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Bordism groups of solutions to differential relations

In terms of category theory, the Gromov homotopy principle for a set valued functor $F$ asserts that the functor $F$ can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor $F$ holds if the functor $F$ can be induced from a (co)homology functor. We examine the bordism principle in the case of functors given by (co)bordism groups of maps with prescribed singularities. Our main result implies that if a family $R$ of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space $B(R)$ such that for each smooth manifold $N$ the cobordism group of maps into $N$ with only $R$-singularities is isomorphic to the group of homotopy classes of maps $[N, B(R)]$.