Morse theory on G-manifolds

Let G be a finite group and M be a compact G-manifold on which the G-action is semifree. For a specific local coefficient system we show that the Bredon cohomology of M can be expressed in terms of the classical equivariant cohomology of the fixed point sets. Using this we define Morse polynomial for an equivariant Morse function f defined on M. If M is a G-manifold such that codimMG⩽2 and iff is an equivariant Morse function on M such that f|MG is also Morse then we show that the Morse polynomial of f completely reflects a G-CW structure of M.