Obstruction classes and Pontrjagin classes (Ex)

Latest revision as of 01:00, 3 June 2014

Take the stable vector bundle $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Question == <wikitex>; Take the stable vector bundle over the i$-sphere corresponding to the generator of $\pi_{4i} BO = \mathbb{Z}$. What's its $i$-th Pontryagin class in $H^{4i}(S^{4i}) \cong \mathbb{Z}$ ? </wikitex> == Answers == <wikitex>; YOUR TEXT HERE ... </wikitex> == Further discussion == <wikitex>; YOUR TEXT HERE ... </wikitex> == References == {{#RefList:}} <!-- Please add further headings according to your needs. --> [[Category:Questions]] [[Category:Study questions]]\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 $H^{4i}(S^{4i}) \cong \mathbb{Z}$.