# Commentarii Mathematici Helvetici

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**Volume 92, Issue 3, 2017, pp. 429–465**

**DOI: 10.4171/CMH/416**

Published online: 2017-07-27

An explicit cycle map for the motivic cohomology of real varieties

Pedro F. dos Santos^{[1]}, Robert M. Hardt

^{[2]}, James D. Lewis

^{[3]}and Paulo Lima-Filho

^{[4]}(1) Instituto Superior Técnico, Lisbon, Portugal

(2) Rice University, Houston, USA

(3) University of Alberta, Edmonton, Canada

(4) Texas A&M University, College Station, USA

We provide a direct construction of a cycle map in the level of representing complexes from the motivic cohomology of real (or complex) varieties to the appropriate ordinary cohomology theory. For complex varieties, this is simply integral Betti cohomology, whereas for real varieties the recipient theory is the bigraded $\operatorname{Gal}(\mathbb C/\mathbb R)$-equivariant cohomology [19]. Using the finite analytic correspondences from [7] we provide a sheaf-theoretic approach to ordinary equivariant $RO(G)$-graded cohomology for any finite group $G$. In particular, this gives a complex of sheaves $\mathbb Zp_{\omega}$ on a suitable equivariant site of real analytic manifolds-with-corner whose construction closely parallels that of the Voevodsky's motivic complexes $$\mathbb Zp_{\mathcal M}$. Our cycle map is induced by the change of sites functor that assigns to a real variety $X$ its analytic space $X(\mathbb C)$ together with the complex conjugation involution.

*Keywords: *Ordinary equivariant cohomology, motivic cohomology, cycle map, finite analytic currents, real varieties

dos Santos Pedro, Hardt Robert, Lewis James, Lima-Filho Paulo: An explicit cycle map for the motivic cohomology of real varieties. *Comment. Math. Helv.* 92 (2017), 429-465. doi: 10.4171/CMH/416