Volume 9, issue 4 (2009)

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Bordism groups of solutions to differential relations

Algebraic & Geometric Topology 9 (2009) 2311–2347
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 $\mathsc{J}$ 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 $\left[W,{\Omega }^{\infty }{\mathbf{B}}_{J}\right]$. 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.

Keywords
differential relation, h-principle, generalized cohomology theory, singularity of a smooth map, jet, fold map, Morin map, Thom–Boardman singularity
Mathematical Subject Classification 2000
Primary: 55N20, 53C23
Secondary: 57R45