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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].

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