Via Poincare duality the vanishing theorem is equivalent to the statement that $cyc_{k}$ is an isomorphism for $k>q$.

Via the Milnor conjecture over $\C$ and over $\R$ we can deduce an isomorphism $\pi_{k}\mcal{R}^{q}(X)\to\pi_{k}\mcal{R}^{q}_{top}(X)$ for $k\geq q$. Here $\mcal{R}^{q}_{top}(X)$ is the group of “reduced topological cocycles”. For a precise definition see Section \[subdual\], but essentially this is a version of the quotient group $\map{X_{\C}(\C)}{\mcal{Z}_{0}(\A^{q}_{\C})}^{G}/\map{X_{\C}(\C)}{\mcal{Z}_{0}(\A^{q}_{\C})}^{av}$ which has reasonable homotopical properties (such as fitting into a homotopy fiber sequence involving $\map{X_{\C}(\C)}{\mcal{Z}_{0}(\A^{q}_{\C})}^{G}$ and $\map{X_{\C}(\C)}{\mcal{Z}_{0}(\A^{q}_{\C})}^{av}$).

The final ingredient for our vanishing theorem is now provided by Corollary \[rdual\] which shows that for $X$ projective, $$\widetilde{\Phi}:\pi_{k}\mcal{R}^{q}(X)\rightarrow \pi_{k}\mcal{R}^{q} _{top}(X)$$ agrees with the cycle map $\pi_{k}\mcal{R}^{q}(X) \to H^{q-k}_{sing}(X(\R);\Z/2)$ for $k\geq 2$.

Here is a short outline of the paper. In the second section we review the equivariant homotopy used in the paper. The third section is dedicated to introducing the topological spaces of cycles we study and proving some basic properties that we use and for which we don’t find exact references in the literature. In the fourth section we prove a Poincare Duality between equivariant morphic cohomology and real Lawson homology. In the fifth section we discuss the cycle maps from equivariant morphic cohomology and Bredon cohomology 