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.
895
questions
2
votes
1answer
27 views
$\text{SO}(4)$ is homeomorphic to $\text{SO}(3)\times S^3$
Is there a reference for a proof that $\text{SO}(4)$ is homeomorphic to $\text{SO}(3)\times S^3$? Since $\text{SO}(4)$ acts transitively on $S^3$ with stabilizer $\text{SO}(3)$, we have a fiber bundle ...
2
votes
0answers
27 views
Induced representations built on sections of an associated vector bundle. Questions on notations
Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions $\varphi$ on this group:
$$
{\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\...
0
votes
1answer
17 views
Image of path under fiber transport of a right G-principal cover is G-equivariant
I have seen many different definitions of $G$-principal covers (which I assume are largely equivalent) so in this context, a right $G$-principal cover is a properly discontinuous action of $G$ on a ...
4
votes
1answer
65 views
Various proofs: pairs of points in a circle = Möbius strip
It is well-known that the space of "pairs of points in a circle" (also called the symmetric square of $S^1$) can be identified with a Möbius strip. I don't know where the idea originates, ...
1
vote
1answer
35 views
A covering with connected tocal space and finite fibres
I have a covering map $p:X\to Y$ with $X$ connected and the fibres are finite.
What could I conclude?
Could I conclude that $Y$ is connected (or only locally)?
Could I conclude that the fibers have ...
1
vote
1answer
29 views
Trivialising homeomorphisms for a principal $G$-bundle as $G$-space morphisms
Let $P$ by a principal $G$-fibre bundle over a locally-compact Hausdorff space $X$. Denote by
$$
h: U \times G \to P|_{U}
$$
a trivialising homeomorphism for a trivialising open set $U \subseteq X$. ...
1
vote
0answers
73 views
Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected?
Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected? What if we substitute $\mathbb{R}^n$ and $\mathbb{R}^m$ with open neighborhoods of ...
1
vote
0answers
32 views
Last step in Voisin's proof of Ehresmann's lemma
I'm reading Voisin's proof of Ehresmann's lemma, given in the book Hodge Theory and Complex Algebraic Geometry I as theorem 9.3; the proof is essentially reproduced in this answer.
To summarize, there'...
2
votes
0answers
23 views
Finding a natural homomorphism between two relative cohomology groups
I'm trying to better understand how relative homology/cohomology works in order to make use of them for a specific example in mind. Here I'll be working with cohomology over the integers.
Let $A \...
0
votes
0answers
30 views
If the bundle of orthonormal frames has a continuos/smooth global section will the bundle of spin frames also have one?
Let $(M,g)$ be a semi-Riemannian manifold with metric of signature $(p,q)$. I believe the signature of the metric is not relevant for this discussion so I leave it arbitrary (corrections to this are ...
0
votes
0answers
32 views
Existence of fibre metric compatible connections on vector bundle
Given a fibre metric $h$ over a vector bundle $E$, how can I show there always exist connections on $E$ such that it is compatible with $h$ in the sense that given any two smooth section $\sigma,\psi$,...
5
votes
1answer
129 views
Compactness about fiber bundle
I am working on a problem (problem 10.19 (d)) in John M. Lee's Introduction to Smooth Manifold.
Assume that $\pi$: $E$ $\rightarrow$ $M$ is a fiber bundle with model fiber $F$, I need to prove if $E$ ...
5
votes
0answers
52 views
Dot product of functions on cosets
Let the measures of locally compact groups $\,K < G\,$ be $\, dk\,$ and $\, dg\,$, correspondingly.
For a Hilbert space $\mathbb{V}$ equipped with a dot product $\,\langle~,~\rangle\,$, introduce ...
1
vote
0answers
21 views
Coproduct in the category of differential bundles
Just for clarity I briefly state the definition I am using. A differential bundle is a differentiable submersion $\pi: E\to B$, where $E$ and $B$ are differential manifolds. Observe that I am not ...
0
votes
0answers
22 views
Applying the Leray-Hirsch theorem on certain manifolds
Let $M$ be a connected closed orientable manifold with Euler characteristic $0$. Let $TM$ be the tangent bundle over $M$ and $SM$ be the induced sphere bundle with fiber $S^{n-1}$.
Looking at page 432 ...
1
vote
0answers
21 views
Extension of a principal bundle
Let $G$ be a Lie group and $M$ a smooth manifold. The universal cover $\tilde M$ is a principal $\pi_1(M)$-bundle over $M$.
Question: Why does any homomorphism $\phi:\pi_1(M)\to G$ induces a principal ...
2
votes
1answer
135 views
Integration along fibers
The following statements are from the book heat kernel and dirac operator chapter 1.
" Let $\pi : M \rightarrow B $ be a fiber bundle with n-dimensional fiber, such that both M and B are ...
2
votes
1answer
50 views
Well definedness of the correspondence between connections on $B$ and sections of $J^1B$
Let $M$ be a smooth manifold and $\pi\colon B\to M$ a fiber bundle on $M$. A (Ehresmann) connection on $B$ is a subbundle $H$ of $TB$ such that $TB=H\oplus V$, where $V$ is the bundle of vertical ...
