Obstruction classes and Pontrjagin classes (Ex)

Take the stable vector bundle $\xi$$<wikitex>; Take the stable vector bundle $\xi$ over the i$-sphere corresponding to a generator of $\pi_{4i}(BO) = \mathbb{Z}$. By defintion, the the primary obstruction to trivialising $\xi^{4i}$ is a cohomology class $x \in H^{4i}(S^{4i}; \pi_{4k-1}(O)) = H^{4i}(S^{4i})$ which generates $H^{4i}(S^{4i}) \cong \mathbb{Z}$. {{beginthm|Exercise}} Determine the $i$-th integral Pontryagin class of $\xi^{4i}$, $p_i(\xi) \in H^{4i}(S^{4i})$, in terms of $x$. {{endthm}} </wikitex> [[Category:Exercises]] [[Category:Exercises with solution]]\xi$ over the $4i$$4i$-sphere corresponding to a generator of $\pi_{4i}(BO) = \mathbb{Z}$$\pi_{4i}(BO) = \mathbb{Z}$. By defintion, the the primary obstruction to trivialising $\xi^{4i}$$\xi^{4i}$ is a cohomology class $x \in H^{4i}(S^{4i}; \pi_{4k-1}(O)) = H^{4i}(S^{4i})$$x \in H^{4i}(S^{4i}; \pi_{4k-1}(O)) = H^{4i}(S^{4i})$ which generates $H^{4i}(S^{4i}) \cong \mathbb{Z}$$H^{4i}(S^{4i}) \cong \mathbb{Z}$.

Exercise 0.1. Determine the $i$$i$-th integral Pontryagin class of $\xi^{4i}$$\xi^{4i}$, $p_i(\xi) \in H^{4i}(S^{4i})$$p_i(\xi) \in H^{4i}(S^{4i})$, in terms of $x$$x$.