Talk:Obstruction classes and Pontrjagin classes (Ex)
 1 Answer
Let , let be the integer k-factorial and recall that is a generator.
Theorem 1.1 [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 1.1 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 .
 2 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.
 3 Desiderata for this page
Ideally this page will also include at least the following information:
- A reference, probably to Baues' book, for obstructions to null-homotopy.
- A precise reference to [Kervaire1959].
- A precise reference to Bott who also proved Theorem 2.1 (perhaps even before Kervaire?)
- Conventions/constructions for settling the sign in Theorem 2.1.
- A reference to Husemoller's Fibre Bundles for a similar result for Chern classes. This also gives an explanation of the result via the Chern character which should be included in the page.
- Further discussion about the same problem for other characteristic classes and other bases spaces.
Diarmuid Crowley 16:21, 17 March 2010 (UTC)