Abstract:A smooth manifold M" imbedded in (n + 2r)-dimensional space En+2r is called weakly complex if a specific reduction of the normal bundle to the unitary group Ur is given.A 'complex cobordism theory' fo...A smooth manifold M" imbedded in (n + 2r)-dimensional space En+2r is called weakly complex if a specific reduction of the normal bundle to the unitary group Ur is given.A 'complex cobordism theory' for such manifolds may be defined and Milnor [4] has shown that two weakly complex closed manifolds belong to the same cobordism class if and only if they have the same Chern numbers.The following theorem says roughly we may kill a nonzero multiple of a characteristic class of a closed weakly complex manifold that does not show up in the Chern numbers.Theorem A. Let uM be a homogeneous polynomial in the Chern classes of a weakly complex manifold M" such that any n-dimensional homogenous polynomial wM with uM as a factor (i.e., wM = uM-vM) has value zero on the basic class of M. Then M is complex cobordant to a manifold M' with uM-of finite order.Further, if the normal bundle of M is trivial over its (2q -l)-skeleton, the same may be assumed for M'.Let A be an n-dimensional complex cobordism class, and let K(X) = Uaíex Ker/^,where/M: M->Bt/ isa classifying map for the(stable) normal bundle, and /]£ : H*(BU; Q) -* H*(M; Q) is the induced homomorphism in the rational coefficients.(For M empty, Ker/^ = H*(BU; Q).) Then Theorem A implies X(A)is an ideal.Let C(A) = H*(BU; Q)/K(X); it is a graded algebra called the rational characteristic ring of A.Theorem B (Poincaré duality for complex cobordism classes).The map Ci(X)®C~i(X)->C(X) induced by cup product gives a pairing of C(X)and C~'(X) to Q; the isomorphism C(X) at Q(X # 0) being given by associating to each monomial in H"(BU; Z) the corresponding Chern number, extending linearly, and then passing to the quotient.(If A = 0, then all the groups C'(X) are zero.)Theorem B is an immediate consequence of Theorem A.Read More