Discover published sets by community

Chern classes are characteristic classes associated with complex vector bundles that are used to gauge the non-triviality of a bundle.

Stiefel–Whitney classes are characteristic classes associated with real vector bundles which give obstructions to the existence of sections of certain types.

The total Stiefel–Whitney class of a vector bundle is the sum of its individual Stiefel–Whitney classes, each corresponding to a different degree.

The first Stiefel–Whitney class determines the orientability of a real vector bundle; if it is zero, the bundle is orientable.

The total Chern class of a complex vector bundle is defined as the sum of the individual Chern classes over all degrees, similar to the total Stiefel–Whitney class for real vector bundles.

The Whitney product formula states that the total Stiefel–Whitney class of a direct sum of two real vector bundles equals the cup product of their total Stiefel–Whitney classes.

Chern classes cannot be defined for real vector bundles directly; they are only defined for complex vector bundles. However, for a real vector bundle, one can consider its complexification and then discuss Chern classes.