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.
1,042
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Convexity in "usual Partition of Unity arguments"
I stumbled upon Problem 13-2 on p.344 in John Lee's Introduction to Smooth Manifolds (2nd Edition) where Lee explains that the proof for the existence of a Riemannian Metric on a manifold is done by a ...
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Why is for cotangent bundles $T^*X_1 \times T^*X_2 \simeq T^*(X_1 \times X_2)$
For smooth manifolds $X_1, X_2$ why can the cotangent bundle $T^*(X_1 \times X_2)$ be identified with $T^*(X_1) \times T^*(X_2)$, i.e. the product of the cotangent bundles.
I feel like this should be ...
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homogeneous circle bundle over a hyperbolic surface
Let $ M $ be the total space of a circle bundle $ S^1 \to M \to \Sigma_g $ for $ g \geq 2 $. Suppose that there exists a transitive action of $ \widetilde{SL_2(\mathbb{R})} $ on $ M $. Must $ M $ be ...
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What is a symplectic structure on a smooth vector bundle
We just had the definition of a symplectic structure on a vector bundle in the lecture and I am having trouble understanding it
Definition: Let $\pi : E \to M$ be a smooth vector bundle. Then a ...
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1
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The total space of the standard quaternionic Hopf fibration of an $S^3$ fiber bundle over $S^4.$
I am trying to understand the paper "Milnor's Construction of Exotic 7-Spheres" by Rachel McEnroe (link). Here's the abstract:
I do not understand why the total space of this fiber bundle ...
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Understanding how the local trivializations were calculated.
I am trying to understand the paper "Milnor's Construction of Exotic 7-Spheres" by Rachel McEnroe (link). Here's the abstract:
But when it comes to calculating the local trivialization of ...
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1
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Understanding the transition map in the case of the proof that $S^3$ bundles over $S^4$
Here is the part of the paper of "Rachel Mcenroe" on Milnor's construction of Exotic 7-spheres:
But I do not understand why the transition map is $\frac{1}{z}$ and it does not include any $\...
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Example of classifying spaces [closed]
I'm new studying algebraic topology and I am studying classifying space for a group $G$, but I cannot find other examples different than $G = \mathbb{Z} / 2$ or $G = U(1)$. I'm interested in the ...
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2
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Why the fiber of the normal bundle of the Klein bottle is $\mathbb{R}^2$ instead of $\mathbb{R}$?
What I know:
If M is an m-dimensional manifold embedded in $\mathbb{R}^{{m+k}}$, the normal bundle $NM$ is with the typical fiber $\mathbb{R}^k$.
My Question:
When I think about a torus ($T$) or a ...
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Normal bundle of a non-orientable manifold
What I know:
A tangent bundle $TM$ of a manifold $M$ is orientable if and only if the manifold $M$ is orientable.
If the first Stiefel-Whitney class $w_1(E) \neq 0$, the vector bundle $E$ is non-...
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The first Stiefel-Whitney class of a trivial line bundle $E=M \times \mathbb{R}$
If the base space $M$ is non-orientable, is the trivial line bundle $E=M \times \mathbb{R}$ also non-orientable? i.e. $w_1(E) \neq 0$. If so, how could it be proved? Could we use $w_k(\xi\times\eta)=\...
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Why is the stalk cohomology of the nearby cycles of a sheaf complex exactly the hypercohomology of the Milnor fiber of that complex?
Let $X$ be a complex manifold and $f\colon X\to\mathbb{C}$ a non-constant holomorphic map. We define $X_t=f^{-1}(t)$. Let $\mathcal{F}^\bullet\in D^b(X)$ and we denote by $\psi_f$ the nearby cycle ...
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correspondence between local section and local trivalization for vector bundle and principle $G$-bundle
For the principle G-bundle $\pi :P\to M$, given a local section $s:U\to P$, it correspond to a local trivialization $$U\times G \to \pi^{-1}(U)\\(x,g) \mapsto s(x)\cdot g$$
where $s(x)\cdot g$ is the ...
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Fundamental group of Seifert fibered space over $S^2$
I am reading p.3 of Saveliev's Invariants for homology 3-spheres. Here is the construction of a Seifert 3-manifold $X=M(b;(a_1,b_1),\dots,(a_n,b_n))$.
Consider the $S^1$-bundle $M\to S^2$ with Euler ...
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Explicit formula of $S^1$-bundle over $S^2$ with euler number $n$
Suppose $M \to S^2$ is an orientable circle bundle with Euler number $n$. Regarding $S^2$ as $D^2_1 \cup_{\text{id}} D^2_2$, since $M$ is trivial over $D^2_i$ ($i=1,2$), we have $M=(S^1\times D^2_1)\...
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List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle [closed]
List all the circle bundles over the circle, sphere, projective plane, torus, and Klein bottle. State the orientability of the total space, the base and the bundle (orientability of a circle bundle is ...
