Monthly Archives: December 2024

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 … Continue reading →