Every o-minimal structure admits an Euler characteristic with values in $\mathbb{Z}$. Our computation showing that $K_0(\mathfrak{R})$ is a quotient of $\mathbb{Z}$ when $\mathfrak{R}$ is an o-minimal expansion of an ordered field may be reversed to define the o-minimal Euler characteristic.
If $\mathfrak{R}$ is an o-minimal expansion of an ordered field, then $K_0(\mathfrak{R})$ is a quotient of $\mathbb{Z}$. In a later post, we will see that, in fact, $K_0(\mathfrak{R})$ is exactly $\mathbb{Z}$.
We have seen how to assign a natural number valued dimension to definable sets in o-minimal structures. There is a second integer valued invariant, which we will call the Euler characteristic. In o-minimal expansions of fields, these two invariants characterize definable sets up to definable bijection.
With this post we discuss the general theory of Euler characteristics of definable sets and will see that when $\mathfrak{R} = (R,<,+,\cdot,\ldots)$ is an o-minimal expansion of a field every Euler characteristic must take values in a quotient of $\mathbb{Z}$.
A vector space $V$ with the closure operation defined by taking a subset $A \subseteq V$ to the linear span of $A$ is a quintessential example of a pregeometry. To any vector space $V$ we may define the dimension of $V$ to be cardinality of a basis, a maximal linearly independent set. It follows from Zorn’s Lemma that bases exist and some simple manipulations permit one to see that all bases have the same cardinality so that $\dim V$ is a well-defined cardinal.
We may transpose the definition of a basis to any pregeometry and then the proofs of the existence of bases and that they all have the same cadinality lift from linear algebra to general pregeometries.
For the sake of building a dimension theory for definable sets in o-minimal structures, we will follow a more model theoretic approach, due to Pillay, to dimensions than what appears in van den Dries’ book. That is, we will show that o-minimal structures are geometric which means that algebraic closure endows the universe of such a structure with a pregeometry, or what is sometimes called a matroid. In the following post we will explain how to define dimensions in pregeometries.
In our proof of the Cell Decomposition Theorem, we took as known the Uniform Finiteness Theorem. In this post, we fill that gap.
Uniform Finiteness Theorem: Let $Y \subseteq R^{n+1}$ be a definable set for which all fibers $Y_a := \{ y \in R : (a,y) \in R \}$ are finite as $a$ ranges through $R^n$. Then there is a number $N = N(Y)$ so that all fibers have size at most $N$.
The Cell Decomposition Theorem might be called the “Fundamental Theorem of O-minimality”. With this theorem we show that from the hypothesis that the definable sets in one variable are particularly simply, i.e. those which admit a quantifier-free definition in the reduct to the language of ordered sets with parameters, it follows that the higher dimensionsal sets are also tame.
You can find the material from my talks in the research seminar here.
A basic principle in tame geometry is that there are no pathological definable functions in o-minimal structures. One precise sense in which this principle is true is given by the Monotonicity Theorem.
Monotonicity Theorem : Let $f: R \to R$ be a definable function in some o-minimal structure $\mathfrak{R}$. Then $f$ is piecewise continuous and is piecewise constant or strictly monotone. That is, we can find $-\infty = a_0 < a_1 < \ldots < a_m = \infty$ so that for $0 \leq i < m$ the function $f \upharpoonright (a_i,a_{i+1})$ is continuous and constant, strictly increasing, or strictly decreasing.
For the most part, we will be following Lou van den Dries’ book Tame Topology and O-minimal Structures for this seminar. We will supplement with material from reseach papers. If there are specific theorems about o-minimality you would like to see this term, let me know. Some possible topics include
