Multiplications in the Complex Bordism Theory with Singularities

In 1967, D. Sullivan [13] introduced a bordism theory based on manifolds with singularities and successively N. Baas [3], [4] studied and reformulated the theory so that it has been given a more accessible ground. This theory is considered as a natural generalization of usual bordism theory, and for each given singularity class & = {Ply Pz, } , a sequence of closed manifolds, one obtains such a theory. Thus there have appeared various interesting generalized homology theories. For example, as Baas [4] shows, there exists a tower of homology theories and natural transformations connecting complex bordism to ordinary singular homology. One of main problems with these theories has been to show whether they are multiplicative or not (Baas [6]). And the purpose of the present paper is to study this problem. For convenience sake, we will restrict ourselves to the case of complex bordism theory MU(J^} * ( ) with singularity class & . In this theory we introduce natural (external) multiplications (§ 3 and § 6)