Euler classes of projective spaces

Question

The Euler classes of spheres $S^d$ is $$ e(S^d)=1+(-1)^d. $$

The Euler classes of real projective space ${\mathbb{P}}_d({\mathbb{R}})$ is $$e({\mathbb{P}}_d({\mathbb{R}}))=d+1 \mod 2.$$

The Euler classes of complex projective space ${\mathbb{P}}_d({\mathbb{C}})$ is $$e({\mathbb{P}}_d({\mathbb{C}}))=d+1.$$

The Euler classes of quaternion projective space ${\mathbb{P}}_d({\mathbb{H}})$ is $$e({\mathbb{P}}_d({\mathbb{H}}))=d+1.$$

1. Are my above statements correct? Can I derive these results more cleanly and uniformly in a systematic approach?

2. How can I relate these results to other characteristic classes?

Answer 1

I believe that you somewhat confuse Euler class and Euler characteristic. For example, Euler class is defined only for the orientable manifolds, where the top cohomology group is Z, while Euler characteristic is defined for any manifold. (Half of the projective spaces are not orientable!)