2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78556Let $\widetilde{JO}(X)=\widetilde{KO}(X)/TO(X)$ 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.15 pages, Latex2e, to appear in Hiroshima Math. J., 2 (1999)Algebraic TopologyK-Theory and Homology55Q50, 55R50 (primary)text