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 1 KO-Pontryagin classes
The KO-Pontryagin classes for oriented vector bundles, i.e. in are defined by setting for for complex line bundles L and then requiring naturality and where . Here and are oriented bundles.
In fact, these properties determine because the group injects into under the complexification of the map which is induced by the restriction to the maximal torus (compare [Anderson&Brown&Peterson1966a]).
 2 References
- [Anderson&Brown&Peterson1966a] D. W. Anderson, E. H. Brown and F. P. Peterson, -cobordism, -characteristic numbers, and the Kervaire invariant, Ann. of Math. (2) 83 (1966), 54–67. MR0189043 (32 #6470) Zbl 0137.42802