Topological Invariance of the Combinatorial Euler Characteristic of O-minimal Sets

We prove the topological invariance of the combinatorial Euler characteristic of ominimal sets with the help of a canonical, topologically defined stratification of o-minimal sets by locally compact ones. Introduction. Let Bn be the collection of semi-algebraic subsets of R, i.e. sets definable by a finite boolean combination of polynomial equalities and inequalities. The Bn, n ∈ N, satisfy (i) any element of B1 is a finite union of open intervals (possibly infinite) and singletons; (ii) if X ∈ Bn and R pr −→ R is any projection, then pr(X) ∈ Bm. (i) is immediate while (ii) is the Tarski-Seidenberg theorem. An o-minimal (short for orderminimal ) structure over R is a non-trivial family of boolean algebras of subsets of R satisfying (i) and (ii). (See van den Dries [vdD98] for the meaning of non-trivial and a minimal set of axioms.) Starting from the 90’s, remarkable examples of o-minimal structures have been discovered, both over the reals and other linearly ordered groups. Over R, one can intuitively think of these as the result of permitting special families of real-analytic functions besides polynomials to serve in the equations and inequalities defining subsets. Given an o-minimal structure S, one has the associated notion of o-minimal function (a function whose graph belongs to S); the product and coproduct of o-minimal sets are o-minimal. Let K(S) be the Grothendieck ring of the category of S-minimal sets and functions. There is a homomorphism (♣) eu : K(S) −→ Z that we will call the combinatorial Euler characteristic. Whenever S contains all semi-algebraic sets, eu is in fact an isomorphism. When S is the collection of semi-linear sets,K(S) is isomorphic to Z⊕ Z and eu is the homomorphism 〈m,n〉 7→ m+ n. The map (♣) has a long history. It starts, of course, with the proof of triangulability of algebraic and analytic varieties by Hironaka and Łojasiewicz. Over real closed fields, triangulations of semi-algebraic sets by open affine cells were constructed by Knebusch and Delfs [DK82]. In the context of semi-linear sets, eu and the determination of the Grothendieck semiring are due to Schanuel [Sch91]. Independently, van den Dries [vdD98] found a remarkable construction of eu that works for all o-minimal structures and avoids the use of triangulations in favor of the more order-theoretic cylindrical cell decompositions. For semi-algebraic sets, these were introduced by Collins; see Basu-Pollack-Roy [BPR06] Ch. 5. For reference, let us recall their definition following