The quasi \(KO_*\)-types of weighted projective spaces (Q1380139)

Associated to a sequence of positive integers, \((q_0,\dots,q_n)\), there is a weighted projective space given by the quotient of \(\mathbb{C}^{n+1}-\underline 0\) by the \(\mathbb{C}^*\)-action, \(w(z_0,\dots,z_n)=(w^{q_0}z_0,\dots,w^{q_n}z_n)\). For each weighted projective space, \(X\), the authors give a construction (in terms of spheres and the cofibre of the Hopf map) of a space, \(Y\), such that \(KO\wedge X\) is homotopy equivalent to \(KO\wedge Y\) as a \(KO\)-module spectrum.