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 The main tool used in their work is the fiber-cofiber construction introduced in topology by T. Ganea.

In the first part of this talk, which is based on joint work with Yu. Berest and A. C. Ramadoss, we will describe a natural generalization of the fiber-cofiber construction in the context of abstract homotopy theory, which we call relative join construction. Then, in the second part, we will show how to apply this relative join construction to obtain analogues of spaces of quasi-invariants which can be identified with homotopy quotient of certain $G$-spaces for a compact Lie group $G$. The rational cohomology rings of these generalized spaces of quasi-invariants have nice algebraic properties that are similar to those of classical quasi-invariants. Moreover, the equivariant K-theory of those $G$-spaces exhibits similar properties.