nLab
universal complex line bundle

Contents

Contents

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 EU(1)E U(1) over the classifying space BU(1)B U(1) via the canonical action of U(1)U(1) on \mathbb{C}.

Under the identification BU(1)P B \mathrm{U}(1) \,\simeq\, \mathbb{C}P^\infty of the U(1)\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 2BU(1)S^2 \longrightarrow B U(1) (the map which represents 1π 2(BU(1))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.

Last revised on January 25, 2021 at 09:47:09. See the history of this page for a list of all contributions to it.