Morse theory in definably complete d-minimal structures

TL;DR

This paper extends Morse theory to definably complete d-minimal structures, showing that Morse functions are generically dense among definable $C^r$ functions on certain manifolds.

Contribution

It proves that in definably complete d-minimal structures, the set of definable Morse functions is open and dense among definable $C^r$ functions on definably compact, normal manifolds.

Findings

Morse functions form an open and dense subset in the space of definable $C^r$ functions.

The result applies to definably compact, definably normal $C^r$ manifolds.

The proof uses properties of definably complete d-minimal structures.

Abstract

Consider a definable complete d-minimal expansion $(F, <, +, \cdot, 0, 1, \dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and $2 \le r <\infty$. We prove that the set of definable Morse functions is open and dense in the set of definable $C^r$ functions on $X$ with respect to the definable $C^2$ topology.