On the homotopy classification of proper Fredholm maps into a Hilbert space

TL;DR

This paper provides a new homotopy classification of proper Fredholm maps from infinite-dimensional Hilbert manifolds into Hilbert spaces, using framed cobordism, offering an alternative to previous methods.

Contribution

It introduces a novel classification approach that does not rely on additional structures like Fredholm structures, expanding understanding of Fredholm map homotopy classes.

Findings

Classification in terms of framed cobordism

Complete classification for index zero maps

Involves Caccioppoli-Smale mod 2 degree and oriented degree

Abstract

We classify the homotopy classes of proper Fredholm maps from an infinite dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli-Smale mod 2 degree and the absolute value of the oriented degree.