Abstract:
In this paper obstructions to the commutativity and associativity of multiplications in cobordism theories with singularities are determined. Obstructions to the existence and commutativity of multiplications are expressed in terms of Steenrod-tom Dieck operations in cobordism.
General theorems are applied to the cobordism theories $SO^*$, $U^*$, $SU^*$, $Sp^*$ and $Sc^*$ with singularities. Associativity of multiplication is proved in those cases where it exists, as well as the existence of a commutative and associative multiplication if the operation of division by 2 can be carried out in the ring of scalars of a theory with singularities.
As an application of the main theorems, a uniqueness theorem for "generalized $K$-theories" is proved.
Bibliography: 15 titles.
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