I do not know if anyone can help me with these doubts of spin structures and characteristic classes.
1) Is there an orientable manifold that is not spin?
2) Is there a finite group $ G $ such that its cohomology ring $ H ^ \ast (BG, \mathbb {Z} / 2) $ is not generated as a ring by the classes $ w_i (E) $ with $ E \rightarrow BG $ ?
Thank you