Some applications of transversality for infinite dimensional manifolds

TL;DR

This paper develops transversality results for a special class of infinite-dimensional manifolds and applies them to establish fundamental theorems like degree theory, rank, invariance of domain, and Bursuk-Ulam type results.

Contribution

It introduces transversality results for $MC^k$-Fréchet manifolds and uses them to prove key theorems in nonlinear Fredholm mappings in infinite dimensions.

Findings

Established transversality results for $MC^k$-Fréchet manifolds

Constructed degree theory for nonlinear Fredholm maps

Proved rank, invariance of domain, and Bursuk-Ulam theorems in this setting

Abstract

We present some transversality results for a category of Fr\'{e}chet manifolds, the so-called $MC^k$-Fr\'{e}chet manifolds. In this context, we apply the obtained transversality results to construct the degree of nonlinear Fredholm mappings by virtue of which we prove a rank theorem, an invariance of domain theorem and a Bursuk-Ulam type theorem.