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.

311 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
42
votes
0answers
979 views

Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
19
votes
1answer
2k views

An intuitive vision of fiber bundles

In my mind it is clear the formal definition of a fiber bundle but I can not have a geometric image of it. Roughly speaking, given three topological spaces $X, B, F$ with a continuous surjection $\pi: ...
15
votes
0answers
143 views

Do Hopf bundles give all relations between these “composition factors”?

Write a fiber bundle $F\to E\to B$ in short as $E=B\ltimes F$ (in analogy with groups). (This is not necessary, but: given another bundle $X\to B\to Y$, we can write $E=(Y\ltimes X)\ltimes F$, but ...
12
votes
0answers
616 views

Visualizing Hopf fibration $S^3\to S^2$ as a based map $S^1\to \mathrm{Map}(S^2,S^2)$

A fiber bundle $F\to E\to B$ may be interpreted as $E$ being a bunch of $F$s arranged in the shape of a $B$. For instance, a Mobius band $M$ is a bunch of line segments $[0,1]$ arranged in the shape ...
11
votes
0answers
200 views

How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
9
votes
0answers
188 views

how to obtain a generalized Morse function out of a fiber bundle?

Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, 1990)", K. Igusa says that if dim $B$$<$dim $M$, then, there exists a smooth ...
9
votes
0answers
155 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
9
votes
0answers
465 views

Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
8
votes
0answers
66 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 \...
8
votes
0answers
295 views

quaternion vector bundle and quaternion grassmannian

Let $\mathbb{H}$ be quaternion numbers. Let $G_n(\mathbb{H}^\infty)$ be the grassmannian of $n$-subspaces of $\mathbb{H}^\infty$. Then $H^*(G_n(\mathbb{H}^\infty),\mathbb{Z})=\mathbb{Z}[q_1,\cdots,q_n]...
8
votes
0answers
269 views

automorphisms of the torus bundle over the circle

Let us consider a $\mathbb{T}^2$-bundle $\xi$ over circle $S^1$. These bundles are completely determined by their monodromy matrix. A fiber-preserving homeomorphism of $\xi$ is called automorphism of ...
7
votes
1answer
112 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} \},\\ & \...
7
votes
0answers
167 views

Clifford Multiplication and Spinor Bundles

I am trying to follow the discussion of Clifford multiplication on page 384 of The Wild World of 4-Manifolds, by Alexandru Scorpan (link, although I hope this will be totally self-contained), and I'm ...
7
votes
0answers
229 views

Action of $Sp(1)$ over the sphere $S^7$

Consider the Hausdorff space $S^4 = \mathbb{HP}^1$ and the topological group $Sp(1)$ (identified with the unit quaternions), the cover $\{V_j\}_{j \in J} = \{U_1, U_2\}$ of $\mathbb{HP}^1$ where $$U_1=...
7
votes
0answers
108 views

Connection in fibre bundle from discontinuous group action

I am trying to understand connections in fibre bundles. I thought of the following problem: Let $\Gamma$ be the discrete group generated by \begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ ...
7
votes
0answers
258 views

How to classify principal bundles over a 2 dimensional surface?

I just want to how much people know about this at the moment? I thought this is elementary and may be of execrise level, but a quick google search showed serious papers written on this subject (like ...
6
votes
0answers
138 views

Example of a Serre fibration between manifolds which is not a fiber bundle?

I'm looking for an example of a map $f : X \to Y$, where $X$ and $Y$ are manifolds (without boundary), and $f$ is a Serre fibration, but $f$ is not a fiber bundle. I know that if $f$ is proper, and $...
6
votes
1answer
93 views

Condition for integrability (foliation)

Let $(M^{n+1}, \langle \cdot, \cdot \rangle)$ be a parallelizable Riemannian manifold with a vector bundle isomorphism $$\varphi : TM \to M \times \mathbb{R}^{n+1}.$$ For $x \in M$, denote by $\...
6
votes
0answers
277 views

Does a left group action on a principal bundle induce an action on associated vector bundles?

Let $G\hookrightarrow P\xrightarrow{\pi}M$ be a principal $G$-bundle with right action $\cdot $ and suppose we are also given a left action $\rho: U\times P\rightarrow P$ of some group $U$ on $P$. ...
6
votes
0answers
1k views

Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
6
votes
0answers
535 views

Sections of endomorphisms of a vector bundle

The following could be rather silly question but I haven'd found it stated explicitly; from the other side, it seems to me, that this fact is used often without comments. The problem is the ...
6
votes
0answers
829 views

Cohomology of fiber bundle with a section

Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map $$ f^{*}:H^{*}(B,\...
6
votes
0answers
203 views

Alternate pullback bundle construction

If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is $$\phi^* F := \{(m,f) \in M \...
5
votes
1answer
130 views

Covariant derivative on the base space

The basic definition of a covariant derivative for a Lie algebra valued n-form $\alpha \in \Omega^n(P)\otimes T_eG$ with $P$ a principle bundle with base space a manifold $M$, and $T_eG$ the Lie ...
5
votes
0answers
174 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) ...
5
votes
0answers
440 views

Proof details of the fact that the unit tangent bundle is compact in $TM$ if $M$ is a compact manifold

Let $M$ to be a manifold $m$-dimensional with a smooth hermitian metric $g$. The tangent bundle of $M$ is given by $TM= \bigcup_{p\in M} T_{p}M$, and the unit tangent bundle is given by $S=\{x \in ...
5
votes
0answers
158 views

Is a principal bundle of a principal bundle still principal?

