Cobordisms of maps with singularities of given class
Let \(P\) be a smooth manifold, and consider maps from \(n\) dimensional smooth manifolds to \(P\) with singularities of a given class. These maps, under the equivalence of cobordism (the cobordism map can only have singularities of a given open class) form a monoid. The main topic of the present paper is the study of this monoid, which depends on \(P\), \(n\), and the allowed singularity types. The main result of the paper describes this cobordism monoid in terms of the image of a certain map between stable homotopy groups. Thus a \(C^{\infty}\) object is identified with a \(C^0\) object. The theorem holds under the assumption of the existence of the homotopy-principle associated with the allowed singularity class. The author studies the occurring map between stable homotopy groups, and proves that the map is surjective provided the allowed singularity class is invariant under the group of contact equivalences of singularities. An explicit homotopy theoretic construction of a classifying space \(B\) is also given, satisfying the following property: the cobordism classes of maps with given singularities into \(P\) are identified with the homotopy classes of continuous maps from \(P\) to the classifying space \(B\). This theorem holds under mild assumptions on the dimension setting, as well as the singularity class. Oriented and non-oriented cases are both studied.
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