Definition 1 : Let $B$ be a topological space. A complex vector bundle of rank $n$ over $B$ consists of the data $(E,B,\pi,\{U_i\}_i,\{\phi_i\}_i)$ where $E$ is a topological space, $\pi:E\to B$ is a continuous map called the projection map, $\{U_i\}_i$ is an open cover of $B$ and $\phi_i:\pi^{-1}(U_i)\to U_i\times \mathbb C^n$ is a homeomorphism such […]

As we know, $S^0\to S^n \to \mathbb RP^n$ is a fiber bundle with the usual covering space projection. I wonder if we can construct a projection $S^n \to S^n$ such that $S^0 \to S^n \to S^n$ becomes a fiber bundle. I need this result to complete the proof in here.

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 just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn’t necessarily surjective, of course, because one or more of the $X_i$ may be empty. Anyway, I noticed that the set $\prod_{i:I} X_i$ can be identified with the set of sections of […]

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations $S^1\to\mathbb RP^{2n+1}\to\mathbb CP^n$ and $S^2\to\mathbb CP^{2n+1}\to\mathbb HP^n$. Question. Does octonionic Hopf fibration $S^7\to S^{15}\to\mathbb OP^1$ give rise to a fibration $\mathbb HP^3\to\mathbb OP^1$? (On one hand, constructing a map $\mathbb HP^3\to\mathbb OP^1$ […]

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that $\phi_U:p^{-1}(U)\longrightarrow U\times \mathbb{K}^n$,$\phi_V:p^{-1}(V)\longrightarrow V\times \mathbb{K}^n$. $(b,x)\in E$. Then $\phi_U(b,x)=(b,u)$, $u\in \mathbb{K}^n$, $\phi_V(b,x)=(b,v)$, $v\in \mathbb{K}^n$. Then $\phi_V\phi_U^{-1}(b,u)=(b,v)$, $v=\lambda(b)u$. Thus for any $(b,x_1\otimes x_2)\in F_b(\xi_1\otimes \xi_2)$, […]

Suppose that $p_1:E_1\to B$, $p_2:E_2 \to B$ are two $C^\infty$ fiber bundles which are $C^1$ isomorphic. That is, there exists a $C^1$ diffeomorphism $f:E_1\to E_2$ satisfying $p_2 \circ f = p_1$. Question: Does it follow that $p_1:E_1\to B$ and $p_2:E_2 \to B$ are $C^\infty$ isomorphic? Motivation: If $M, N$ are two $C^\infty$ manifolds which are […]

Let $\pi: E \to M$ a fiber bundle and $\Gamma(M,E)$ the space of smooth sections of the bundle with topology induced by the Whitney topology on $C^{\infty}(M,E)$. Assume that each fiber is contractible. Is $\Gamma(M,E)$ necessarily contractible?

For principal $G$-bundles with $G$ a Lie group there exists a principal $G$-bundle $EG \to BG$ such that we have a bijection $$ [X,BG] \leftrightarrow \text{(principal $G$-bundles over X)} $$ $$ f \mapsto f^* EG $$ where $[X,BG]$ is the set of homotopy classes of maps from $X$ to $BG$. As a result of this, […]

A $\Bbb K$ vector bundle over a space $B$ is a space $E$ and a continuous map $p:E\to B$ so that $p^{-1}(b)$ is a topological $\Bbb K$ vector space for any $b\in B$. One always includes local triviality in the definition, that is one demands that for any $b\in B$ there is a neighbourhood $U$ […]

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