Constructible functions on 2-dimensional analytic manifolds (Q1884143)

The authors present a characterization of the sums of signs of global analytic functions on a real analytic manifold \(M\) of dimension 2. They prove that for dimension \(2\) the set of these sums of signs of global analytic functions is the intersection between two classes of constructible functions: the semianalytically constructible ones and the analytically constructible ones. Recall that a function \(\varphi:M\to{\mathbb Z}\) is semianalytically constructible if it is constant on each element of a finite partition of \(M\) into global semianalytic sets. On the other hand, an analytically constructible function is a linear combination, over the integers, of Euler's characteristics of fibers of proper analytic morphisms. Even if \(M\) is a compact \(2\)-dimensional manifold the previous classes are not contained one in the other, and as it was pointed above their intersection is the set of the sum of signs of global analytic functions. Similar results were known previously for the algebraic case [\textit{M. Coste} and \textit{K. Kurdyka}, Topology 37, 393--399 (1998; Zbl 0942.14031); \textit{A. Parusiński} and \textit{Z. Szafraniec}, in: Singularities symposium -- Lojasiewicz 70. Banach Center Publ. 44, 175--182 (1998; Zbl 0915.14032)] and also for the Nash compact case [\textit{I. Bonnard}, Manuscr. Math. 112, 55--75 (2003; Zbl 1025.14010)]. The authors also improve this last case avoiding the compactness asumption for dimension \(2\). In fact, they prove that unlike the algebraic case, obstructions at infinity are not relevant for analytic and Nash manifolds of dimension \(2\): {a function is a sum of signs (of functions of the suitable type) on a \(2\)-dimensional manifold \(M\) if and only if this is true on each compact subset of \(M\)}.