A full characterization is given of those compact Lie groups G with the property that every G-map X→X on a finite-dimensional G-complex X of finite orbit type, XG = Ø, is (non-equivariantly) essential. For arbitrary G, conditions are given on the G-space X which guarantee this property. Finally, conditions are given for the non-existence of a G-map X→Y inducing a homotopy equivalence XG≃YG on the fixed point sets. These results have applications to critical point theory of almost G-invariant functionals.