# 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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### Alternative Geometric Construction of Minkowski Space Metric

In the Minkowski Space, at each point, we define a tangent space, and using the metric construct, we calculate a real number that shows the infinitesimal invariant interval between two points. This ...
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### Why $K_\nabla$ is the same as $K_{\omega_S}$?

Let $G\subset GL_n(\mathbb{R})$ be a Lie subgroup and let $M$ be a manifold. Consider $S$ a $G-$structure on $M$ and $\nabla$ a connection compatible with $S$. We know that if $E=E(P,V)$ is the vector ...
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### On a definition of covering space in terms of fiber bundles

I'm currently taking a course in (smooth) fiber bundles. I am not very used to covering spaces, but I'm trying to understand them in the language of fiber bundles. I would like to know whether the ...
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### Flat Bundle vs Trivial bundle

In R.W. Sharpe's Differential Geometry, a flat fibre bundle is defined as a bundle whose transition functions are constant. I don't understand the difference between this and a trivial bundle. Because,...
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### The equivalence relation when constructing the associated bundle

When constructing from a typical fibre $F$, an action of a Lie group $G$ on that fibre and a $G$-principal bundle $P$ over some base space $M$ the associated bundle $P[F]$, one does this by ...
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### When does the map from Fundamental Group to Holonomy Group Injective?

We know that over a rank $n$ vector bundle $E$ (with base space being $M$, a manifold), if the connection $\nabla$ is flat, then the parallel transport along loop $\gamma:I\to M$ for $E$ will only ...
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### Definition of the Principal Bundle for smooth manifolds?

The fiber bundle is defined as A fiber bundle is defined as the tuple $(E, B, \pi)$ where $\pi: E \to B$ is a continuous surjective map from topological space $E$ to topological space $B$. ...
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### Right G-Space (Husemoller, Fiber Bundles)

Hussemoller's definition of a right G-space confuses me at a few key points: For a topological group G, a right G-space is a space $X$ together with a map $X \times G \xrightarrow{\quad }X$. The image ...
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### B-Isomorphic? (Husemoller, Fiber Bundles). Illustrated diagrams included.

Wikipedia gives the definition of a bundle map as the arrow $\left( \xrightarrow{\quad \varphi \quad } \right)$ in the commuting diagram below: Hussemoller gives the definition of a bundle map as the ...
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### If Chern classes are only defined for vector bundles, why can a $U(1)$ principal bundle have associated Chern classes?

When people talk about Chern classes of $U(1)$ principal bundles, do they really mean the Chern class of the associated vector bundle? As far as I can understand, Chern classes exist only for complex ...
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### Discrete cocycle datum of a principal $G$-bundle

Let $X$ be the topological realization of a finite simplicial complex, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. Let's recall the standard fact that more generally for any numerable ...
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### Gysin sequence from Serre spectral sequence vs Thom isomrophism

Let $n\geq 1$ and let $\pi:E\to B$ be a fiber bundle with fiber $S^n$ with $B$ simply connected. We will fix an orientation of $p$. An easy analysis of the Serre spectral sequence shows that there is ...
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### "Open" Cell? (Hatcher and Husemoller)

Have $\mathbb D^n$ be some n-dimensional unit disk. Have $\partial S$ denote the boundary of some space $S$. Let infix $(-)$ denote a difference between collections of objects. In Dale Husemoller's ...
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### Are function spaces over a shrinking set vector bundles?

I have function spaces $F(S_t) \subseteq \{f : S_t \to \mathbb{R}\}$ defined over shrinking sets $S_t\subset S_s$ for $t> s$. I have a trivial fiber bundle $[0,\infty) \times F(S_0)$ where the ...
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### $\pi$ is a fiber bundle over $B$... Difficulty interpreting some Wikipedia definitions

Let (i)$\left(F \xrightarrow{\quad \quad} E \xrightarrow{\quad \pi \quad}B \right)$ be any fiber bundle. Wikipedia has a different parlance when defining some fiber bundles. Take this excerpt from the ...
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### How can de Rham cohomology find obstructions?

A differential form is closed iff its exterior derivative is $0$. A differential $k$-form $w$ is exact iff there exists a differential $(k-1)$-form $\eta$ so that $\hbox{d}\eta = \omega$. Every exact ...
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### Requirements for trivial fiber bundles

I am starting to learn about fiber bundles, and wanted to understand the following. We consider a fiber bundle $(E,B,\pi,F)$ where $E,B,F$ are smooth manifolds, and $\pi : E \to B$ is a continuous ...
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### Flat Maurer-Cartan connection iff flat Berry connection

I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$. The first is the canonical or $H$-...
### How to find the cocycle and sections of the covering $\pi:S^{1}\rightarrow S^{1}$ such that $\pi(z)=z^{2}$ with $F=\mathbb{Z}_{2}$
I am following these lectures on principal fiber bundles. Here, a principal fiber bundle is defined as a fiber bundle of which total space $P$ has a right free action of some Lie group and which is ...