KO-Characteristic classes
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− | The KO-Pontryagin classes $\pi^j$ are defined by setting | + | The KO-Pontryagin classes $\pi^j$ for oriented vector bundles, i.e. in $KO(BSO)$ are defined by setting |
− | $\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$ for $j \ge 2$ | + | $\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$ for $j \ge 2$ for complex line bundles L and then requiring naturality and |
− | for complex line bundles L and then requiring naturality and | + | $\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$ where $\pi_s =\sum_j \pi^j s^j$ . |
− | $\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$ | + | |
− | where $\pi_s =\sum_j \pi^j s^j$ . | + | |
Here $\xi$ and $\eta$ are oriented bundles. | Here $\xi$ and $\eta$ are oriented bundles. | ||
+ | |||
In fact, these properties determine $\pi^j$ because the group $KO(BSO(m))$ injects into $K(BT^{[m/2]})$ under the | In fact, these properties determine $\pi^j$ because the group $KO(BSO(m))$ injects into $K(BT^{[m/2]})$ under the | ||
complexification of the map which is induced by the restriction to the maximal torus $T^{[m/2]}$ | complexification of the map which is induced by the restriction to the maximal torus $T^{[m/2]}$ | ||
− | (compare {{cite|Anderson&Brown& | + | (compare {{cite|Anderson&Brown&Peterson1966a}}). |
− | + | ||
− | + | ||
== References == | == References == |
Revision as of 12:49, 27 January 2010
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]).
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
This page has not been refereed. The information given here might be incomplete or provisional. |