Properties of Chern Classes
Given a complex vector bundle V over a topological space X, the Chern classes of V are a sequence of elements of the cohomology of X. The th Chern class of V, which is usually denoted ck(V), is an element of
- H2k(X;Z),
the cohomology of X with integer coefficients. One can also define the total Chern class
Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example.
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“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
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