bundle of spheresの例文
- Since the tangent bundle of the sphere is stably trivial but not trivial, all other characteristic classes vanish on it, and the Euler class is the only ordinary cohomology class that detects non-triviality of the tangent bundle of spheres : to prove further results, one must use secondary cohomology operations or K-theory.
- For instance, the tangent bundle of spheres is stably trivial but not trivial ( the usual inclusion of the sphere "'S " "'n " ?" "'R " "'n " + 1 has trivial normal bundle, thus the tangent bundle of the sphere plus a trivial line bundle is the tangent bundle of Euclidean space, restricted to "'S " "'n ", which is trivial ), thus other characteristic classes all vanish for the sphere, but the Euler class does not vanish for even spheres, providing a non-trivial invariant.
