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*.

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

Last revised on March 5, 2024 at 00:33:51. See the history of this page for a list of all contributions to it.