The quasi \(KO_*\)-types of CW-spectra \(X\) with \(KU_0 X\cong Z/2^m\) and\(KU_1X\cong Z/2^n\) (Q2717462)

Two CW spectra \(X, Y\) are said to be quasi \(KO_*\)-equivalent if \(KO \wedge X\) and \(KO \wedge Y\) are isomorphic as \(KO\)-module spectra. In earlier papers the author has described the possible quasi \(KO_*\)-types of spectra with \(KU_* X\) free or of the form \(\text{free }\oplus {\mathbb Z}/2^m {\mathbb Z}\) [Osaka J. Math 27, No. 3, 465-498 (1990; Zbl 0719.55007); 36, No. 4, 747-765 (1999; Zbl 0953.55008)]. These questions have also been studied by \textit{A. K. Bousfield} [J. Pure Appl. Algebra 66, No. 2, 121-163 (1990; Zbl 0713.55007)]. The present paper extends this description to the case in which \(KU_* X\) is as described in the title. The method is to construct small spectra of the required sort, calculate their \(KC\)-homology and involution and then show these enumerate all possible quasi \(KO_*\)-types. The list of types is too long to give here.