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

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