Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of problem? What mathematical tool and concept should be required? And, could someone suggest some reference books related to the problem? (I’m not sure my […]

I recently read the principal $G$-bundle construction on a smooth manifold $M$, where $G$ is a Lie group. To understand them better, I am looking for some applications. Can the principal $G$-bundle help us get some usual bundle constructions, for example tensor product of two vector bundles, the pullback bundle etc? Right now, the constructions […]

Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally isomorphic to $M \times G$ with the obvious right action of $G$ on $M \times G$? […]

I’ve finally gotten around to learning about principal $G$-bundles. In the literature, I’ve encountered (more than) four different definitions. Since I’m still a beginner, it’s unclear to me whether these definitions are equivalent or not. I would appreciate any clarification. All maps and group actions are assumed continuous. Definition 1: A principal $G$-bundle is a […]

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if each point of $B$ has neighborhood $U$ such that there is a homeomorphism $\phi:U\times F\to p^{-1}(U)$ such that $p(\phi\langle b,y\rangle)=b$ for all […]

A classifying space for a group $G$ is a topological space $BG$ with a principle $G$-bundle $p : EG \to BG$ where $EG$ is contractile, so that $BG = EG/G$. A classifying space is universal in the sense that if $q : E \to B$ is a principle $G$-bundle, then there is a continuous map […]

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 transformations of $P$) acts on the space of sections of the associated bundle $P\times_G F$ as follows: if $\alpha\in\text{Aut}P$ […]

It seems to me that given a vector bundle, the associated principal bundle is univocally determined. In fact one has to construct a principal bundle given the base, the fibre (the group $G$ in which the transition functions of the vector bundle take values) and a local trivialization whose associated transition functions satisfy the cocycle […]

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such that $(p, r)$ is a principal $G$-bundle, then it follows that $B \cong E/G$ where $E/G$ […]

For two manifolds $M$ and $N$ I’m trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then $\widetilde{M}$ and $\widetilde{N}$ the orientation double covers of $M$ and $N$ are disconnected. But $\widetilde{M} \times \widetilde{N}$ is the […]

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