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.
Throughout this post, $\mathfrak{R}$ is an o-minimal structure.
Definition 1: We define by recursion on $n$ what it means for $\mathcal{C}$ to be a cylindrical decomposition of a definable set $X \subseteq R^n$. For $n = 0$, either $\mathcal{C} = \varnothing = X$ or $\mathcal{C} = \{ R^0 \}$ and $X = R^0$. For $n+1$, $\mathcal{C}$ is a partion of $X$ into cells and $\pi \mathcal{C} := \{ \pi C : C \in \mathcal{C} \}$ is a cylindrical decomposition of $\pi X$ where $\pi:R^{n+1} \to R^n$ is the projection to the first $n$ coordinates.
Remark 2: From our proof of the cell decomposition theorem, every definable set admits a cylindrical decomposition. In fact, given finitely many definable sets $X_1, \ldots, X_m \subseteq R^n$ we can find a cylindrical decomposition $\mathcal{C}$ of $R^n$ so that for each $i$ the set $\mathcal{C} \upharpoonright X_i := \{ C \in \mathcal{C} : C \subseteq X \}$ is a cylindrical decomposition of $X_i$.
Definition 2: For a definable set $X \subseteq R^n$ and a cylindrical decomposition $\mathcal{C}$ of $X$, we define $\chi_{\mathcal{C}}(X) := \sum_{C \in \mathcal{C}} (-1)^{\dim C}$.
Our aim is to define $\chi(X) := \chi_{\mathcal{C}}(X)$ for any cylindrical decomposition $\mathcal{C}$ of $X$. For this to give a well-defined value we need to show that it is independent of the choice of decomposition. We do so in stages. We start by showing the independence of this value when $X$ is itself a cell.
Proposition 3: For any cell $X \subseteq R^n$ and a cylindrical dcomposition $\mathcal{C}$ of $X$, $\chi_{\mathcal{C}}(X) = \chi_{\{ X \}} (X)$, which is by definition $(-1)^{\dim X}$.
Proof: We work by induction on $n$.
When $n = 0$, we have $X = \{ \ast \} = R^0$ and there is only one decomposition of $X$, namely $\mathcal{C} = \{ X \}$. Hence, the claimed equality holds as the two expressions are identical.
In the inductive case of $n+1$ we have two cases to consider.
Suppose that $X = \Gamma_Y(f)$ is the graph of the continuous definable function $f:Y \to R$ on the cell $Y \subseteq R^n$. The map $\pi \upharpoonright X: X \to Y$ is a definable homeomorphism taking cells in $X$ to cells $Y$. In particular, the association $C \mapsto \pi C$ from $\mathcal{C}$ to $\pi \mathcal{C}$ is a bijection and it preserves dimensions. Thus, $$\begin{align} \chi_{\mathcal{C}} (X) &= \sum_{C \in \mathcal{C}} (-1)^{\dim C} \\ &= \sum_{C \in \mathcal{C}} (-1)^{\dim \pi C} \\ &= \sum_{C’ \in \pi \mathcal{C}} (-1)^{\dim C’} \\ &= \chi_{\pi \mathcal{C}} (Y) \\ &= (-1)^{\dim Y} \\ &= (-1)^{\dim X} \end{align}$$