0
votes
0answers
43 views
How does BDiff$(F)$ classify smooth $F$-bundles and why is it topologized with the Whitney $C^\infty$-topology?
We consider a smooth fiber bundle $p:E \rightarrow B$ with fiber $F$. I want to understand how such a fiber bundle is classified by a map $B\rightarrow $BDiff$(F)$. Namely, we should have a bijection
$...
1
vote
0answers
21 views
Isometries of fiber bundles
Let $F\to S\overset{\pi}{\to} B$ a fiber bundle or Riemannian manifolds with totally geodesic fibers (i.e. $F_x\cong\pi^{-1}(x)\subset S$ is a totally geodesic submanifold for each $x\in B$).
Question:...
4
votes
1answer
63 views
What does it mean geometrically that $H^*(B\mathbb{Z}_2) = \mathbb{Z}[a]$?
I found cited in a physics paper that $H^*(B\mathbb{Z}_2) = \mathbb{Z}_2[a]$ and I wanted to know what geometric shapes this corresponded to. Notice the answer is a polynomial ring. Here $\mathbb{Z}...
3
votes
0answers
14 views
Bijection between quotient space of principal smooth and topological bundle.
I was reading the notes on bundles "Differential Topology of Fiber Bundles" by Karl-Hermann Neeb and on the final pages (138-139) "Homotopy theory of bundles" the author author ...
1
vote
0answers
22 views
Lie-Algebra Bundle and Wigner-Inönü Contraction
Here it is stated, that for a Lie-Algebra Bundle the isomorphism-class of the Lie-Algebra is locally constant, if the Lie-Algebra is rigid. Especially it is stated, that semisimple Lie-Algebras are ...
3
votes
0answers
29 views
About uniqueness of lifted paths in the phath lifting theorem in fiber bundles
Im trying to understand the path lifting theorem (in the context of locally trivial fiber bundles presented in the book "Manifolds, Tensor Analysis and Applications", by Jerrold E. Marsden ...
6
votes
1answer
162 views
Realising the non-trivial orientable $S^2$-bundle over $T^2$ as a quotient of $S^2\times T^2$
Using the fact that $\operatorname{Diff}^+(S^2)$ deformation retracts onto $SO(3)$, one can show that over any connected, closed, smooth surface, there are two orientable $S^2$-bundles, the trivial ...
2
votes
0answers
25 views
Fibration of $\mathbb{P}\left( \mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus n}\right)$ by global sections
Consider the bundle $\mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2} \to \mathbb{CP}^1$. The collection $A:=\left\{ \left( \bar{b}- \bar{a}\zeta, a +b\zeta \right) \mid (a,b) \in \mathbb{C}^2 \...
0
votes
1answer
52 views
Do the fiber bundles over an Abelian category form an Abelian category? [closed]
Assume I have an Abelian category $C$. Is the category of fiber bundles, where the fibers are objects of $C$ Abelian?
2
votes
1answer
44 views
A subbundle of $(0,1) \times \mathbb{R}^m$ over $(0,1)$ is trivial if derivatives of all sections are in it
Let $K \subseteq (0,1) \times \mathbb{R}^m$ be a subbundle of rank $k$ of the trivial bundle $(0,1) \times \mathbb{R}^m$ over $(0,1)$, that is, $K$ is a differentiable submanifold of $(0,1) \times \...
1
vote
1answer
49 views
Surface bundle with flat connection and monodromy representation
I saw the following statement:
An $S_g$-bundle $p: E\to B$ admits a flat connection if and only if the
monodromy representation $\rho:\pi_1(B)\to\text{Mod}(S_g)$ lifts to $\rho':\pi_1(B)\to\text{Diff}...
2
votes
0answers
68 views
Principal $G$-bundle maps and sections of Associated Bundles
These days, I am making my way towards the classification result about homotopy classes of maps from a CW-complex $Y$ to $BG$ and isomorphism classes of principal $G$-bundles over $Y$, where $G$ is a ...
0
votes
0answers
29 views
canonically induces structure on Poincaré dual space
Let Poincaré dual $PD(a)$ be a smooth, possibly non-orientable surface which represents a class in $H_2(M^3,Z/2)$ Poincaré dual to $a ∈ H^1(M^3, Z/2)$.
What are the logic dependences of the ...
2
votes
0answers
39 views
Recover a vector bundle by (some) restrictions
Problem: Consider a smooth, projective variety $X$ in $\mathbb{P}^n$, consider a vector bundle $E\xrightarrow{\pi}X$ os rank $r$. Suppose that $\mathbb{P}^k$ is a subvariety of $X$, and we know that ...
1
vote
0answers
27 views
Metric tensor of a space as an induced metric from embedding in complex space, and dilations thereof.
Apologies, I know the title's a mouth-full.
We can consider the standard unit three-sphere $S^{3}$ as being embedded in $\mathbb{C}^{2}$ with coordinates:
$$\Phi=\begin{array}{c}
\phi^{1}+i\phi^{2}\\
\...
