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An Atiyah-Hirzebruch spectral sequence for \(KR\)-theory. (English) Zbl 1109.14024
The Atiyah-Hirzebruch spectral sequence relates singular cohomology to complex topological \(K\)-theory. Bloch, Lichtenbaum, Friedlander, Levine and Voevodsky have shown that there is a similar spectral sequence converging from motivic cohomology to algebraic \(K\)-theory, see [M. Levine, \(K\)-Theory 37, No. 1–2, 129–209 (2006; Zbl 1117.19003)].
In this paper an analogous spectral sequence \(H^{p,-\frac{q}{2}}(X,{\underline{\mathbb Z}}) \Rightarrow KR^{p+q}(X)\) is constructed. It converges from \(RO({\mathbb Z}/2{\mathbb Z})\)-graded equivariant cohomology to Atiyah’s \(KR\) theory. The importance of this spectral sequence come from the fact that motivic homotopy theory over \({\mathbb R}\) is naturally related to \({\mathbb Z}/2{\mathbb Z}\)-equivariant homotopy theory. The proof uses Postnikov towers together with an identification of their homotopy fibers with suitable Eilenberg-MacLane spaces. The paper contains also a discussion of étale variants.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19L47 Equivariant \(K\)-theory
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