# An explicit cycle map for the motivic cohomology of real varieties

### Pedro F. dos Santos

Instituto Superior Técnico, Lisbon, Portugal### Robert M. Hardt

Rice University, Houston, USA### James D. Lewis

University of Alberta, Edmonton, Canada### Paulo Lima-Filho

Texas A&M University, College Station, USA

## Abstract

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.

## Cite this article

Pedro F. dos Santos, Robert M. Hardt, James D. Lewis, Paulo Lima-Filho, An explicit cycle map for the motivic cohomology of real varieties. Comment. Math. Helv. 92 (2017), no. 3, pp. 429–465

DOI 10.4171/CMH/416