Obstruction classes and Pontrjagin classes (Ex)

Revision as of 18:15, 17 March 2010

1 Question

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 $\xi$ over the i$-sphere corresponding to a generator of $\pi_{4i}(BO) = \mathbb{Z}$. {{beginthm|Question}} What is the $i$-th integral Pontryagin class of $\xi^{4i}$, $p_i(\xi) \in H^{4i}(S^{4i}) \cong \mathbb{Z}$ ? </wikitex> == Answer == <wikitex>; Let $a_j = (3 - (-1)^j)/2$, let $k!$ be the integer k-factorial and let $x \in H^{4i}(S^{4i})$ be a generator. {{beginthm|Theorem|{{cite|Kervaire1959}}}} There is an identity $$ p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$ {{endthm}} </wikitex> == Further discussion == <wikitex>; ... </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 an obstruction class $x \in H^{4i}(S^{4i})$ which generates $H^{4i}(S^{4i}) \cong \mathbb{Z}$.

Question 1.1.

What is the $i$-th integral Pontryagin class of $\xi^{4i}$, $p_i(\xi) \in H^{4i}(S^{4i})$ ?

2 Answer

Let $a_j = (3 - (-1)^j)/2$, let $k!$ be the integer k-factorial and let $x \in H^{4i}(S^{4i})$ be a generator.

Theorem 2.1 [Kervaire1959]. There is an identity

$$\displaystyle p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$

3 Further discussion

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4 References

• [Kervaire1959] M. A. Kervaire, A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. MR0107863 (21 #6585) Zbl 0124.16302