Obstruction classes and Pontrjagin classes (Ex)
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== Question == | == Question == | ||
<wikitex>; | <wikitex>; | ||
− | Take the stable vector bundle over the $4i$-sphere corresponding to | + | Take the stable vector bundle $\xi$ over the $4i$-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>; | <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> | </wikitex> | ||
== Further discussion == | == Further discussion == | ||
<wikitex>; | <wikitex>; | ||
− | + | ... | |
</wikitex> | </wikitex> | ||
Revision as of 18:07, 17 March 2010
Contents |
1 Question
Take the stable vector bundle over the -sphere corresponding to a generator of .
Question 1.1.
What is the -th integral Pontryagin class of , ?2 Answer
Let , let be the integer k-factorial and let be a generator.
Theorem 2.1 [Kervaire1959]. There is an identity
3 Further discussion
...
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