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

Krzysztof Jan Nowak¹

¹ Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Annales Polonici Mathematici, Tome 93 (2008) no. 3, pp. 231-246

Résumé

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 ${\mathbb R}$ of reals. A ($C^{\infty}$) smooth definable function $\varphi: U \rightarrow {\mathbb R}$ on an open set $U$ in ${\mathbb R}^{n}$ determines two closed subsets $$ W := \{ u \in U: \varphi(u) \leq 0 \}, \ \quad Z := \{ u \in U: \varphi(u) = 0 \}. $$ We shall investigate the links of the sets $W$ and $Z$ at the points $u \in U$, 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: $$ \chi (\mathop{\rm lk}\nolimits (u;W)) = \sum_{i=1}^{r} \mathop{\rm sgn} \sigma_{i}(u), \ \quad \frac{1}{2} \, \chi (\mathop{\rm lk}\nolimits (u;Z)) = \sum_{i=1}^{s}\mathop{\rm sgn} \zeta_{i}(u). $$ 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.

Détail

DOI : 10.4064/ap93-3-4

Keywords: purpose paper carry o minimal settings results about euler characteristic algebraic analytic sets consider polynomially bounded o minimal structure field mathbb reals infty smooth definable function varphi rightarrow mathbb set mathbb determines closed subsets varphi leq quad varphi shall investigate links sets points which defined definable homeomorphism proven euler characteristic those links being local topological invariant expressed finite sum signs global smooth definable functions chi mathop nolimits sum mathop sgn sigma quad frac chi mathop nolimits sum mathop sgn zeta present version functions depending smoothly parameter analytic these formulae has worked out nowel immediate consequence euler characteristic each link zero set even generalizes o minimal setting classical result sullivan about real algebraic sets