Bordism groups of solutions to differential relations

Abstract

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 $\mathcal{J}$ of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space ${\Omega }^{\infty }{\mathbf{B}}_J$ such that for each smooth manifold $W$ the cobordism group of maps into $W$ with only $J$–singularities is isomorphic to the group of homotopy classes of maps $[W,{\Omega }^{\infty }{\mathbf{B}}_J]$. The spaces ${\Omega }^{\infty }{\mathbf{B}}_J$ are relatively simple, which makes explicit computations possible even in the case where the dimension of the source manifold is bigger than the dimension of the target manifold.