On the Euler characteristic of the links of a set determined by smooth definable functions
Volume 93 / 2008
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 ${\mathbb R}$ 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.
