In a recent work, Yu. Berest and A. C. Ramadoss formulated and studied the realization problem for rings of quasi-invariants of finite reflection groups in terms of classifying spaces of compact Lie groups. They solved the realization problem in the rank one case using the fiber-cofiber construction introduced in topology by T. Ganea. In this talk, we will introduce a new class of topological G-spaces generalizing the classical flag manifolds G/T of compact connected Lie groups. These spaces --- which we call the m-quasi-flag manifolds F_m(G,T) --- are topological realizations of the rings Q_m(W) of m-quasi-invariant polynomials of finite reflection groups.