# 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 $X$.

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30 votes
1 answer
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### Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
• 25.3k
45 votes
5 answers
3k views

### Failure of isomorphisms on stalks to arise from an isomorphism of sheaves

It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an ...
• 127k
2 votes
1 answer
1k views

• 1,753
9 votes
1 answer
2k views

### Pull-back of sections of vector bundles

I'm sure this is a silly question but I'm stuck at the concept of pulling back sections of a vector bundle. Let $\pi:E\to X$ be a vector bundle on a variety $X$ and $f:Y\to X$ a morphism. We have a ...
• 13.7k
9 votes
1 answer
450 views

### Lawson and Michelsohn's proof of the splitting principle for oriented vector bundles

I'm trying to understand the proof of the following result in Lawson & Michelsohn's Spin Geometry: Proposition 11.2. Let $E$ be an oriented real vector bundle of dimension $2n$ over a manifold $X$...
• 43.8k
9 votes
1 answer
1k views

### Are vector bundles isomorphic when their transition functions are homotopic?

I'm in the process of trying to understand the equivalence between two different approaches to vector bundles, namely defining them to be smooth surjections $E\to B$ that are locally isomorphic to ...
8 votes
1 answer
956 views

### Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G

If $X$ is a scheme and $\mathcal{F}$ is a locally free sheaf of rank $n$, then in Vakil's book the total space of $\mathcal{F}$ is defined to be $Spec(\text{Sym}^\bullet \mathcal{F}^\vee)$, the ...
• 2,589
8 votes
2 answers
2k views

### Tangent bundle : is a manifold

I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector ...
• 191
8 votes
1 answer
4k views

### If a line bundle admits a non-vanishing section then it is trivial

Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$. I have to prove that the line bundle above is trivial. Idea: ...
• 2,727
8 votes
3 answers
5k views

### Understanding the trivialisation of a normal bundle

I've been looking for a definition of "trivialisation of normal bundle". I think I understand what a vector bundle, fibre bundle and a local trivialisation of either is. I also know what a tangent ...
• 44.9k
8 votes
2 answers
2k views

### Triviality/non-triviality of line/circle bundle over $S^3$

I am interested in whether I can find non-trivial bundles in the case of a fiber bundle with base space $S^3$ and fiber either $\mathbb{R}$ or $S^1$. I know that in the case of a principal bundle ...
• 81
8 votes
1 answer
895 views

### 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 ...
• 383
8 votes
1 answer
916 views

### Isomorphism between $\operatorname{Hom}(E,F)$ and $E^*\otimes F$, $E$ and $F$ vector bundles.

I need some help in finding the solution to the following problem: Let $E$, $F$ be vector bundles (of finite rank, if needed) over a manifold $M$. Consider the set $\operatorname{Hom}(E,F)$ of all ...
• 4,704
8 votes
1 answer
3k views

### Isomorphism class of vector bundle over $\mathbb S^1$.

I'm currently self-studying through the book Differential forms in Algebraic Topology by Bott & Tu and got stuck on an exercise 6.10, which asks to compute $\textrm{Vect}_k(\mathbb S^1),$ the ...
• 523
7 votes
2 answers
1k views

### "Basis extension theorem" for local smooth vector fields

Let $\pi: E \to M$ be a smooth vector bundle of rank $n$, and suppose $s_1, \ldots, s_m$ are independent smooth local sections over an open subset $U \subset M$. Can I prove the "basis extension ...
• 561