Quasi \(KO_*\)-types of \(CW\)-spectra \(X\) with \(KU_*X \cong Free \oplus Z/2^m\) (Q1567116)

A CW-spectrum \(X\) is said to be quasi \(KO_*\)-equivalent to a CW-spectrum \(Y\) if there exists a map \(Y\to KO \wedge X\) inducing an isomorphism \(KO_*Y \cong KO_*X\). This is an equivalence relation, while \(KO_*\)-equivalence is not. It is known that if \(X\) and \(Y\) are quasi \(KO_*\)-equivalent, then \(KU_*X\) is isomorphic to \(KU_*Y\) as an abelian group with involution \(\psi_C^{-1}\). In this paper, a reverse is studied. That is, the problem is what the quasi \(KO_*\)-type of \(X\) is if the structure of \(KU_*X\) with the conjugation \(\psi_C^{-1}\) is given. Bousfield first studied the case where \(KU_*X\) is free, and showed \(X\) is a wedge sum of some suspensions of \(S\), \(C(\eta)\) and \(C(\eta^2)\), where \(S\) and \(C(x)\) denote the sphere spectrum and a mapping cone of \(x\), respectively, and \(\eta\) is the Hopf map. The author had developed a method to study the problem, and classified the case where \(KU_*X\) is the direct sum of free groups and \(\mathbb{Z}/2^m\) with \(KU_1X= 0\). In this paper, the author shows Bousfield's theorem by the author's method which is easier, and classifies the above case without the condition \(KU_1X= 0\) by considering the \(S\)-duality. The answer is complicated but it is the direct sum of the above spectra and about ten kinds of finite spectra which have at most four cells.