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Generalizing a property of the two sphere
There are many circle bundles over the sphere $ S^2 $ (in fact infinitely many) but all of them are principal.
Do there exist any other manifolds besides $ S^2 $ for which (nontrivial bundles exist ...
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Let $(E,B,\pi)$ be vector bundle and $W \subseteq E$. Prove $W = \pi^{-1}(W \cap B)$.
I am having trouble proving a statement for my differential geometry class. The problem is as follows:
Let $(E,B,\pi)$ be a vector bundle and let $\Psi : W \longrightarrow W' \times F'$ be a vector ...
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Homotopy groups of infinite dimension lens space
Let $n > 1$. We define the infinite dimensional lens space $L$ as follows. Let $S^{\infty}$ be the unit sphere in the infinite dimensional complex vector space $C^{\infty}$, and let $\mathbb{Z}/n = ...
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Link of zero set of a quasi homogeneous polynomial in $\Bbb C[x,y,z]$ has a Seifert fibered structure
I have a question about Seifert fibered 3-manifold while reading this survey paper: https://arxiv.org/pdf/math/0602562.pdf.
In Example 22 (p.10), it is said that for integers $a,b,c\geq 1$, set $S_{...
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A simple example of a $G$-principal bundle that cannot be lifted to an $F$-principal bundle along an epimorphism $F\to G$.
I am trying to understand spin structures, in particular how they may fail to exist. To start with, I would like to see a most simple example of a $G$-principal bundle that cannot be lifted to an $F$-...
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$\{(u,z)\in S^1\times D^2:u^cz^d=w\}$ is $S^1$ where $w\in D^2-\{0\}$ and $c,d$ are integers with $0\leq c<d$ and $(c,d)=1$
According to Definition 16 of https://www.intlpress.com/site/pub/files/_fulltext/journals/pamq/2008/0004/0002/PAMQ-2008-0004-0002-a001.pdf, for a pair of integers $c,d$ with $0\leq c<d$ and $(c,d)=...
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"Tightening up" a map in the fibre of a principal torus bundle
Suppose $p_1:E_1\to B$ and $p_2: E_2\to B$ are two compact principal $\mathbb{T}^d$-bundles over the same base $B$. Suppose there exists a fibre-preserving map $F:E_1\to E_2$ that covers the identity $...
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Fiber bundle orientability vs manifold orientability
I read this question about vector bundles
Bundle orientability vs manifold orientability
In the answer to this question the last sentence states the following (I think fairly well known) result about ...
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"tensor product" of two metrics
Consider we have a Reiamnnian metric $g$ on a closed manifold $M$. We also have a fiberwise metric $h$ on a locally trivial fiber bundle $E$.
How do we use these two metric to define metric on $\Omega^...
- 1,104
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Standard metric on adjoint bundle
Consider we have a principal $G$-bundle $P$ over a closed manifold $V$. Denote $\mathfrak{g}_P$ by the associated bundle $P\times_G \mathfrak{g}$ where $G$ acts by adjoint action. Denote $\mathscr{G}$ ...
- 1,104
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Property of exterior covariant derivative
Let $(M,g)$ be an oriented riemannian manifold, $(E,h)$ be an hermitian fiber bundle over $M$ and $\nabla$ a connection compatible with respect to the metric. Is it possible to define a wedge product ...
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Projectivization of a bundle is invariant under tensoring with a line bundle
I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)\cong P(E\otimes L)$".
By bundle you can as well ...
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Action of the diffeomorphism group on a tensor bundle
I have two questions in two paragraphs I'm reading from the book Heat kernels and Dirac operators page 17.
$\textbf{ paragraph 1}$
Let $TM$ be the tangent bundle of $M$. A section $X \in \Gamma(M,TM)$...
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What is fiber space?
I'm reading a notes about algebraic topology.It uses a phrase 'fiber space' without any definition.I think maybe it's a generalization of fiber bundle.So I want to ask if there's anyone who knows what ...
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Approximating continuous functions by those with range in a dense subset
Let $A$ be a Banach space and $A_0\subseteq A$ a dense subset (linear subspace if necessary). Write
$$F_A=\{\text{continuous functions } f\colon[0,1]\to A\},$$
$$F_{A_0}=\{\text{continuous functions } ...
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What is / why the connection one-form from a physics point of view?
Take the Yang-Mills gauge theory for example. Gauge field $A$ is the pullback of the connection one-form to the base manifold. Other concepts of gauge theory also find their definition in fiber ...
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Triviality of Sp(TM)
Let M be a symplectic manifold of dimension $2n$ and $TM$ denote its tangent bundle. Let Sp(TM) denote the bundle over M whose fibers are linear maps preserving symplectic structure on M. Is Sp(TM) ...
