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 $\mathbb{C}$.

Under the identification $B \mathrm{U}(1) \,\simeq\, \mathbb{C}P^\infty$ of the $\mathrm{U}(1)$-classifying space with the infinite complex projective spaces, this is the dual tautological line bundle on the latter.

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.

Properties

Zero-section into Thom space is weak equivalence

See at zero-section into Thom space of universal line bundle is weak equivalence.

Related concepts

• Möbius strip

• tautological line bundle, canonical line bundle

• complex oriented cohomology theory