Circle Bundle Article Index for
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Information About

Circle Bundle





RELATIONSHIP TO ELECTRODYNAMICS

  • F being Cohomologous to zero. In particular, there always exists a 1-form ''A'' such that


  • F = dA


Given a circle bundle ''P'' over ''M'' and its projection

:\pi:P o M

one has the Homomorphism

  • :H^2(M,\mathbb{Z}) o H^2(P,\mathbb{Z})




CLASSIFICATION


The Isomorphism Class es of circle bundles over a manifold ''M'' are in one-to-one correspondence with the elements of the second Integral Cohomology Group H^2(M,\mathbb{Z}) of ''M''. This isomorphism is realized by the Euler Class .

Equivalently, the isomorphism classes correspond to homotopy classes of maps to the infinite-dimensional Complex Projective Space CP^\infty, which is the classifying space of U(1) . See Classifying Space For U(n) .

  • . In this situation, the Euler class of the circle bundle or real two-plane bundle is the same as the first Chern Class of the line bundle.


See also: Wang Sequence .