Morse functions definable in d-minimal structures
Masato Fujita
TL;DR
The paper addresses extending Morse theory to definable $\mathcal{C}^p$ functions in a d-minimal setting by proving that definable $\mathcal{C}^p$ Morse functions on a definable $\mathcal{C}^p$ submanifold $M$ form a dense subset of $\mathcal{D}^p(M)$ under the definable $\mathcal{C}^p$ topology. The approach develops a definable differential-geometric framework—including multi-valued graphs, a definable Sard-type theorem, and a zero-approximation lemma—and constructs Morse perturbations via a finite induction over a cover of $M$, ensuring nondegenerate critical points with distinct critical values. The main contribution is the density (and accompanying technical lemmas) of definable Morse functions in d-minimal structures, generalizing known o-minimal results; the paper also highlights limitations by presenting a counterexample where Morse-ness is not an open condition in certain d-minimal settings. This work enables Morse-theoretic methods on definable manifolds within broader logical structures, expanding the toolkit for studying definable geometry and topology.
Abstract
Fix a d-minimal expansion of an ordered field. We consider the space $\mathcal D^p(M)$ of definable $\mathcal C^p$ functions defined on a definable $\mathcal C^p$ submanifold $M$ equipped with definable $\mathcal C^p$ topology. The set of definable $\mathcal C^p$ Morse functions is dense in $\mathcal D^p(M)$.