Density of Morse functions on sets definable in o-minimal structures

Abstract

We present a tameness property of sets definable in o-minimal structures by showing that Morse functions on a definable closed set form a dense and open subset in the space of definable C functions endowed with the Whitney topology. Introduction. In Morse theory it is proved that the topological shape of a space can be described via data given by Morse functions defined on the space. For Morse theory on compact smooth manifolds we refer the readers to the book by Milnor [Mi], and for Morse theory on singular spaces to the book by Goresky and MacPherson [GM]. [GM] proves the density and openness of Morse functions on closed Whitney stratified subanalytic sets in the space of smooth functions endowed with the Whitney topology (see also [Mo], [Mi], [La], [Be], [P], [O] and [Br]). In this note we present similar results for definable sets in o-minimal structures. The definitions of o-minimal structures and Morse functions are given in Section 1, and the main theorems are stated and proved in Section 2. The proofs are based on Sard’s theorem and an approximation theorem in the o-minimal context. The proofs of Proposition 1 and Theorem 2 are essentially the same as the corresponding proofs in [GM], with “subanalytic” replaced by “definable”. Since the proofs are short, they are included for completeness. Note that the spiral {(x,y)∈R2 :x=e−φ 2 cosφ,y=e−φ 2 sinφ, φ≥0}∪{(0,0)} or the oscillation {(x,y)∈R2 :y=xsin(1/x),x>0}∪{(0,0)} has no Morse functions, even though the first one is a closed Whitney stratified set (see Remark 1 in 1.4). Therefore, in some sense, our results show a tameness property of definable sets. A part of this note was presented at the Conference on Real Algebraic Geometry and its Applications, at the ICTP, Trieste, 2003. The author 2000 Mathematics Subject Classification: 32B20, 14P10, 14B05.