KO-Characteristic classes

KO-Pontryagin classes

The KO-Pontryagin classes $\pi^j$$== KO-Pontryagin classes == <wikitex>; The KO-Pontryagin classes $\pi^j$ are defined by setting $\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$ for $j \ge 2$ for complex line bundles L and then requiring naturality and $\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$ where $\pi_s =\sum_j \pi^j s^j$ . Here $\xi$ and $\eta$ are oriented bundles. In fact, these properties determine $\pi^j$ because the group $KO(BSO(m))$ injects into $K(BT^{[m/2]})$ under the complexification of the map which is induced by the restriction to the maximal torus $T^{[m/2]}$ (compare {{cite|Anderson&Brown&Peterson1966}}). For $J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$. Such a class gives a map $MSpin\to ko \langle m\rangle$. == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\pi^j$ are defined by setting $\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$$\pi^0(L) = 1, \pi^1(L)=L-2, \pi^j(L) = 0$ for $j \ge 2$$j \ge 2$ for complex line bundles L and then requiring naturality and $\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$$\pi_s(\xi + \eta) = \pi_s(\xi)\pi_s(\eta)$ where $\pi_s =\sum_j \pi^j s^j$$\pi_s =\sum_j \pi^j s^j$ . Here $\xi$$\xi$ and $\eta$$\eta$ are oriented bundles. In fact, these properties determine $\pi^j$$\pi^j$ because the group $KO(BSO(m))$$KO(BSO(m))$ injects into $K(BT^{[m/2]})$$K(BT^{[m/2]})$ under the complexification of the map which is induced by the restriction to the maximal torus $T^{[m/2]}$$T^{[m/2]}$ (compare [Anderson&Brown&Peterson1966]). For $J=(j_1,\dots, j_n)$$J=(j_1,\dots, j_n)$ we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$$\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$$n(J)=\sum_i j_i$. Such a class gives a map $MSpin\to ko \langle m\rangle$$MSpin\to ko \langle m\rangle$.

References

• [Anderson&Brown&Peterson1966] D. W. Anderson, E. H. Brown and F. P. Peterson, Spin cobordism, Bull. Amer. Math. Soc. 72 (1966), 256–260. MR0190939 (32 #8349) Zbl 0156.21605

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