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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