International Congress of Mathematicians

60 M. Levine2. Let 1Z d%m (X) be the subgroup of Z*(X) generated by cobordism cycles ofthe form (/ : Y -ï X,n*Li,... ,n*L r , Mi,..., M s ), where TX : Y -ï Z is asmooth morphism in Smj, the L, are line bundles on Z, and r > dim/. Z. LetZ,(X) = Z4X)/TZ dim (X).3. Add the Gysin isomorphism: If L —¥ Y is a line bundle and s : Y —¥ Lis a section transverse to the zero-section with divisor i : D —¥ Y, identify(/ : Y -• X,Li,...,L r ,L) with (/o»:D4 X,i*L u ... ,i*L r ). We letn„(X) denote the resulting quotient of Z_^(X). Note that on n„(X) we have,for each line bundle L —¥ X, the Chern class operatorCi(L) :£,(*)-• 0,-1 (*)(/ : F -+ X,Li,...,L r ) ^ (f : F -+ X,L U . ..,L r ,fL)as well as push-forward maps /» : Ü»(X) —¥ Q*(X') for / : X —^ X' projective.4. Impose the formal group law: Regrade L by setting L„ := L _n . Let 0»(X) bethe quotient of L» ®Q» (X) by the imposing the identity of maps L» ®I2» (F) —^L» ®Q,(X)(id ® /») o F L (ci(L),ci(M)) = id ® (fi oci(L®Mj)for f :Y —¥ X projective, and L, M line bundles on Y. Note that, having imposedthe relations in 1Z d%m , the operators ci(L), ci(M) are locally nilpotent,so the infinite series FL(CI(L), ci(Mj) makes sense.As the notation suggests, the most natural construction of 0 is as an orientedBorel-Moore homology theory rather than an oriented cohomology theory; the transitionto an oriented cohomology theory on Sm/. is given as in remark 1.2(1). Theproof of theorem 3.6 uses resolution of singularities [4] and the weak factorizationtheorem [1] in an essential way.Remark 3.7. In addition to the properties of 0» listed in theorem 3.6, 0»(X) isgenerated by the classes of "elementary" cobordism cycles (/ : Y —t X).4. Degree formulasIn the paper [12], Rost made a number of conjectures based on the theoryof algebraic cobordism in the Morel-Voevodsky stable homotopy category. Manyof Rost's conjectures have been proved by homotopy-theoretic means (see [3]); ourconstruction of algebraic cobordism gives an alternate proof of these results, andsettles many of the remaining open questions as well. We give a sampling of someof these results.4.1. The generalized degree formulaAll the degree formulas follow from the "generalized degree formula". We firstdefine the degree map Q*(X) —t Q*(k).