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.
1,114
questions
4
votes
0
answers
25
views
A topological Ehresmann's theorem
A proper local homeomorphism is a covering space (with some mild conditions on the spaces involved). I want to know about the following generalization, which I believe is false but cannot come up with ...
- 7,742
1
vote
1
answer
61
views
Pullback of a covering map along the covering map.
Consider $H$ as a subgroup of $G$, we can push them to the level of classifying spaces
$p: BH \rightarrow BG$. This is a covering space with fiber $[G:H]$.
What is the pullback of this covering map ...
- 87
0
votes
0
answers
15
views
Fiber of continuous surjection from higher to lower dimensional unit balls
Lets $B_k=\{x\in\mathbb{R}^k|\Vert{x}\Vert\le1\}$ denotes the k-dimension closed unit ball.
Suppose $f:B_n\rightarrow{B_m}$ is a continuous surjective map where m<n. For an interior point y of $B_m$...
- 11
0
votes
0
answers
19
views
Bundle homomorphisms and sections of Hom-bundle
I reading "Complex Topological K-theory" by Efton Park and came across exercise 1.4.
Let $V,W$ be vector bundles over a compact Hausdorff space $X$.
a) Show that the collection $Hom(V,W)$ of ...
- 437
1
vote
1
answer
109
views
Riemann curvature tensor "as a" vector in the total space?
I have been reading through Wald's book on General Relativity, often using other sources to help gain a deeper understanding of the mathematics. Through this supplemental learning, I encountered ...
- 315
0
votes
0
answers
16
views
How to show that any section here will give rise to family of functions and how to specify the compatibility conditions
This question was asked in my exercises of differential geometry and I am completely struck on the problem.
So, I am posting it here in hope I will get some leads:
Question: Let $M$ be a manifold of ...
- 69
1
vote
1
answer
37
views
Erlangen-Style approach to Homogneoues Fiber Bundles
In connection with a philosophy of physics project I have recently been looking at fiber bundles from an Erlangen-inspired perspective. My inspiration is the following well-known result: Every smooth ...
0
votes
1
answer
34
views
Uniqueness (up to homeomorphism) of the fiber of a locally trivial fiber bundle with connected base
Let $\pi:E\rightarrow M$ be a locally trivial fiber bundle, i.e. for every $x\in M$ there is a neighbourhood of $U_x\ni x$, $U_x\subset M$, a space $F_x$, and a homeomorphism $\phi_x:\pi^{-1}(U_x)\...
- 182
2
votes
1
answer
136
views
How to push up / lift a connection one-form from the base manifold to the total space?
I am following this YouTube lecture by Schuller where he finds the appropriate formalism for the quantum mechanics in the physical curved space.
Everything makes sense to me but at the very end I see ...
3
votes
0
answers
42
views
Are there theorems in fiber bundle land and differential geometry land that make calculations in electromagnetism easier?
Some time ago while thinking about life and such, I thought to myself does recasting electromagnetism in bundle theory make certain calculations easier? To be precise are there theorems in ...
- 445
2
votes
2
answers
56
views
Classification of surface bundles and an isomorphism between $[S^n, BDiff(X)]$ and $\pi_{n-1}(Diff(X))/\pi_0(Diff(X))$
I'm trying to learn about the classification of surface bundles (in the smooth case, over a circle), and I might be missing some prerequisites. I am somewhat familiar with classifying spaces and ...
- 1,068
0
votes
1
answer
38
views
Notation clarification in fibre bundles
I am trying to comprehend the following material, I have some doubts mostly regarding the notations.
Chern Classes:
Throughout this section, we will use $\mathbb{Z}$ coefficients and let $\xi=(E, X, \...
- 107
1
vote
1
answer
56
views
Is two concentric $S^1$s topologically the same as a $S^1$?
In the case of a fibre bundle with a discrete fibre, in particular as given in the example below,
The covering map $S^n \rightarrow \mathbb{R} \mathbb{P}^n$ gives a fibre bundle $\{\pm 1\} \...
