The \(K_ *\)-local type of the smash product of real projective spaces (Q674603)

If \(KO\) and \(KU\) denote the real and complex \(K\)-spectrum and \(S_K\) denotes the \(K_*\)-localization of the sphere spectrum \(S\), then two CW spectra \(X\) and \(Y\) have the same \(K_*\)-local type if and only if \(X\) is quasi-\(S_{K_*}\)-equivalent to \(Y\). The author obtains the \(K_*\)-local types of the smash products of the real projective spaces \(\mathbb{R}\mathbb{P}^n\) and the stunted real projective spaces \(\mathbb{R}\mathbb{P}^n/\mathbb{R}\mathbb{P}^m\). The calculations are very involved and technical.