Euler Class - Properties

Properties

The Euler class satisfies these properties, which are axioms of a characteristic class:

Functoriality
If is another oriented, real vector bundle and is continuous and covered by an orientation-preserving map, then . In particular, .
Whitney sum formula
If is another oriented, real vector bundle, then the Euler class of the direct sum is given by

Its distinguishing feature is that it detects the existence of a non-vanishing section:

Normalization
If possesses a nowhere-zero section, then .

It also satisfies:

Orientation
If is with the opposite orientation, then .

Note that unlike other characteristic classes, it is concentrated in a single dimension, which depends on the rank of the bundle: — there are no . In particular, and, but there is no . This reflects the fact that the Euler class is unstable, as discussed below.

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