Poincaré duality and cobordism

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' 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.