Fold maps and immersions from the viewpoint of cobordism

TL;DR

This paper characterizes cobordism classes of simple fold maps using geometric invariants and computes related cobordism groups, revealing their structure through stable homotopy groups and natural homomorphisms.

Contribution

It provides a complete description of cobordism classes of simple fold maps via geometric invariants and computes the associated cobordism groups.

Findings

Cobordism group of simple fold maps is isomorphic to a sum of stable homotopy groups.

Explicit computation of ranks of oriented bordism groups of simple fold maps.

Description of natural maps between cobordism groups of immersions and fold maps.

Abstract

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented (right-left) bordism groups of simple fold maps.