Consider now that case that $X = (f,g)_Y$ where $Y \subseteq R^n$ is a cell and $f$ and $g$ are continuous definable functions on $Y$ (or identically $\pm \infty$) with $f < g$. For each $C’ \in \pi \mathcal{C}$, the preimage $(\pi \upharpoonright Y)^{-1} C’$ is equal to $(f \upharpoonright C’, g \upharpoonright C’)_{C’}$ and is given with the decomposition $\mathcal{C}_{C’} := \{ C \in \mathcal{C} : \pi C = C’ \}$. As these cells partition $X \cap (C’ \times R) = (f \upharpoonright C’, g \upharpoonright C’)_{C’}$ we must have definable continuous functions $f \upharpoonright C’ = \alpha_1 < \alpha_2 < \ldots < \alpha_{m_{C’}} = g \upharpoonright C’$ so that
$$\mathcal{C}_{C’} = \{ \Gamma_{C’}(\alpha_i) : 1 < i < m_{C’} \} \cup \{(\alpha_i,\alpha_{i+1})_{C’} : 1 \leq i < m_{C’} \}$$
We compute $$\begin{align} \chi_{\mathcal{C}}(X) &= \sum_{C \in \mathcal{C}} (-1)^{\dim C} \\ &= \sum_{C’ \in \pi \mathcal{C}} \sum_{C \in \mathcal{C}_{C’}} (-1)^{\dim C} \\ &= \sum_{C’ \in \pi \mathcal{C}} \left( \sum_{i=2}^{m_{C’}} (-1)^{\dim \Gamma_{C’}(\alpha_i)} + \sum_{i=1}^{m_{C’} – 1} (-1)^{\dim (\alpha_i,\alpha_{i+1})_{C’} } \right) \\ &= \sum_{C’ \in \pi \mathcal{C}} \left( \sum_{i=2}^{m_{C’}} (-1)^{\dim C’} + \sum_{i=1}^{m_{C’} – 1} (-1)^{\dim C’ + 1 } \right) \\ &= \sum_{C’ \in \pi \mathcal{C}} (-1)^{\dim C’} \left( (m_{C’} – 2) – (m_{C’} – 1) \right) \\ &= \sum_{C’ \in \pi \mathcal{C}} (-1)^{\dim C’} (-1) \\ &= \chi_{\pi \mathcal{C}} (\pi X) (-1) \\ &= (-1)^{\dim \pi X} (-1) \\ &= (-1)^{\dim \pi X + 1} \\ &= (-1)^{\dim X} \end{align} $$ as required. $\Box$
Proposition 5: If $\mathcal{C}$ and $\mathcal{D}$ are cylindrical decompositions of $X \subseteq R^n$ and $\mathcal{C} \subseteq \mathcal{D}$, then $\chi_{\mathcal{C}})(X) = \chi_\mathcal{D}(X)$.
Proof: For any $C \in \mathcal{C}$, let us set $\mathcal{D}_C := \{ D \in \mathcal{D} : D \subseteq C \}$. This set $\mathcal{D}_C$ is a cylindrical decomposition of $C$.
Let us compute $\chi_\mathcal{D}(X)$.
$$\begin{align} \chi_{\mathcal{D}}(X) &= \sum_{D \in \mathcal{D}} (-1)^{\dim D} \\ &= \sum_{C \in \mathcal{C}} \sum_{D \in \mathcal{D}_C} (-1)^{\dim D} \\ &= \sum_{C \in \mathcal{C}} \chi_{\mathcal{D}}(C) \\ &= \sum_{C \in \mathcal{C}} (-1)^{\dim C} \\ &= \chi_\mathcal{C}(X) \end{align} $$ as required. $\Box$
Lemma 6: If $\mathcal{C}_1$ and $\mathcal{C}_2$ are cylindrical decompositions of definable sets $X_1 \subseteq R^n$ and $X_2 \subseteq R^n$, respectively, then there is a cylindrical decomposition $\mathcal{D}$ of $R^n$ with $\mathcal{C}_1 \cup \mathcal{C}_2 \subseteq \mathcal{D}$, so that in particular the restriction $\mathcal{D} \upharpoonright X_i := \{ D \in \mathcal{D} : D \subseteq X_i \}$ for $i = 1$ or $2$ is a cylindrical decomposition of $X_i$ refining $\mathcal{C}_i$.
