# 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.

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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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### 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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### 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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### 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 $\mathscr{G}$ ...
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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 ...
1 vote
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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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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 } ... • 4,150 1 vote 1 answer 65 views ### 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 ... • 165 2 votes 1 answer 71 views ### 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) ... • 285 1 vote 1 answer 35 views ### 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 ... 0 votes 0 answers 40 views ### 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 ... • 165 1 vote 0 answers 26 views ### 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 ... • 633 2 votes 0 answers 49 views ### 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... 4 votes 1 answer 43 views ### 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(\... • 17.4k 2 votes 1 answer 41 views ### 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 ... • 956 1 vote 1 answer 64 views ### 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^\... 0 votes 0 answers 74 views ### 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 ... • 2,908 5 votes 1 answer 106 views ### 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 ... • 2,908 0 votes 1 answer 37 views ### 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 ... 0 votes 0 answers 22 views ### 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 votes 3 answers 123 views ### 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. ... • 5,723 3 votes 0 answers 34 views ### 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 ... • 321 5 votes 0 answers 79 views ### 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)) ... 1 vote 1 answer 40 views ### 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 ... • 85 0 votes 0 answers 70 views ### 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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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 ...