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2008 | 93 | 3 | 231-246
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On the Euler characteristic of the links of a set determined by smooth definable functions

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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 ℝⁿ determines two closed subsets
W := {u ∈ U: φ(u) ≤ 0}, Z := {u ∈ U: φ(u) = 0}.
We 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:
$χ(lk(u;W)) = ∑_{i=1}^{r} sgn σ_{i}(u)$, $1/2χ(lk(u;Z)) = ∑_{i=1}^{s} sgnζ_{i}(u)$.
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.
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  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
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