and equivariant applications of the Beilinson-Lichtenbaum conjecture. The sixth section is devoted to the proof of our main vanishing Theorem. One of the main technical ingredients of this proof is left for section seven where we reinterpret the cycle map \[tcycm\] from reduced Lawson homology groups to the singular homology in a manner needed to prove our vanishing theorem. The paper ends with two appendixes where we prove and recollect a few results on topological monoids used in the paper.

The authors would like to thank Eric Friedlander, Christian Haesemeyer and Mark Walker for helpful discussions.

0.5cm *Notation:* By a quasi-projective $k$-variety we mean a reduced and separated quasi-projective scheme of finite type over a field $k$. We write $Sch/k$ for the category of quasi-projective $k$-varieties and $Sm/k$ for the subcategory of smooth quasi-projective $k$-varieties. Except in section 2, $G$ always denotes $Gal(\C/\R)$ and $\sigma\in G$ denotes the nontrivial element.

Equivariant Homotopy and Cohomology
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We recall the basic definitions and theorems we need from equivariant homotopy theory. For more details see [@May:equi]. In this paper we will only work with $G=\Z/2$, but since no simplification results in the basic definitions, we let $G$ denote an arbitrary finite group. The 