# KO-Characteristic classes

## 1 KO-Pontryagin classes

The KO-Pontryagin classes $\pi^j$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\pi^j$ for oriented vector bundles, i.e. in $KO(BSO)$$KO(BSO)$ 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&Peterson1966a]).