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Questions tagged [vector-bundles]

For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$ (for example $X$ could be a topological space, a manifold, or an algebraic variety): to every point $x$ of the space $X$ we associate (or "attach") a vector space $V(x)$ in such a way that these vector spaces fit together to form another space of the same kind as $X$ (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over $X$. Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles.

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Tangent Bundle of Product Manifold

Suppose $M,N$ are manifolds, and consider the product $M\times N$. From this answer, I know that: $T_{(m,n)}(M \times N) \cong T_m M \oplus T_n N $ Can we conclude that $T(M\times N) \cong T(M) \oplus T(N)$
Mark B
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Why aren't complex vector bundles isomorphic to their duals?

For real vector bundles the argument goes like this: pick a metric on the bundle (i.e. a continuously varying metric on the fibers, which exists if the base space is paracompact), and map a vector $v$ to $\langle v, \cdot \rangle$, which is an iso…
Emilio Ferrucci
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Understanding the canonical line bundle $H$, and the fact that $(H \otimes H)\oplus 1 \simeq H \oplus H$

I am trying to understand Example 1.13 of Hatcher's book on vector bundles and K-Theory (page 24). The cannonical line bundle $H \to \mathbb{C}P^1$ satisfies the relation $(H \otimes H)\oplus 1 \simeq H \oplus H$. The total space, $H$, is given by…
Juan S
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Tensor Product of Vector Bundles

In Hatcher's book on Vector Bundles he states (page 14) that It is routine to verify that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element, the trivial line…
Juan S
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Vector bundles and principal $G$-bundles

I am trying to understand the notion of a principal $G$-bundle versus a vector bundle. Here $G$ is a Lie group. Supposedly, principal $G$-bundles are a generalization of vector bundles. My problem here is that most sources, for example the wikipedia…
Thomas
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Rank $n$ vector bundle with $n$ pointwise linearly independent sections is trivial

I have an $n$-dimensional vector bundle $E \to X$ and sections $s_1, \dots, s_n : X \to E$ such that for any $x \in X$, the elements $s_1(x), \dots, s_n(x)$ are linearly independent in the vector space $E_x$. By using the theorem of fiberwise…
Soren123
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Alternate definition of vector bundle?

Recall the usual definition of a $k$-dimensional vector bundle (everything is assumed to be continuous/smooth/etc depending on the category): A $k$-dimensional vector bundle is a triple $(E,B,\pi)$, where $\pi\colon E \to B$, satisfying the…
Avi Steiner
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Why is BO(n) the same as the classifying space for real vector bundles?

I can construct the classifying space for real vector bundles as the Grassmanian $Gr(n, \mathbb R^\infty)$, and I can construct $BO(n)$ as well. They happen to be the same. But is there an easier way of seeing this, perhaps by noting something about…
Kevin Yin
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How to understand the coordinate transition map in $E\otimes E'$?

This question seems embarrassingly basic. Let $X$ be some manifold and $E,E'$ two vector bundles over $X$ with rank $n$ and $n'$ respectively. Now assume to possible refinement we have $E,E'$ defined on the some coordinate atlast $\{U_{i}\}$ with…
Bombyx mori
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Difference between $\mathfrak{X}(M)\otimes_{\mathbb{R}}\mathfrak{X}(M)$ and $\Gamma(TM\boxtimes TM)$

Let $M$ be a smooth manifold and $\mathfrak{X}(M)$ the real vector space consisting of vector fields over $M$. Let $TM\boxtimes TM$ be the vector bundle over $M\times M$ whose fiber at $(x,y)\in M\times M$ is $T_xM\otimes T_yM$. Finally, let…
Josh Burby
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How do I get the correct coordinate transition functions for the dual bundle?

Suppose $E$ is a vector bundle over $X$, and $X$ is equipped with an atlas of charts $U_{i}$ that is locally finite. Now assume up to possible refinement we have local trivilization $f_{i}: E_{U_{i}}\rightarrow U_{i}\times \mathbb{R}^{n}$, where…
Bombyx mori
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Real structure on complex vector bundle

In Chapter 19 of Homotopical Topology by Fomenko and Fuchs, there is an exercise asserting that a complex vector bundle $\xi$ is the complexification of a real vector bundle if and only if there is an isomorphism $\xi\cong\bar\xi$. I have been…
Nikhil Sahoo
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How to show $\det(E)\cong M\times \mathbb{R}$ when $M$ is orientable?

If we have an orientable bundle $E\rightarrow M$, then the transition maps can be adjusted by Gram-Schmidt process to be in $SO(n,\mathbb{R})$. So the determinant bundle $\det E$ is isomorphic to $M\times \mathbb{R}$ with a trivial section…
Bombyx mori
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Why is this bundle homomorphism a isomorphism?

I got trouble with an exercise: Suppose $E$ and $E'$ are smooth vector bundles over a smooth manifold $M$,and $F:E\rightarrow{E'}$ is a bijective smooth bundle homomorphism over $M$ Prove:$F$ is a smooth bundle isomorphism i.e. $F^{-1}$ is also…
C Weid
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Surjective map from a trivial bundle to any vector bundle

I was reading over some notes on vector bundles which make use of the following fact: If $X$ is a $n$-manifold and $V$ is a real vector bundle on $X$ of rank $k$, then there exists a surjective map of vector bundles from the trivial bundle $X…
JHF
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