Obstruction classes and Pontrjagin classes (Ex)
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== Answer == | == Answer == |
Revision as of 18:20, 7 February 2012
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 from the case using Bott periodicity and multiplicativity:
- 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
The integrality condition for the Chern character (and the additional factor of 2 for complexifications of real vector bundles in dimensions ) also follows from the Atiyah-Singer Index Theorem.
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