Proof: Any cell decomposition of $R^n$ subjacent to $\mathcal{C}_1 \cup \mathcal{C}_2$ constructed from our proof of cell decomposition suffices. $\Box$
Proposition 7: For any definable set $X$ and pair of cylindrical decompositions $\mathcal{C}$ and $\mathcal{D}$ of $X$ we have $\chi_{\mathcal{C}}(X) = \chi_{\mathcal{D}}(X)$.
Proof: By Lemma 6 we can find a cylindrical decomposition $\mathcal{E}$ of $X$ refining both $\mathcal{C}$ and $\mathcal{D}$. Applying Proposition 5 twice we have $\chi_{\mathcal{C}}(X) = \chi_{\mathcal{E}}(X) = \chi_{\mathcal{D}}(X)$, as required. $\Box$
Definition 8: For a definable set $X$ we define $\chi(X) := \chi_\mathcal{C}(X)$ for any choice of a cylindrical decomposition $\mathcal{C}$ of $X$.
By Proposition 7, the value of $\chi(X)$ does not depend on the choice of the decomposition $\mathcal{C}$.
Proposition 9: The function $\chi:\text{Def}(\mathfrak{R}) \to \mathbb{N}$ satisfies the following.
- $\chi(\varnothing) = 0$
- $\chi(\{ \ast \}) = 1$
- $\chi(X \times Y) = \chi(X) \cdot \chi(Y)$ for any pair of definable sets $X$ and $Y$, and
- $\chi(X \dot{\cup} Y) = \chi(X) + \chi(Y)$ for any pair of disjoint definable subsets $X$ and $Y$ of $R^n$.
Proof: For the first point, any cylindrical decomposition $\mathcal{C}$ of the empty set is itself empty so that $$ \begin{align} \chi(\varnothing) &= \chi_{\mathcal{C}}(\varnothing) \\ &= \chi_{\varnothing}(\varnothing) \\ &= \sum_{C \in \varnothing} (-1)^{\dim C} \\ &= 0 \end{align} $$
For the second point, the only cylindrical decomposition $\mathcal{C}$ of the singleton $\{ \ast \}$ is $\mathcal{C} = \{ \{ \ast \} \}$ so that $$ \begin{align} \chi(\{ \ast \}) &= \chi_{\mathcal{C}}(\{ \ast \}) \\ &= \chi_{\{ \{ \ast \} \} }(\varnothing) \\ &= \sum_{C \in \{ \{ \ast \} \} } (-1)^{\dim C} \\ &= (-1)^{\dim \{ \ast \}} \\ &= (-1)^0 \\ &= 1 \end{align} $$
For the third point, let $\mathcal{C}$ be a cylindrical decomposition of $X$ and $\mathcal{D}$ be a cylindrical decomposition of $Y$. Then $\mathcal{C} \times \mathcal{D} := \{ C \times D : C \in \mathcal{C} \text{ and } D \in \mathcal{D} \}$ is a cylindrical decomposition of $X \times Y$. We compute $$\begin{align} \chi(X \times Y) &= \chi_{\mathcal{C} \times \mathcal{D}} (X \times Y) \\ &= \sum_{E \in \mathcal{C} \times \mathcal{D}} (-1)^{\dim E} \\ &= \sum_{C \in \mathcal{C}} \sum_{D \in \mathcal{D}} (-1)^{\dim (C \times D)} \\ &= \sum_{C \in \mathcal{C}} \sum_{D \in \mathcal{D}} (-1)^{\dim C + \dim D} \\ &= \sum_{C \in \mathcal{C}} \sum_{D \in \mathcal{D}} (-1)^{\dim C} \cdot (-1)^{\dim D} \\ &= \left( \sum_{C \in \mathcal{C}} (-1)^{\dim C} \right) \left( \sum_{D \in \mathcal{D}} (-1)^{\dim D} \right) \\ &= \chi_{\mathcal{C}}(X) \cdot \chi_{\mathcal{D}}(Y) \\ &= \chi(X) \cdot \chi(Y) \end{align}$$
