Obstruction classes and Pontrjagin classes (Ex)
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<wikitex>; | <wikitex>; | ||
Let $a_j = (3 - (-1)^j)/2$, let $k!$ be the integer k-factorial and recall that $x \in H^{4i}(S^{4i})$ is a generator. | Let $a_j = (3 - (-1)^j)/2$, let $k!$ be the integer k-factorial and recall that $x \in H^{4i}(S^{4i})$ is a generator. | ||
− | {{beginthm|Theorem|{{cite|Kervaire1959}}}} | + | {{beginthm|Theorem|{{cite|Kervaire1959}}}} \label{thm:1} |
There is an identity | There is an identity | ||
$$ p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$ | $$ p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$ | ||
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where $y\in H^{2i}(S^{2i})\cong \mathbb{Z}$ is a generator. | where $y\in H^{2i}(S^{2i})\cong \mathbb{Z}$ is a generator. | ||
{{endthm|Theorem}} | {{endthm|Theorem}} | ||
− | + | === Justification === | |
− | A way to prove the Theorem is to use the Chern character | + | A way to prove the Theorem \ref{thm:1} is to use the Chern character |
$$\tilde K_0(X)\to \tilde H^{ev}(X;\mathbb{Q})$$ | $$\tilde K_0(X)\to \tilde H^{ev}(X;\mathbb{Q})$$ | ||
from complex topological $K$-theory. It can be defined using the explicit formula | from complex topological $K$-theory. It can be defined using the explicit formula |
Revision as of 18:33, 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.
Justification
A way to prove the Theorem 2.1 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