- 107
1
vote
1
answer
38
views
How is $\pi^{-1}\left(U_i\right)$ different from $U_i \times F$ in fiber bundles?
On defining a fibre bundle, it is argued that the projection map $\pi$ requires to be satisfied the condition that there is a homeomorphism $h$ such that the first coordinate coincide.
Definition (...
- 107
1
vote
0
answers
29
views
Principal bundle Isomorphism between the Hopf fibration and the orthornomal frame bundle of the tautological line bundle.
As usual, the Hopf fibration $S^3\to \mathbb{C}P^1$ is a principal $S^1$ bundle. Now, since we can embed the tautological complex line bundle $L$ in the trivial bundle $\mathbb{C}P^1 \times \mathbb{C}...
- 53
0
votes
0
answers
59
views
Second Chern class $c_2$ of $SU(n)$ bundle over $S^4$ v.s. $\pi_3(SU(n))=\mathbb{Z}$
Let us compare the four statements:
Consider the $SU(2)$ fiber over the $S^4$. Such that this $SU(2) = S^3$ fiber over $S^4$ gives a second Chern class $c_2$ on the $S^4$ with $c_2=1$.
Consider the $...
- 5,689
1
vote
1
answer
64
views
First Chern class $c_1$ of $U(1)$ bundle over $S^2$ v.s. $\pi_1(S^1)=\mathbb{Z}$
Let us compare the four statements:
Consider the $U(1)$ fiber over the $S^2$. Such that this $S^1$ fiber over $S^2$ gives a
first Chern class $c_1$ on the $S^2$ with $c_1=1$.
Consider the $S^1$ ...
- 5,689
1
vote
1
answer
96
views
Surjectivity of J homomorphism
Can somebody say something about surjectivity of the $J$ homomorphism $J_{7,7}$ : $\pi_{7}$SO($7$) $\rightarrow$ $\pi_{14}(S^7)$ ? Husemoller in Fibre Bundles says this :
Where $M_{n}^{1}$ denotes ...
- 23
1
vote
1
answer
67
views
Are measurable bundles fiber bundles?
I bumped into a definition of measurable bundle from page 101 of an introduction to smooth ergodic theory by Barreira and Pesin.
Let $E$, $X$ be measurable spaces and $\pi:E\to X$ be a measurable ...
- 91
1
vote
1
answer
49
views
Definition of vector bundle
Let $E$, $B$, $F$ be topological spaces, and $p:E\to B$ a continuous surjection. These data define a fiber bundle with fiber $F$ if, for every $b\in B$, exists an open set $b\in U\subset B$ and a ...
- 232
1
vote
0
answers
52
views
Chern class and multiplication structure
Let $c_i$ be the i-th Chern class of vector bundle $F→E→B$.
By Leray-Hirsch formula ,we have $H^*(PE)$ isomorphic to $H^*(B) \bigotimes\mathbb{Z}｛x^1,…,x^{n-1}｝$ as $H^*(B)-module$. Where $PE$ is ...
- 105
1
vote
0
answers
44
views
Multiplication structure of $E^∞$ and $H^*$
Given a fiber bundle $F→E→B$. Suppose the associated spectral sequence convergence and $E^∞$ isomophic to $H^*(E)$ as abelian groups.
My question is when they are isomorphic as rings?Do we have any ...
- 105
1
vote
1
answer
64
views
A morphism of fiber bundles induces a continuous map of the fibers
Let $E$, $B$, $F$ be topological spaces, and $p:E\to B$ a continuous surjection. According to my course, this data define a fiber boundle if, for any $b\in B$, exist a open neighborhood $U$ of $b$ and ...
- 232
2
votes
0
answers
41
views
Extending a section of vector bundle to union of open sets
I'm having some trouble with one part of Bott–Tu Theorem 6.8, which proves that homotopic maps induce isomorphic vector bundles under pullback.
The setup is that $\pi:Y\times I\to Y$ is the projection,...
- 1,312
2
votes
1
answer
36
views
Definition of a bundle associated to a character
In page 31 of the book "Heat kernels and Dirac operators" we find the following paragraph
The bundle of densities $| \wedge|$ is very closely related to the bundle of volume forms $\wedge^n ...
