Non-linear proper Fredholm maps and the stable homotopy groups of spheres

TL;DR

This paper classifies non-linear proper Fredholm maps between Hilbert spaces using stable homotopy groups of spheres, revealing their structure and relationships, and discusses the oriented case with a bijection to these groups.

Contribution

It establishes a classification of non-linear proper Fredholm maps via stable homotopy groups, identifying the kernel and the bijection in the oriented case.

Findings

Surjective map from stable homotopy groups to Fredholm maps

Kernel of the map is explicitly determined

Oriented maps correspond bijectively to stable homotopy groups

Abstract

We classify non-linear proper Fredholm maps between Hilbert spaces, up to proper homotopy, in terms of the stable homotopy groups of spheres. We show that there is a surjective map from the stable homotopy groups of spheres to the set of non-linear proper Fredholm maps up to proper homotopy and determine the non-trivial kernel. We also discuss the case of oriented non-linear proper Fredholm maps which are in bijection with the stable homotopy groups of spheres.