International Congress of Mathematicians

64 M. LevineProof. For CH*, this uses localization, theorem 4.9 and resolution of singularities.For Ko, one writes down an integral Chern character, which gives the inverseisomorphism by the Grothendieck-Riemann-Roch theorem.D6. Higher algebraic cobordismThe cohomology theory represented by the P 1 -spectrum MGL in the Morel-Voevodsky  1 -stable homotopy category [9, 13] gives perhaps the most natural algebraicanalogue of complex cobordism. By universality, Q"(X) maps to MGL 2n ' n (X);to show that this map is an isomorphism, one would like to give a map in the otherdirection. For this, the most direct method would be to extend 0* to a theory ofhigher algebraic cobordism; we give one possible approach to this construction here.The idea is to repeat the construction of 0», replacing abelian groups withsymmetric monoidal categories throughout. Comparing with the Q-construction,one sees that the cobordism cycles in 1Z. d%m (X) should be homotopic to zero, butnot canonically so. Thus, we cannot impose this relation directly, forcing us tomodify the group law by taking a limit.Start with the category Z(X) 0 , with objects (/ : F —t X, L\,..., L r ), whereF is irreducible in Sni/., / is projective, and the L, are line bundles on F. Amorphism (/ : F —t X, L\,..., L r ) —t (/' : F' —t X, L[,..., L' r ) in Z(X) 0 consist ofa tuple (*L' via ), let i' : D' -t Y' be the map induced by , s' : Y' -t V thesection induced by s, andip D : (/ o i : D -> X, i*Li,..., i*L r ) -> (/' o i' : D' -> X, i'*L[,..., i'*L' r ),the morphism induced by tp. We impose the relation V°7L,S = 1L', S ' °'