Obstructions to the existence of fold maps

TL;DR

This paper investigates the conditions under which smooth manifolds admit fold maps, identifying obstructions via characteristic classes and relating the problem to vector field existence on tangent bundles.

Contribution

It characterizes obstructions to fold map existence using characteristic classes and Postnikov invariants, linking singularity elimination to topological invariants.

Findings

Obstructions are described in terms of characteristic classes.

Primary and secondary obstructions are identified.

Connections to vector fields on tangent bundles are established.

Abstract

We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be interpreted as primary and secondary obstructions to the elimination of certain singularities. We also discuss the relationship between the existence problem of fold maps and that of vector fields of stabilized tangent bundles.