to B$ with homeomorphic fiber $F=p^{-1}(b)$.Therefore I'm wonder if the converse is true,that is ...
4
votes
1answer
82 views
Short Exact Sequences as Fiber Bundles
I am very much a visual thinker, and so to me, to feel that I have understood a concept, it is rather important for me to be able to "see a picture of it in my head". Now I recognize that for many ...
0
votes
1answer
34 views
SCHEME-THEORIC FIBER
I am reading INTERSECTION THEORY (Fulton), I am in page 15 and I do not understand the meaning of : SCHEME-THEORIC FIBER (I know what is a fiber, but what does scheme-theoric means?
1
vote
0answers
39 views
Grassmann manifold
Grassmann manifold $G_{k,n}$ is the set of k-dimensional subspaces of $\mathbb{R^n}$. Let’s consider the set of k-frames $V_{k,n}$.
I want to show that
$$ G \to V_{k,n} \to^{\pi} G_{k,n} $$ can be ...
5
votes
1answer
72 views
Is a Bundle whose Fibers are Homeomorphic a Fiber Bundle?
To avoid problems with the notoriously inconsistent notation/formalisms of geometry, I will define bundles and fiber bundles (as I call them) below:
We define a bundle $E \xrightarrow{\pi} B \ $ as a ...
2
votes
1answer
98 views
Topology of fiber bundles
Everything is over $\mathbb C$. Let $X\to \mathbb P^1$ be a fiber bundle with fiber some smooth projective variety $F$. If we have another such bundle $Y \to \mathbb P^1$, then, for which $p,q$ we ...
1
vote
1answer
58 views
Why the pullback bundle is a submanifold?
Is $f^*E$ (the pullback of a smooth bundle) an embedded submanifold of $N\times E$?
I know that it's well defined and at least immersed:
Let $\pi: E \longrightarrow M$ a smooth fiber bundle and $f:...
2
votes
0answers
96 views
$\mathbb{P}^2(\mathbb{R})$ can't be a smooth fiber bundle of a $1$-dimensional smooth manifold
This question was stated during a lesson where I asked my professor if every two dimensional smooth manifold arises as a tangent bundle of some one dimensional smooth manifold. My professor said that ...
0
votes
2answers
39 views
Fiber bundles of $G$-spaces
So if $G$ is a topological group and $H,\ J$ are closed subgroups such that $H\lhd J$, then the principal bundle $G/H\to G/J$ is trivial iff it has a global section.
I have questions about the ...
