On the cobordism groups of cooriented, codimension one Morin maps

TL;DR

This paper completely computes the cobordism groups of cooriented, codimension one Morin maps, revealing their structure in relation to stable homotopy groups and the Kahn-Priddy map, with extensions to cusp and more complex Morin maps.

Contribution

It provides a full computation of cobordism groups for cooriented fold maps and extends results to cusp and complex Morin maps, linking them to stable homotopy theory.

Findings

Odd torsion matches stable homotopy groups of spheres.

2-primary part is the kernel of the Kahn-Priddy map.

Results extend to cusp and complex Morin maps.

Abstract

Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy group of the infinite dimensional real projective space onto the 2-primary part of the stable homotopy group of spheres). Analogous results - modulo small primes - are obtained for cusp maps and more complicated Morin maps as well.