Editing Characteristic class (section)

==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: <math>c(V \oplus 1) = c(V)</math>. More abstractly, it means that the cohomology class in the [[classifying space]] for <math>BG(n)</math> pulls back from the cohomology class in <math>BG(n+1)</math> under the inclusion <math>BG(n) \to BG(n+1)</math> (which corresponds to the inclusion <math>\mathbf{R}^n \to \mathbf{R}^{n+1}</math> and similar). Equivalently, all finite characteristic classes pull back from a stable class in <math>BG</math>.

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 <math>H^k(X)</math> (hence pulls back from <math>H^k(BO(k))</math>, so it can't pull back from a class in <math>H^{k+1}</math>, as the dimensions differ.