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 () 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: , . 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-4,
author = {Krzysztof Jan Nowak},
title = {On the Euler characteristic of the links of a set determined by smooth definable functions},
journal = {Annales Polonici Mathematici},
volume = {93},
year = {2008},
pages = {231-246},
zbl = {1137.14043},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-4}
}
Krzysztof Jan Nowak. On the Euler characteristic of the links of a set determined by smooth definable functions. Annales Polonici Mathematici, Tome 93 (2008) pp. 231-246. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap93-3-4/