Stability
In the language of stable homotopy theory, the Chern class, Stiefel-Whitney class, and Pontryagin class are stable, while the Euler class is unstable.
Concretely, a stable class is one that does not change when one adds a trivial bundle: . More abstractly, it means that the cohomology class in the classifying space for pulls back from the cohomology class in under the inclusion (which corresponds to the inclusion and similar). Equivalently, all finite characteristic classes pull back from a stable class in .
This is not the case for the Euler class, as detailed there, not least because the Euler class of a k-dimensional bundle lives in (hence pulls back from, so it can’t pull back from a class in, as the dimensions differ.
From the perspective of the splitting principle, this corresponds to the stability of symmetric polynomials, and the instability of alternating polynomials, specifically the Vandermonde polynomial, which represents the Euler class.
Read more about this topic: Characteristic Class
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—David Hume (1711–1776)
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