A note on the localization of J-groups

Abstract

Let $\widetilde{JO}(X)=\widetilde{KO}(X)/TO(X)$ be the J-group of a connected finite CW complex X. We Obtain two computable formulas of $TO(X)_{(p)}$, the localization of $TO(X)$ at a prime p. Then we show how to use these two formulas of $TO(X)_{(p)}$ to find the J-orders of elements of $\widetilde{KO}(CP^m)$, at least the 2 and 3 primary factors of the canonical generators of $\widetilde{JO}(CP^m).$ Here $CP^m$ is the complex projective space.