Obstruction classes and Pontrjagin classes (Ex)
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There is an identity | There is an identity | ||
$$ p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$ | $$ p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$ | ||
− | {{endthm}} | + | |
+ | Similarly, if $\eta$ denotes the complex vector bundle over $S^{2i}$ corresponding to a generator of $\pi_{2i}(BU)\cong \mathbb{Z}$, then its Chern class is given by | ||
+ | $$ c_i(\eta) = \pm (i-1)! \cdot y\in H^{2i}(S^{2i}),$$ | ||
+ | where $y\in H^{2i}(S^{2i})\cong \mathbb{Z}$ is a generator. | ||
+ | {{endthm|Theorem}} | ||
+ | |||
+ | A way to prove the Theorem is to use the Chern character | ||
+ | $$\tilde K_0(X)\to \tilde H^{ev}(X;\mathbb{Q})$$ | ||
+ | from complex topological $K$-theory. It can be defined using the explicit formula | ||
+ | $$ ch(\xi)= \sum_{k>0} s_k(c_1(\xi),\dots,c_k(\xi))/k! $$ | ||
+ | for a virtual complex vector bundle $\xi$, where $s_k$ are the Newton polynomials. In the case $X=S^{2n}$ two special things occur: | ||
+ | |||
+ | #The Chern character is injective with image $H^{2n}(S^{2n};\mathbb{Z})$. (This follows inductively from the case $n=0$ using Bott periodicity.) | ||
+ | #A calculation shows that the image of a (virtual) complex vector bundle $\xi$ over $S^{2n}$ is given by $\pm c_n(\xi)/(n-1)!$. | ||
+ | |||
+ | Hence, $c_i(\eta)$ is given by $\pm (n-1)!$ times a generator. This establishes the second part of the Theorem. | ||
+ | |||
+ | The first part follows using the definition $p_i(\xi)=c_{2i}(\xi\otimes_\mathbb{R} \mathbb{C})$ together with the fact that complexification induces a map | ||
+ | $$ - \otimes_\mathbb{R} \mathbb{C}\colon \widetilde{KO}^0(S^{4i})\to \tilde K^0(S^{4i}) $$ | ||
+ | which is given by multiplication by $a_i$, i.e. is a isomorphism in degrees $8i$ and multiplication by 2 in degrees $8i+4$. | ||
</wikitex> | </wikitex> | ||
Revision as of 15:52, 22 March 2010
Contents |
1 Question
Take the stable vector bundle over the -sphere corresponding to a generator of . By defintion the the primary obstruction to trivialising is an obstruction class which generates .
Question 1.1.
What is the -th integral Pontryagin class of , ?2 Answer
Let , let be the integer k-factorial and recall that is a generator.
Theorem 2.1 [Kervaire1959]. There is an identity
Similarly, if denotes the complex vector bundle over corresponding to a generator of , then its Chern class is given by
where is a generator.
A way to prove the Theorem is to use the Chern character
from complex topological -theory. It can be defined using the explicit formula
for a virtual complex vector bundle , where are the Newton polynomials. In the case two special things occur:
- The Chern character is injective with image . (This follows inductively from the case using Bott periodicity.)
- A calculation shows that the image of a (virtual) complex vector bundle over is given by .
Hence, is given by times a generator. This establishes the second part of the Theorem.
The first part follows using the definition together with the fact that complexification induces a map
which is given by multiplication by , i.e. is a isomorphism in degrees and multiplication by 2 in degrees .
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