nLab
universal complex line bundle

Idea

The universal complex line bundle is the complex universal vector bundle of rank 1, hence the complex line bundle which is associated to the circle group-universal principal bundle $E U(1)$ over the classifying space $B U(1)$ via the canonical action of $U(1)$ on $\mathbb{C}$.

Its pullback bundle along the canonical inclusion $S^2 \longrightarrow B U(1)$ (the map which represents $1 \in \pi_2(B U(1)) \simeq \mathbb{Z}$) is the basic complex line bundle on the 2-sphere.

Related concepts

• Möbius strip

• canonical line bundle

• complex oriented cohomology theory