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

Krzysztof Jan Nowak. "On the Euler characteristic of the links of a set determined by smooth definable functions." Annales Polonici Mathematici 93.3 (2008): 231-246. <http://eudml.org/doc/280917>.

@article{KrzysztofJanNowak2008,
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 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.},
author = {Krzysztof Jan Nowak},
journal = {Annales Polonici Mathematici},
keywords = {Euler characteristic; link of a set at a point; smooth definable functions; polynomially bounded o-minimal structures; sum of the signs of global smooth functions},
language = {eng},
number = {3},
pages = {231-246},
title = {On the Euler characteristic of the links of a set determined by smooth definable functions},
url = {http://eudml.org/doc/280917},
volume = {93},
year = {2008},
}

TY - JOUR
AU - Krzysztof Jan Nowak
TI - On the Euler characteristic of the links of a set determined by smooth definable functions
JO - Annales Polonici Mathematici
PY - 2008
VL - 93
IS - 3
SP - 231
EP - 246
AB - 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.
LA - eng
KW - Euler characteristic; link of a set at a point; smooth definable functions; polynomially bounded o-minimal structures; sum of the signs of global smooth functions
UR - http://eudml.org/doc/280917
ER -