# Talk:Obstruction classes and Pontrjagin classes (Ex)

## Contents |

## [edit] 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.

### [edit] Justification

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 .

## [edit] 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.

## [edit] 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)

## [edit] 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