= Chern class in complex cobordism
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{title2=$c_j^{MU}(E)\in MU^{2j}(X)$}
= Chern classes in complex cobordism
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{synonym}
The complex orientation of <complex cobordism> defines Chern classes of actual complex bundles by the splitting principle and elementary symmetric expressions in the first Chern classes of line summands. For an actual rank-$n$ complex bundle, its top Chern class is its <Euler class in a generalized cohomology theory>. Chern classes extend stably by the Whitney formula; this stable extension must not be confused with the Euler class of a merely stably complex real bundle.
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