Real Algebraic and Analytic Geometry |

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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