variety of dimension $d$. Then $$\pi_{n}\frac{\mcal{Z}_{p}(X_{\C})^{av}}{2\mcal{Z}_{p}(X_{\C})^{av}} = 0$$ for $n\geq 2d-2p+1$.

An essential ingredient in the proof of our vanishing theorem is the Milnor conjecture proved by Voevodsky in [@Voev:miln]. The Milnor conjecture relates motivic cohomology and etale cohomology while real morphic cohomology naturally compares with Bredon cohomology. We need to know that these cycle maps are suitably related which is done in Theorem \[cyccomp\],

\[dg\] Let $X$ be a smooth quasi-projective real variety. The diagram commutes $$\xymatrix{
 L^{q}H\R^{q-k,q}(X;\Z/2) \ar[d]_-{\Phi} && \ar[ll]_{\iso}\H^{2q-k,q}_{\mcal{M}}(X;\Z/2)  \ar[d]^-{cyc}  \\
H^{q-k,q}(X_{\C}(\C);\underline{\Z/2})  \ar[r] & H_{G}^{2q-k}(X_{\C}(\C);\Z/2)\ar[r] &  H^{2q-k}_{et}(X;\mu_{2}^{\otimes q}),
}$$ where $H^{p-q,q}(X_{\C}(\C);\underline{\Z/2})$ denotes Bredon cohomology and $H_{G}^{p}(X_{\C}(\C);\Z/2)$ denotes Borel cohomology.

This suggests that there are possible advantages in replacing the map on Chow groups of real cycles into Borel cohomology with the map into Bredon cohomology since in many respects Bredon cohomology behaves better than Borel cohomology. An application of this idea will be given in a forthcoming paper.

Together with the mod-2 Beilinson-Lichtenbaum conjecture for real and complex varieties (which is a consequence of the Milnor conjecture by [@SV:BK]) we conclude an equivariant Beilinson-Lichtenbaum type theorem for an equivariant extension of Friedlander-Walker’s real morphic cohomology groups (see Definition \[emco\]).

Let $X$ be a smooth quasi-projective real variety 