Obstruction classes and Pontrjagin classes (Ex)
From Manifold Atlas
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− | Take the stable vector bundle $\xi$ over the $4i$-sphere corresponding to a generator of $\pi_{4i}(BO) = \mathbb{Z}$. | + | Take the stable vector bundle $\xi$ over the $4i$-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| | + | {{beginthm|Exercise}} |
− | + | Determine the $i$-th integral Pontryagin class of $\xi^{4i}$, $p_i(\xi) \in H^{4i}(S^{4i})$, in terms of $x$. | |
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{{endthm}} | {{endthm}} | ||
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− | + | [[Category:Exercises with solution]] | |
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Latest revision as of 01:00, 3 June 2014
Take the stable vector bundle over the -sphere corresponding to a generator of . By defintion, the the primary obstruction to trivialising is a cohomology class which generates .
Exercise 0.1. Determine the -th integral Pontryagin class of , , in terms of .