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Book Sections Panoramas et synthèses Year : 2007

Arc-symmetric sets and arc-analytic mappings.

Krzysztof Kurdyka
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Abstract

Arc-symmetric sets and arc-analytic functions were introduced by the first-named author. One of the motivation was a striking difference between real algebraic (or analytic) geometry and the geometry over the field of complex numbers or more generally over an algebraically closed field. In the complex case the topology is adquate to the algebra; for instance the irreducible sets are connected in the strong topology. Over reals the irreducible nosingular cubic $C=\{x^3-x=y^2\}$ has two connected components: $E_c$ which is compact and $E_n$ which is noncompact. Now let us take the cone $\tilde C=\{x^3-x z^2=y^2\}$ over the curve $C$, geometrically we place the curve $C$ in the plane $\{z=1\}$ in $\R^3$ and we draw lines trough the origin and the points of $C$. We have $\tilde C= \tilde E_c \cup \tilde E_n$, where $\tilde E_c$ and $\tilde E_n$ are the cones corresponding to the curves $E_c$ and $E_n$. Clearly $\tilde C$ is a connected irreducible algebraic subset of $\R^3$. Now blow up the origin in $\R^3$ and observe that the strict transform of $\tilde C$ is just $C\times \R= (E_c \times\R) \cup (E_n\times \R)$, so it has again two connected components and is irreducible as an algebraic set. So we see that the "components" $\tilde E_c$ and $\tilde E_n$ persist. In other words from geometrico-topological point of view the set $\tilde C$ is not irreducible but the components we want to exhibit are finer than the connected components of $\tilde C$. Actually our arc-symmetric semialgebraic sets will allow us to detect the "components" $\tilde E_c$ and $\tilde E_n$.

Domains

Geometric Topology [math.GT]

Dates and versions

hal-00368863 , version 1 (17-03-2009)

Identifiers

• HAL Id : hal-00368863 , version 1

Cite

Krzysztof Kurdyka, Adam Parusinski. Arc-symmetric sets and arc-analytic mappings.. Arc spaces and additive invariants in real algebraic and analytic geometry, 24, Société Mathématique de France, Paris,, pp.33--67, 2007, Panoramas et Synthèses, vol 24. ⟨hal-00368863⟩

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