\(KO\)-homologies of a few cells complexes (Q1316518)

Let \(X\) and \(Y\) be CW-spectra; \(X\) is quasi \(KO_ *\)-equivalent to \(Y\) if there exists a map \(h: Y\to KO\wedge X\) such that the composite map \((\mu\wedge 1)\circ (1\wedge h): KO\wedge Y\to KO\wedge X\) is an equivalence (where \(\mu\) denotes the multiplication of the real \(K\)- spectrum \(KO\)). Also, \(X\) is said to be stably quasi \(KO_ *\)-equivalent to \(Y\) if there exists \(i\) such that \(X\) is quasi \(KO_ *\)-equivalent to the \(i\)-fold suspended spectrum \(\Sigma^ i Y\). In this paper the author determines the stable quasi \(KO_ *\)-type of any complexes with 3- or 4-cells (see Theorems 5.3 and 5.4).