- 850
0
votes
0
answers
29
views
Bott Tu local constant presheaf
I´m reading Bott Tu Differential forms in algebraic topology and I have a question. In the page 169 we wnat calculate cohomology of a fiber bundle $\pi:E\to M$ with fiber $F$, and we define the ...
- 59
2
votes
2
answers
133
views
Product in the category of smooth vector bundles
Is the product in the category of smooth vector bundles just the direct product of smooth vector bundles?
More precisely, if $\pi:E \rightarrow B$ and $\pi': E' \rightarrow B'$ are two smooth vector ...
2
votes
1
answer
108
views
Connection on pullback principal bundle
For a Lie group $G$, let $\pi: E \rightarrow M$ be a principal $G$-bundle over a smooth manifold $M$. Let $\omega: TE \rightarrow Lie(G)$ be a connection 1-form on $E$. Let $f:N \rightarrow M$ be ...
1
vote
0
answers
35
views
Descomposition of Tangent bundled in sum of line bundles
Let $P \rightarrow M $ a G-structure where $G\subset GL(n)$ is a group of diagonal matrix. Prove that tangent bundle is isomorphic at $TM=\xi_1 \oplus \xi_2 \oplus \dots \oplus \xi_n$, where $\xi_j$ ...
0
votes
1
answer
39
views
"Cohomology Operations" - Steenrod-Epstein: frame fields on $S^{n-1}$ as sections of a fibre bundle
In the book "Cohomology Operations" by Steenrod and Epstein, at page 56, the author says that the existence of a field of $k$-frames (i.e. a function that maps a point of the sphere to a $k$-...
- 98
0
votes
1
answer
42
views
Bijective fiber bundle map is a bundle isomorphism proof attempt
We are working in the continuous category.
Definition 1: A bundle map between two fiber bundles $\pi_1:E_1\rightarrow B$, $\pi_2:E_2\rightarrow B$ is a continuous map $f:E_1\rightarrow E_2$ such that $...
0
votes
1
answer
46
views
Prove that $f:\mathbb{R}\rightarrow S^1$ is a fiber bundle of fiber $\mathbb{Z}$
I'm asked to prove that $f:\mathbb{R}\rightarrow S^1$, given by $f(t)=(\cos t,\sin t )$ is a fiber bundle of fiber $\mathbb{Z}$. I know that for every $p\in S^1$ I have to find an open set $U\subseteq ...
- 360
0
votes
0
answers
40
views
Principal Bundle Definition Question Action restriction
Let $G$ be a topological group and $\pi: P\rightarrow M$ a fiber bundle with fiber $G$ (working in continuous category). We say that $\pi$ is a principal G-bundle if
(1) $G$ acts freely on the right ...
0
votes
1
answer
126
views
Intuition of curvature
The only physical intuition of curvature that I know: parallel transport along a closed loop doesn't close (e.g. parallel transport of a tangent vector on a sphere).
The Ambrose-Singer theorem says &...
- 223
1
vote
0
answers
45
views
Are all $\mathbf{P^1}$-fibrations projective bundles? [duplicate]
Maybe there is a very simple answer to my question or a counterexample - I don't see it:
Let $\mathcal{X}\rightarrow C$ be a flat, smooth family with fibres $\cong \mathbf{P^1}$ of dim $2$ over a ...
- 1,529
4
votes
1
answer
77
views
When is a principal bundle with group $G \times H$ the product of a principal bundle with group $G$ and one with group $H$?
In general, it is not true that a fiber bundle with a product fiber is the product of two fiber bundles with the factors as fibers (think of vector bundles). However, I read somewhere that every 2-...
- 199
2
votes
1
answer
64
views
Surjectivity of bundle projection
In some textbooks, a fiber bundle is defined as follows: for topological spaces $E$, $B$ and $F$, we say that a continuous map $\pi \colon E \to B$ is a fiber bundle with fiber $F$ if for any point $b ...
