Equivariant semialgebraic homotopies

Let M and N be semialgebraic G spaces. When G is a compact Lie group, we find a one to one correspondence between the set of semialgebraic G homotopy classes of semialgebraic G maps from M to N, with the set of topological G homotopy classes of continuous G maps from M to N. We also deal with the equivariant semialgebraic version of a theorem of J.H.C. Whitehead.