THE EQUIVARIANT HUREWICZ MAP

Let G be a compact Lie group, Y be a based G-space, and V be a G-representation. If pi(V)G(Y) is the equivariant homotopy of Y in dimension V and H(V)G(Y) is the equivariant ordinary homology group of Y with Burnside ring coefficients in dimension V, then there is an equivariant Hurewicz map h: pi(V)G(Y) --> H(V)G(Y). One should not expect this map to be an isomorphism, since H(V)G(Y) must be a module over the Burnside ring, but pi(V)G(Y) need not be. However, here it is shown that, under the obvious connectivity conditions on Y, this map induces an isomorphism between H(V)G(Y) and an algebraically defined modification of pi(V)G(Y). The equivariant Freudenthal Suspension Theorem contains a technical hypothesis that has no nonequivariant analog. Our results shed some light on the behavior of the suspension map when this rather undesirable technical hypothesis is not satisfied.