- 217
0
votes
0
answers
49
views
Explicit Map from Stiefel manifold to Milnor Join
I am reading about universal $G$-bundles. There is a statement in the book "Fibre Bundles" by Dale Husemoller, which I would like to understand/prove. The statement is
Here $V_k(F^{mk}) = \{...
0
votes
0
answers
53
views
Trivialization of the orthonormal frame bundle
Given an $n$-dimensional manifold $M$ with Riemannian metric $\eta$, the orthonormal frame bundle over $M$ is defined as the set of all orthonormal frames over all points in $M$. Denote such bundle by ...
- 1,945
1
vote
1
answer
33
views
Scalar fields with definite weight under conformal maps as sections of some bundle
Let $(M,g)$ be a smooth Riemannian manifold. A scalar field in $(M,g)$ is merely a map $\phi:M\to \mathbb{R}$. Under a diffeomorphism $f:(M,g)\to (M,g)$ it transforms to $\phi' = \phi\circ f$. We then ...
- 25.1k
0
votes
1
answer
60
views
An orientable line bundle is always trivial?
In general, there exist non-trivial orientable bundles. But if we only consider line bundles, then orientable bundles are always trivial bundles. (Similarly, non-trivial bundles are always non-...
- 151
1
vote
1
answer
37
views
Why are transition maps of principal bundles given by multiplications of elements of the group?
It seems to be a common fact that transition maps of principal bundles are given by multiplication by a group element (in this post,for example.)
The setup is this: Consider a $G$-bundle $E \overset{\...
- 2,371
0
votes
1
answer
42
views
CW structures compatible with fiber bundle structures
Suppose $F \to E \to M$ is a vector bundle whose base space $M$ and fiber space $F$ are both CW complexes. Let us call a CW structure on $E$ a “compatible with the bundle structure” if, for any cell $...
- 1,653
5
votes
2
answers
163
views
Intuition behind associated bundles seems natural but I can't see how it's working out "intuitively"!
Suppose $L_1\rightarrow P \xrightarrow{\pi} M$ is a $G$-principal bundle with fibers diffeomorphic to $L_1 = G$, and that $\sigma$ is an effective action of $G$ on $L_2$, i.e. if $g.x := \sigma_g(x) = ...
- 766
0
votes
0
answers
27
views
How to find first nontrivial fibration in Whitehead tower of the $n$-sphere?
Consider the Whitehead tower of the $n$-sphere $X = S^n$:
$$... \rightarrow X' \stackrel{p}{\rightarrow} X,$$
where $X'$ is $n$-connected. Explicitly, what is $X'$ and what is the fibration $p$? How ...
- 507
1
vote
1
answer
72
views
Relation between Lens spaces and Hopf fibrations
Is a Lens space $L(p,q)$, for some particular values of $p$ and/or $q$, homeomorphic to an $S^3$ arising as a Hopf fibration?
For instance, from 1903.04942:
[...] the $S^1$ bundle over $S^2$ with ...
- 111
0
votes
0
answers
44
views
composition of principal bundles
I am struggling to figure out what is wrong with my following conclusion:
Let $A \overset{f}\longrightarrow B$ be a principal $F$-bundle and $B \overset{g}\longrightarrow C$ be a principal $G$-bundle....
- 1
1
vote
1
answer
97
views
If base space of fiber bundle is simply connected then so is the total space?
Let $F \to E \to B$ be a fiber bundle, and assume $E$ is connected and $F$ is simply connected. If the base space $B$ is simply connected, then is the total space $E$ also simply connected?
I think ...
- 507
1
vote
1
answer
55
views
Can I build fibre bundles over bundles ad infinitum?
Given a manifold, I can construct a fiber bundle over it, e.g. the tangent bundle. This fiber bundle is also a manifold, so I can construct a bundle over it, e.g. another tangent bundle. This is again ...
- 381
0
votes
0
answers
37
views
Existence and uniqueness of Riemannian manifold from prescribed connection [duplicate]
Say we start with $\mathbb{R}^d \times \mathbb{R}^d$, which we interpret as a fiber bundle. Then we imagine having an orthonormal basis for each fiber. Finally, we prescribe a connection; that is, ...
- 189