Transversality and Lipschitz-Fredholm maps

TL;DR

This paper extends transversality theory to Lipschitz-Fredholm maps on bounded Fréchet manifolds, demonstrating stability and submanifold structures, and generalizing Smale's theorem using Sard's theorem.

Contribution

It introduces a transversality framework for Lipschitz-Fredholm maps in Fréchet manifolds, including stability results and submanifold existence, extending classical theorems.

Findings

Lipschitz-Fredholm maps of fixed index are transversally stable.

Extended Smale transversality theorem using generalized Sard's theorem.

Established submanifold structure on preimages of transversal submanifolds.

Abstract

We study transversality for Lipschitz-Fredholm maps in the context of bounded Fr\'{e}chet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fr\'{e}chet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the generalized Sard's theorem for this category of manifolds. We also provide an answer to the well known problem concerning the existence of a submanifold structure on the preimage of a transversal submanifold.