On the Euler characteristic of the links of a set determined by smooth definable functions

Abstract

Abstract. The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field R of reals. A (C ∞ ) smooth definable function ϕ : U → R on an open set U in R n determines two closed subsetsWe shall investigate the links of the sets W and Z at the points u ∈ U , which are well defined up to a definable homeomorphism. It is proven that the Euler characteristic of those links (being a local topological invariant) can be expressed as a finite sum of the signs of global smooth definable functions:We also present a version for functions depending smoothly on a parameter. The analytic case of these formulae has been worked out by Nowel. As an immediate consequence, the Euler characteristic of each link of the zero set Z is even. This generalizes to the o-minimal setting a classical result of Sullivan about real algebraic sets.

Fonctions régulues

“…Corollaire 4.12). Nous allons en donner une démonstration directe, en nous inspirant de l'idée donnée dans [27] pour établir la noethérianité de la topologie quasi-analytique. Soit F = Z(f ) avec f ∈ R 0 (R n ).…”

Defining the smooth points of a quotient in polynomially bounded o -minimal structures

“…The DCC is proved for polynomially bounded o-minimal structures over the real field with smooth cell decomposition in [2]. The most general proof we have found is sketched by Nowak in the appendix of [7]. This proof removes the need for smooth cell decomposition by constructing a finite covering of the zero set of a smooth function with smooth definable manifolds.…”

Decomposition into special cubes and its applications to quasi-subanalytic geometry

“…Let R be the o-minimal substructure generated by those global smooth functions from R; R-semianalytic sets are those from the boolean algebra generated by the sets of the form {x ∈ R N : f (x) = 0} with f (x) being a global smooth R-function on R N , and R-subanalytic sets are projections of R-semianalytic sets. We proved in [10] that the ring of global smooth definable functions is topologically noetherian. Nevertheless, the question whether the complement theorem holds for R-subanalytic sets or, equivalently, whether the structure R is model-complete, seems to be much more difficult and is yet unsolved.…”