International Congress of Mathematicians

66 M. Levineis an isomorphism. Since (li") ®L ^»(X)) TO = Q TO (X) for m > n, we are done. DThe categories 0 (X) are covariantly functorial for projective maps, contravariantfor smooth maps (with a shift in the grading) and have first Chern classnatural transformations ci(L) : flm (X) —^ Q^tl1 (X) for L —^ X a line bundle.We conjecture that the inverse system used to define 0 TOjr (X) is eventuallyconstant for all r, not just for r = 0. If this is true, it is reasonable to define thespace BO TO (X) as the homotopy limitBÜm(X) := holimBQ^)(X).nOne would then have 0 TOjr (X) = 7r r (TiO TO (X),0) for all m,r; hopefully the propertiesof 0» listed in theorem 3.6 would then generalize into properties of the spacesBflm(X).References[i[2[3;[4;[6;[r[9[io;[H[12:[is;D. Abramovich, K. Karu, K. Matsuki, J. Wlodarczyk, Torification and factorizationof birational morphisms, preprint 2000, AG/9904135.J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures inMathematics. University of Chicago Press, Chicago, Ill.-London, 1974.S. Borghesi, Algebraic Morava K-theories and the higher degree formula,preprint May 2000, www.math.uiuc.edu/K-theory/0412/index.html.H. Hironaka, Resolution of singularities of an algebraic variety over a field ofcharacteristic zero. I, II, Ann. of Math., (2) 79 (1964), 109^203; ibid. 205^326.M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math.France, 83 (1955), 251-274.M. Levine et F. Morel, Cobordisme algébrique I, II, C.R. Acad. Sci. Paris, SérieI, 332 (2001), 723-728; ibid. 815-820.M. Levine et F. Morel, Algebraic cobordism, I, preprint Feb. 2002,www.math.uiuc.edu/K-theory/0547/index.html.M. Levine, Algebraic cobordism, II, preprint June 2002,www.math.uiuc.edu/K-theory/0577/index.html.F. Morel, V. Voevodsky, Â 1 homotopy of schemes, Publications Mathématiquesde PI.H.E.S, volume 90.I. Panin, Push-forwards in oriented cohomology theories of algebraic varieties,preprint Nov. 2000, www.math.uiuc.edu/K-theory/0459/index.html.D. Quillen, Elementary proofs of some results of cobordism theory using Steenrodoperations, Advances in Math., 7 (1971), 29^56.M. Rost, Construction of splitting varieties, preprint, 1998.V. Voevodsky, Â 1 -homotopy theory, Proceedings of the International Congressof Mathematicians, Vol. I (1998). Doc. Math. Extra Vol. I (1998), 579^604.