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Applications of Atiyah-Hirzebruch spectral sequences for motivic cobordism. (English) Zbl 1086.55005
When V. Voevodsky’s proof of the Milnor conjecture first appeared in the 1996 preprint
[www.math.uiuc.edu/K-theory/0170] it was immediately clear to those familiar with Adams spectral sequence calculations in unitary cobordism and in Brown-Peterson cohomology that here was an exciting new context in which that self-same type of calculation was taking place. During the intervening decade the complete gamut of cobordism-related generalised cohomology theories and cohomology operations has been developed and used with great effect to study smooth schemes over fields and their algebraic cycles. These generalised cohomology theories generally have the “algebraic” prefix, as in algebraic cobordism, algebraic BP-theory, algebraic Morava K-theories, E-theories and so on.
As in algebraic topology, the principal theory is algebraic cobordism, which comes in two flavours as in [V. Voevodsky, Proc. Symp. Pure Math. 67, 283–303 (1999; Zbl 0941.19001)] and [M. Levine and F. Morel, C. R. Acad. Sci. Paris 332, 723–728 (2001; Zbl 0991.19001) and ibid. 815–820 (2001; Zbl 1009.19002)]. M. Hopkins and F. Morel constructed a motivic version of the Atiyah-Hirzebruch spectral sequence relating motivic cohomology with coefficients in $$MU^{*}$$ to algebraic cobordism.
The author gives a guided tour of many applications of this spectral sequence to computations of the algebraic motivic cohomology theories mentioned above. En route he gives an extremely useful concise exposition of motivic cohomology and the $$A^{1}$$-homotopy and stable homotopy categories of [F. Morel and V. Voevodsky, Publ. Math., Inst. Hautes Etud. Sci. 90, 45–143 (1999; Zbl 0983.14007)]. The applications are too numerous to be covered here but I mention the section on algebraic cobordism, which is important because of the relation to Chow groups. The algebraic cobordism theories of Levine-Morel and Voevodsky are not known to coincide but there is a comparison map relating them. The author uses the motivic version of the Atiyah-Hirzebruch spectral sequence to study this map and also to study the algebraic BP-theory of the classifying space of an algebraic group.

##### MSC:
 55P42 Stable homotopy theory, spectra 57T25 Homology and cohomology of $$H$$-spaces 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
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