0
votes
0answers
35 views
Basis of vertical vector fields in a fiber bundle from generators of the Lie algebra of the structural group
Let $E$ be a fiber bundle, $B$ its base manifold, $F$ its fiber and and $G$ its structural group of dimension $n$ greater than the dimension $r$ of $F$. We note $z$ the points of $E$ and $x$ the ...
0
votes
0answers
27 views
Normal bundle and determinant line bundle to the submanifold: Induced structures
Can someone explain step by step why this is true? (intuition)
The normal bundle to the submanifold $PD(A) ≡ N_2 ⊂ M_3$ for oriented $M_3$ can be realized as determinant line bundle $\det TN_2$.
The ...
3
votes
2answers
104 views
Lifting a map to the total space of a circle bundle
Let $\pi:P \to M $ be a smooth circle bundle, so $S^1$ is the fibre, $f:N \to M$ a smooth map. I would like to know what are the necessary and sufficient conditions for $f$ to lift to a map $g:N \to P$...
2
votes
1answer
51 views
Computing the euler number of an oriented disc bundle over a sphere
Suppose $E\to S^2$ is an (smooth) oriented disk bundle. Let $D_1$ and $D_2$ denote the upper and lower hemispheres of $S^2$, respectively. Then $E|_{D_1}$ and $E|_{D_2}$ are trivial, so they are both ...
0
votes
1answer
81 views
Obstruction to a Spin structure on a bundle ξ, and ξ ⊕ $n$ det ξ
In Ref, it says that:
The obstruction to putting a Spin structure on a bundle $ξ (= Rn → E → B)$ is $w_2(ξ) \in H^2(B;Z/2Z)$.
Pin± structures is that
Pin− structures on ξ correspond to Spin ...
2
votes
1answer
67 views
Decomposing a normal bundle
I'm reading a paper on symplectic geometry. I reached to a point where the author uses normal bundles ( which I know the definition but I've never work with). He actually decomposes a given normal ...
0
votes
0answers
61 views
G torsor v.s. Principal G-bundle
G torsor v.s. Principal G-bundle
Are G torsors always Principal G-bundles?
How are G torsors related to Principal G-bundles?
Is that G torsor primarily for algebraic geometry?
While Principal G-...
1
vote
0answers
38 views
“Horizontal” Foliation on a fiber bundle
Before I ask my question, I would like to build up the setting.
Let $\pi:E \rightarrow M$ be a smooth fiber bundle. Since $\pi$ is a smooth submersion, there is an induced foliation $\mathcal{F}(\pi)...
0
votes
0answers
17 views
Reference about integration along the fiber and bundles
I am reading $\textit{Differential Forms in Algebraic Topology}$ by Bott. This book is superb but I would like to find other references (maybe more recent) that concern, in particular, topics about ...
0
votes
0answers
13 views
How to picture a horizontal lift using two different fibers?
Motivation: I am trying to define a connection that relates two different fiber bundles sharing a base space B, and think about parallel transport using both of them.
Although I am still new to this ...
3
votes
0answers
53 views
Bott&Tu, Definition of the Euler class of a vector bundle
I have a question while reading chapter 11 of Bott&Tu, Differential Forms in Algebraic Topology. The book first defines the Euler class of an oriented sphere bundle (a fiber bundle with fiber $S^n$...
5
votes
2answers
97 views
Non isomorphic principal $G$-bundles
Remark: throughout this post we work in the smooth category, so that all manifolds, bundles, maps etc are assumed to be smooth.
An exercise asks me to show that there exists no principal $S^1$-bundle ...
1
vote
0answers
39 views
Fibre bundles and good pairs
Consider a fibre bundle $E\rightarrow B$. I was wondering when we can say that the pair $(E,B)$ is a good pair? By a good pair $(A,X)$ I mean two spaces $X\subset A$ such that $X$ is closed and is a ...
1
vote
1answer
55 views
Understanding a mistake I made regarding homomorphism between certain function spaces
Let $M$ be a closed $n$-dimensional smooth manifold. Let $X = \amalg_{p \in M} \text{Aut}(T_p M)$ be a fiber bundle over $M$ topologized in the natural way. It can be shown that the space of sections ...
0
votes
0answers
8 views
Components of vector field and pushforward through a section
For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $p\in M$ and $u=\phi (p)$.
Suppose we have $E$ has coordinates $(x,y)$ and we have the map $\psi :E\mapsto E$ given by
$\...
0
votes
1answer
38 views
Local trivialization of a fiber bundle associated to a principal bundle
Let $(P,\pi,M,G)$ be a principal bundle, where $P$, $M$, $G$ are smooth manifolds (total space, base space and fiber=structure group respectively, so $G$ is both fiber and structure group)
the ...
1
vote
1answer
49 views
Some questions about Bott & Tu - Differential Forms in Algebraic Topology, chapter 11.
I am reading chapter 11 of Bott & Tu - Differential forms in algebraic topology. And I have some questions about this section.
1: Let $\pi:E\to M$ be a sphere bundle with fiber $S^n$. For each $x\...