Equivariant semi-topological invariants, Atiyah's \(KR\)-theory, and real algebraic cycles (Q2844851)

By generalizing the method of \textit{E. M. Friedlander} et al. [Math. Ann. 330, No. 4, 759--807 (2004; Zbl 1062.19004)] in the semi-topological Atiyah-Hirzebruch spectral sequence relating the morphic cohomology and the semi-topological K-theory of smooth complex variety, the authors constructed a version for smooth real varieties. Moreover, mimicing the complex analogue, they compare it with the known motivic Atiyah-Hirzebruch spectral sequence, the topological Atiyah-Hirzebruch spectral sequence and Dugger's spectral sequence for real varieties.NEWLINENEWLINEGeneralizing Suslin's Conjecture on morphic cohomology of smooth complex varieties, the authors formulated its equivariant and real versions, and showed that they are all equivalent to each other. This enables them to prove equivariant Suslin's Conjecture in codimension one for all smooth real varieties and computes real morphic cohomology in codimension one.NEWLINENEWLINEAs an application, they compute the semi-topological \(K\)-theory of certain real varieties. They also give another proof of the 2-adic Lichtenbaum-Quillen Conjecture over \(\mathbb{R}\), which was originally proved by Karoubi-Weibel and Rosenschon and Ostvaer.NEWLINENEWLINEIn the end, they compare Poincaré Duality relating equivariant morphic cohomology and dos Santos' equivariant Lawson homology groups, and Poincaré Duality relating the Bredon cohomology and usual homology.