• What is a line bundle?
• What does it mean to tensor two line bundles?
• Why are line bundles invertible?
• Can we draw pictures?

§ Why are bundles invertible?

Because locally, they're locally modules. This leads us to

§ Why are modules invertible?

All modules are invertible when tensored with their dual.

To simplify further, let's move to linear algbera from ring theory; consider the field $\mathbb R$. Over this, we have a vector space of dimension $1$, $\mathbb R$. Now, if we consider $\mathbb R \otimes \mathbb R^*$, this is isomorphic to $\mathbb R$ since we can replace $(r, f) \mapsto f(r)$. This amounts to the fact that we can contract tensors.

So, $\mathbb R \otimes \mathbb R^* \simeq \mathbb R$. Generalize to bundles.

§ References

• Fiber bundles at physics travel guide