Let $\left(P_1,\pi_1,M,G_1\right)$ and $\left(P_2,\pi_2,P_1,G_2\right)$ be two principal bundles, where $M$, $P_1$ and $P_2$ are differential manifolds, and $G_1$ and $G_2$ are Lie groups. With these ...
5
votes
0answers
165 views

Connection on fiber bundle of generic fiber

I am looking for references about connections on a generic fiber bundle. A lot of books deal with connections on vector bundles, some books as the Kobayashi-Nomizu generalize the concept of ...
5
votes
1answer
952 views

A fiber product is a fiber bundle

Let $F,B$ be topological spaces. A fiber bundle $E$ over the basis $B$ with fiber $F$ is a topological space $E$ endowed with a continuous surjection $\pi:E\to B$ such that there exists an open cover $...
5
votes
0answers
142 views

Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
5
votes
0answers
120 views

What is the geometric meaning of powers of the first Stiefel-Whitney class?

If $M$ is an $n$-dimensional manifold, I know $\\w_1^n(M)$ is a Stiefel-Whitney number of the manifold, but what does its vanishing geometrically mean? More generally, does the Stiefel-Whitney height ...
5
votes
1answer
120 views

Torsion in $H^2(X,\mathbb{Z})$ induced by torsion in $H_1(X,\mathbb{Z})$

Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $$ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow H^{k}(X,\mathbb{Z})\...
4
votes
0answers
101 views

Hatcher's proof of Leray–Hirsch theorem

Questions: I think the definition of $\Phi$ relies on the assumption (b), but the left and right $\Phi$ from the commutative diagram don't have such an assumption. I wonder how is the left and right ...
4
votes
0answers
65 views

Visualizing a fiber bundle

Define the torus $$(S^1)^3=\left\{ (e^{2\pi ir_1},e^{2\pi ir_2},e^{2\pi ir_3});\ \ r_1,r_2,r_3\in \mathbb R\right\}$$ and consider the dense subgroup $$Q:=\left\{ (e^{2\pi ir_1},e^{2\pi \alpha r_1},...
4
votes
1answer
64 views

Identifying sheaves of sections of fiber bundles

Given a topological space $X$, there's a fundamental category equivalence between local homeomorphisms to $X$ and sheaves of sets over $X$. One direction takes a local homeomorphism to its sheaf of ...
4
votes
0answers
80 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 ...
4
votes
0answers
133 views

Mapping Tori and Monodromy

I have a question regarding the right setup for mapping tori. Let me give the definitions that I use first. Let $I$ denote the interval $[0,2\pi]$ and let $I^*$ denote the quotient of $I$ by the ...
4
votes
0answers
90 views

Why is a Dirac operator involutive only if the curvature and torsion are in the image of the kernel of the symbol under exterior multiplication?

I am trying to understand the classic paper by Atiyah, Hitchin, and Singer https://www.jstor.org/stable/79638 and I'm getting stuck on part of the proof of proposition 3.1. The proposition is ...
4
votes
0answers
333 views

Roots of a canonical line bundle on a compact Riemann surface

Suppose we have a compact Riemann surface $X$ of genus $g$. Let $K$ denote the canonical line bundle on $X$, it's well known that $deg\ K=2g-2$. A square root of $K$ by definition is a holomorphic ...
4
votes
0answers
217 views

Computing Stiefel Whitney classes

I am computing the cohomology of $BO(2) \times B0(3)$ and I would like to identify the Stiefiel Whitney classes of this space. For instance, I know $H^*(BO(2);\mathbb{Z}/2)\cong \mathbb{Z}/2[w_1,w_2]$...
4
votes
0answers
49 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 $...
4
votes
0answers
247 views

Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If $e^{f_{...
4
votes
0answers
485 views

Curvature of a principal bundle and the exterior covariant derivative

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...
4
votes
0answers
64 views

Locally a product equals fiber bundle?

I know the definition of a fiber bundle as a map $p:E \rightarrow B$ such that the preimage of any open set in $B$ is diffeomorphic to a product and fits in a commutative diagram with the projection $...
4
votes
0answers
126 views

Reference request for studying on Fiber bundles

I am looking for some material (e.g. references, books, notes) to get started with Fiber bundles and vector bundles. Can someone help me? Thanks.
4
votes
0answers
142 views

Fiber Bundle of Manifolds

I want to show the following: Let $E \rightarrow B$ be a fiber bundle with fiber $F$. Show that if $B$ and $F$ are manifolds, then so is $E$. Solving this problem seems easy enough simply by ...
4
votes
0answers
340 views

What is the commutator of a horizontal and vector field for a connection on a Fiber bundle?

I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator ...
4
votes
0answers
313 views

Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
4
votes
1answer
303 views

Fiber Bundle: Hairbrush

I am trying to understand the hairbrush example of a fiber bundle from the Wikipedia article on fiber bundles. If I am understanding this, in the hairbrush example E is the hairbrush, ie. all the ...
4
votes
0answers
186 views

Lie group quotient bundle with image of normalizer as structure group

In Glen Bredon "Topology and Geometry", Ch. II-13, I am stuck on the following "Problem 1. Finish the proof at the end of this section that $G\rightarrow G/H$ is a bundle. Also show that this is a ...