Real Algebraic and Analytic Geometry
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193. Krzysztof Jan Nowak:
On the Euler characteristic of the links of a set determined by smooth definable functions.

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Submission: 2006, February 24.

Abstract:
Consider a polynomially bounded, o-minimal structure on the field $\matR$ of reals. A smooth (i.e.\ of class $C^{\infty}$) definable function $\varphi: U \longrightarrow \matR$ on an open set $U$ in $\matR^{n}$ determines two closed subsets $$W := \{ u \in U: \varphi(u) \leq 0 \} \ \ {\rm and} \ \ Z := \{ u \in U: \varphi(u) = 0 \}.$$ We shall investigate the links of the sets $W$ and $Z$ at the points $u \in U$ (being local topological invariants well-defined up to a definable homeomorphism). It is proven that the Euler characteristic of those links can be expressed as a finite sum of signs of global smooth definable functions on $U$: $$\chi ({\rm lk}\, (u;W)) = \sum_{i=1}^{r} {\rm sgn}\, \sigma_{i}(u) \ \ \mbox { and } \ \ \frac{1}{2} \, \chi ({\rm lk}\, (u;Z)) = \sum_{i=1}^{s}{\rm sgn}\, \zeta_{i}(u).$$ Also presented is a version for functions depending smoothly on a parameter. The analytic case of these formulae has been worked out by Nowel~\cite{24}.
As an immediate consequence, the Euler characteristic of the links of the zero set $Z$ is even. This generalizes to the o-minimal setting Sullivan's result~\cite{32} about real algebraic sets. Research concerning topological invariants of algebraic sets has been conducted by many mathematicians, as for instance \cite{1,2,4,8,9,10,19,20,24,25,26,32,33,34}. A natural challenge arises: to transfer some of the results to o-minimal geometry. In this paper we endeavour to make a small step towards its pursuit.

Mathematics Subject Classification (2000): 32S50, 03C64, 26E10, 14P15, 32B20.

Keywords and Phrases: Euler characteristic, link of a set at a point, smooth definable functions, polynomially bounded o-minimal structures, sum of signs of global smooth functions.

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