Algebraic & Geometric Topology

Bordism groups of solutions to differential relations

Rustam Sadykov

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

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 4 (2009), 2311-2347.

Dates
Received: 25 December 2006
Revised: 18 May 2009
Accepted: 19 May 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513797085

Digital Object Identifier
doi:10.2140/agt.2009.9.2311

Mathematical Reviews number (MathSciNet)
MR2558312

Zentralblatt MATH identifier
1179.57044

Subjects
Primary: 55N20: Generalized (extraordinary) homology and cohomology theories 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 57R45: Singularities of differentiable mappings

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

Citation

Sadykov, Rustam. Bordism groups of solutions to differential relations. Algebr. Geom. Topol. 9 (2009), no. 4, 2311--2347. doi:10.2140/agt.2009.9.2311. https://projecteuclid.org/euclid.agt/1513797085


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