Morse theory in 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.

arXiv admin note: text overlap with arXiv:2301.04264