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On the motivic \(\pi_0\) of the sphere spectrum. (English) Zbl 1130.14019
Greenlees, J. P. C. (ed.), Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, September 9–20, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1833-9/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 131, 219-260 (2004).
This paper starts with a leisurely and pleasant exposition, focusing on the parallels between homotopy theory of topological spaces and motivic homotopy theory. The outset is that the stable homotopy groups \(\pi_i(S^0)\) of the (topological) sphere spectrum vanish when \(i<0\), is the ring of integers when \(i=0\) and are finite when \(i>0\). The algebraic counterpart of \(i=0\) is considered in this paper. The case \(i<0\) has already been solved by Voevodsky, and the question for \(i>0\) is believed to be much more difficult.
In the context of \(\mathbb A^1\)-homotopy invariance, the author nicely motivates the move from smooth schemes to simplicial sheaves and \(S^1\)-spectra on the Nisnevich site. The author then goes on to give an exposition of various central \(S^1\)-stable results before introducing \(\mathbb P^1\)-spectra and motives. In the last section the author gives results about the motivic homotopy groups of the motivic sphere spectrum and of \(\mathbb G_m^{\wedge n}\): for a perfect field \(k\) of characteristic not equal to \(2\) there are isomorphisms of rings \[ K^{MW}_*(k)\cong [S^0,\mathbb G_m^{\wedge *}]_{\mathbb P^1}, \] and so in particular \(GW(k)\cong [S^0,S^0]_{\mathbb P^1},\) where \(K^{MW}_*(k)\) is the Milnor-Witt ring, and \(GW(k)\) is the Grothendieck-Witt ring of quadratic forms. The proof is sketched and some applications are given.
For the entire collection see [Zbl 1050.57001].

14F35 Homotopy theory and fundamental groups in algebraic geometry
14F42 Motivic cohomology; motivic homotopy theory