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Open problems in the motivic stable homotopy theory. I. (English) Zbl 1047.14012
Bogomolov, Fedor (ed.) et al., Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press (ISBN 1-57146-090-X). Int. Press Lect. Ser. 3, No. I, 3-34 (2002).
One of the main topics of the International Press Conference at the University of California at Irvine in June 1998 was the recent spectacular progress in motivic cohomology and motivic homotopy theory. A good deal of this progress is due to the pioneering contributions of Vladimir Voevodsky, who received the fields medal in 2002 for his outstanding work. In the article under review, which reflects the mini-course he taught at this conference, the author discusses a number of new ideas and results in this context, out of which he derives sixteen precise conjectures concentrated around the notion of slice filtration and the related concept of rigid motivic homotopy groups. The relevant homotopy theory under discussion, the so-called $$\mathbb{A}^1$$-homotopy theory, was introduced by the author at the ICM 1998 in Berlin [V. Voevodsky, $$\mathbb{A}^1$$-homogopy theory, Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, Vol. I, 579–604 (1998; Zbl 0907.19002) and further developed by F. Morel and V. Voevodsky [$$\mathbb{A}^1$$-homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)]. The main conjectures proposed in the present paper concern a number of analogues of topological constructions in the motivic framework,including existence theorems and convergence properties of certain motivic spectral sequences. One central object, in the entire complex of conjectures, is the so-called slice filtration within the stable motivic homotopy category associated with a Noetherian scheme. This crucial new concept is discussed in the first section of the paper, and the conjectures explained in the sequel make its fundamental importance evident. As for the respective conjectures asserted in the present paper, which are conceptually too intricate and many-sided to be expounded here, the author also points out promising strategies of proof for them, mainly so in the last section of his highly important, inspiring and challenging work.
No doubt, the research program outlined in this article must be seen as an essential catalyst for further research in motivic homotopy theory.
For the entire collection see [Zbl 1033.11002].

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14F35 Homotopy theory and fundamental groups in algebraic geometry 14F25 Classical real and complex (co)homology in algebraic geometry 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 19D45 Higher symbols, Milnor $$K$$-theory 18G55 Nonabelian homotopical algebra (MSC2010)