Bordism groups of solutions to differential relations

Sadykov, Rustam¹

¹Department of Mathematics, University of Toronto, Toronto ON, Canada

Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2311-2347

Résumé

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 J of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space Ω∞BJ 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,Ω∞BJ]. The spaces Ω∞BJ 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.

Détail

DOI : 10.2140/agt.2009.9.2311

Keywords: differential relation, h-principle, generalized cohomology theory, singularity of a smooth map, jet, fold map, Morin map, Thom–Boardman singularity

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