universal complex line bundle




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.


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.