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Using compactness in the proof of Fiber Bundles are Serre Fibrations
In the proof that fiber bundles have the homotopy lifting property for disks (i.e. it's a Serre fibration), Hatcher writes:
I know how to use compactness to show we can subdivide $I^n\times I$ enough ...
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Why is the frame bundle of the Möbius strip the Z2 bundle?
In the wiki figure it says "The frame bundle $\mathcal{F}(E)$ of the Möbius strip $E$ is a non-trivial principal $\mathbb {Z} /2\mathbb{Z}$-bundle over the circle."
Shouldn't the frame ...
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What is the "correct" definition of a $G$-bundle/a structure group?
Let $G$ be a topological group. Let $p:E\to B$ be a fibre bundle with fibre $F$ and let $G$ act continuously on $F$.
The way one would typically define whether $E$ has $G$ as its structure group is by ...
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Characteristic Classes of Fiber Bundles of Closed, Aspherical Manifolds with Isomorphic but Non-Congruent Group Extensions
Suppose I have two closed, aspherical manifolds, $F$ and $B$, with $K = \pi_1(F)$ and $Q = \pi_1(B)$, and two group extensions $1 \to K \to G_1 \to Q \to 1$ and $1 \to K \to G_2 \to Q \to 1$ with $G_1$...
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How can a decomposable element be transgressive without violating Leibniz rule?
Consider the fibration $K(\Bbb Z_2,1)\to \ast\to K(\Bbb Z_2,2)$. As we know from Serre (see Hatcher's SSAT), $H^*(K(\Bbb Z_2,n);\Bbb Z_2)$ is the polynomial ring on generators $\operatorname{Sq}^I(\...
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Is the inverse of a local map also local?
Let $E,F\rightarrow X$ be two smooth vector bundles over a manifold $X$ and suppose we have a (linear) map $T:\Gamma(E)\rightarrow \Gamma(F)$ which is invertible and local. I would like to know if the ...
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What is the correspondence between gauge field terminology and bundle terminology in electromagnetism?
In electromagnetism, the electromagnetic field tensor can be expressed as $$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$
If we let $A= A_\mu dx^\mu$, since $F= \frac{1}{2} F_{\mu \nu} dx^\...
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Fiber bundles, non-hyperbolic surfaces and Thurston geometries
Let $ M $ be a compact 3-manifold which is the total space of a fiber bundle (with connected nontrivial fiber and base) where neither the base nor the fiber is a hyperbolic surface. Then $ M $ always ...
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Fiber bundles, hyperbolic surfaces and Thurston geometries
Let $ M $ be a 3-manifold which is the total space of a fiber bundle (with connected nontrivial fiber and base) with either the fiber or base a hyperbolic surface. Then does $ M $ always admit a ...
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Non existence of a preferred Horizontal subspace on a bundle. Why not ? (Basics)
If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=T^2$ (2-torus) and $B=S^1$. Can I choose to put my finger on the identity element of the group over a point ...
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How to generate an associated connection
I know that we can generate an affine connection on a GL(n) bundle.If we have a fibre bundle whose structure group G is the subgroup of GL(n) with diagonalizable matrices M whose eigenvalues are +1 or ...
2
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3
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Fiber bundle map is proper if the model fiber is compact
This is one direction of problem 10-19 (c) from John Lee's Introduction to Smooth Manifolds.
Suppose $\pi: E \to M$ is a fiber bundle with fiber $F$.
Show that $\pi$ is a proper map if $F$ is compact.
...
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Do all fiber bundles admit a finite-dimensional structure group?
Every fiber bundle has the group of homeomorphisms from the fiber to itself as structure group. However, this is a very huge group. I was wondering whether the structure group can always be reduced to ...
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Classifying the homotopy classes of lifts
Lifts of $f \colon B \to X$ into a fibration $F \to E \to X$ can be identified with sections of the pullback bundle $f^*(E) \to B$.
I want to try to compute the path components $\pi_0(\Gamma(f^*(E))$ ...
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Why is it useful to identify sections on an associated bundle $P \times_G E$ with functions ${C^\infty(P,E)}^G$
Let $\pi:P\rightarrow M$ be a Principal $G$-bundle and let $P\times_G E$ be its associated bundle via the representation $\rho:G\rightarrow GL(E)$.
We know that we can identify the set of sections of ...
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70
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Fibers of a principal $G$- bundle are diffeomorphic to $G$.
Definition: A principal bundle $\pi:P \rightarrow M$ with structure group $G$ is a fiber bundle $P$ with a right action of the Lie group $G$ on the fibers, such that
$$\pi(pg)= \pi(p), \quad p \in P , ...
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Question about fiber bundles
The following is from page 14 of the book heat kernels and Dirac operators
Definition: let $\pi: \mathcal{E} \rightarrow M $ be a smooth map from a manifold $\mathcal{E}$ to a manifold $M$. We say ...
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