The context of this paper is the motivic stable homotopy category over an algebraically closed field of characteristic zero. In this context, there is an Adams spectral sequence based on mod2 motivic cohomology, which has first been studied in [F. Morel, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 11, 963–968; MR1696188 (2000d:11056)]. It converges to the homotopy groups of a nilpotent completion of the motivic sphere spectrum, and these can be related to the motivic stable stems. In this paper, the authors prove the stated convergence result and present the results of their calculations. These allow them to work up to the 34-stem. Thereby, the authors are not only able to reveal the existence of motivic homotopy classes without classical analogues, but it is particularly noteworthy that they also succeed in giving new proofs about known differentials in the classical mod 2 Adams spectral sequence; see Section 8.7. Although the authors point out that their methods fail in the case of the Kervaire invariant one problem, this nourishes the hope that motivic methods may be used to obtain other results in this direction as well. Reviewed by Markus Szymi
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