# Tagged Questions

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### Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
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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 $\pi: ... 1answer 82 views ### A question about the definition of fibre bundle The canonical definition of fibre bundle is the following: Let$B,X,F$be three topological spaces and$\pi:X\rightarrow B$a continuous surjective map; then$(X,F,B,\pi)$is a fibre bundle on$B$... 0answers 153 views ### Fiber Bundle: Hairbrush I am trying to understand the hairbrush example of a fiber bundle from the Wikipedia article on fiber bundles. If I am understanding this, in the hairbrush example E is the hairbrush, ie. all the ... 2answers 347 views ### Which spheres are fiber bundles? The Hopf fibration is a fiber bundle with total space$S^3$, and there are similar constructions for$S^7$and$S^{15}$. Are there any other ways to regard a sphere as a nontrivial fiber bundle? My ... 1answer 98 views ### Restrictions for Principal Bundles on Manifolds I have some manifold$M$and am wondering what kind of Principal Bundles I am allowed to construct on it. To be more precise, what are the restrictions when trying to construct principal Bundles ... 2answers 291 views ### Given a fiber bundle$F\to E\overset{\pi}{\to} B$such that$F,B$are compact, is$E$necessarily compact? Consider a (locally trivial) fiber bundle$F\to E\overset{\pi}{\to} B$, where$F$is the fiber,$E$the total space and$B$the base space. If$F$and$B$are compact, must$E$be compact? This ... 1answer 162 views ### Is a sub-bundle of a vector bundle a vector bundle? Could anyone please help me with this question? (1) Let (E, p, B) be a vector bundle where E is the total space, B is the base, and p is the structure map, that is, p:E->B. Now suppose E' is a ... 1answer 163 views ### Fiber bundle M x M - diagonal Under what conditions for a space$M$does the projection map to the first factor$p: M \times M - \Delta \rightarrow M$has the local triviality condition, i.e. is a fiber bundle? Where$\Delta$... 1answer 308 views ### Principal and fiber bundles as defined by Husemoller In his book 'Fiber Bundles' Husemoller defines principal bundles and fiber bundles quite differently from how they are usually defined. Specifically: Definition: a right$G$-space$X$is called ... 2answers 868 views ### What is 'an identification map'? From Husemöller's 'Fiber Bundles' (slightly rephrased): Proposition: Consider a bundle$\xi: E \to B$, and a mapping$f: B' \to B$. Then for any$s \in \Gamma(\xi)$there is a$\sigma: B' \to ...
Let $B$ and $F$ be compact Hausdorff spaces. Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$. I think this induces a fiber ...