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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
##### Keywords:
equivariant cohomology; $$KR$$ theory
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##### References:
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