Finally, consider the additivity condition. By Lemma 6, we may take $\mathcal{C}$ to be a cylindrical decomposition of $X \cup Y$ for which the restrictions $\mathcal{C}_X$ and $\mathcal{C}_Y$ are cylindrical decompositions of $X$ and $Y$, respectively. Thus, $$\begin{align} \chi(X \cup Y) &= \chi_{\mathcal{C}}(X \cup Y) \\ &= \sum_{C \in \mathcal{C}} (-1)^{\dim C} \\ &= \sum_{C \in \mathcal{C} : C \subseteq X } (-1)^{\dim C} + \sum_{C \in \mathcal{C} : C \subseteq Y } (-1)^{\dim C} \\ &= \sum_{C \in \mathcal{C}_X } (-1)^{\dim C} + \sum_{C \in \mathcal{C}_Y } (-1)^{\dim C} \\ &= \chi_{\mathcal{C}_X}(X) + \chi_{\mathcal{C}_Y}(Y) \\ &= \chi(X) + \chi(Y) \end{align} $$ $\Box$
It follows from Proposition that we may weaken the requirement that $\mathcal{C}$ be a cylindrical decompostion in the definition of $\chi(X)$.
Corollary 10: If $X$ is a definable set and $\mathcal{C}$ is a finite partition of $X$ into cells, then $\chi(X) = \sum_{C \in \mathcal{C}} (-1)^{\dim C}$.
Proof: Using the additivity in Proposition 9 and the calculation of $\chi(C)$ for $C$ a cell of Proposition 3, we have $$\begin{align} \chi(X) &= \chi( \dot{\bigcup}_{C \in \mathcal{C}} C) \\ &= \sum_{C \in \mathcal{C}} \chi(C) \\ &= \sum_{C \in \mathcal{C}} (-1)^{\dim C} \end{align} $$ $\Box$
Proposition 9 does not include the condition that if $X$ and $Y$ are definable sets which are in definable bijection, then $\chi(X) = \chi(Y)$, which we need to establish in order to show that $\chi$ is an Euler characteristic.
Our definition of cells and of cylindrical decompositions depends on a choice of the ordering of the coordinates. In order to show that $\chi$ is invariant under definable bijections, we will show that cells may be further decomposed into cells which look like cells when the coordinates are permuted.
Definition 11: For $n \in \mathbb{N}$ and $\sigma \in \text{Sym}(n)$ a permutation of $\{ 1, \ldots, n \}$, we write $\sigma:R^n \to R^n$ for the map $(x_1, \ldots, x_n) \mapsto (x_{\sigma(1)}, \ldots, x_{\sigma(n)})$.
Lemma 12: If $n \in \mathbb{N}$, $1 \leq i < n$, $\sigma$ is the transposition $(i~i+1)$, and $C \subseteq R^n$ is a cell, then $C$ admits a finite partition $\mathcal{C}$ into cells so that for every $C \in \mathcal{C}$, $\sigma C$ is also a cell.
Before we prove Lemma 12, let us recall the definition of the kind of a cell.
Definition 13: We defined the kind of a cell $C \subseteq R^n$ by recursion on $n$. For $n = 0$, $\text{kind}(C) = \varnothing$. For $n+1$, if $C = \Gamma_{C’}(f)$ is the graph of a continuous definable function on the cell $C’ \subseteq R^n$, then $\text{kind}(C) := \text{kind}(C’) \smallfrown 0$. If $C = (f,g)_{C’}$ is the interval between two continuous definable functions on the cell $C’ \subseteq R^n$, then $\text{kind}(C) := \text{kind}(C’) \smallfrown 1$.
Proof of Lemma 12: We argue by induction on $n$. If $n \leq 1$, then the lemma is trivial. Consider now the case of $n+1$ (with $n > 0$).
Let us consider first the case where $i < n$. We have that $C = (f,g)_{C’}$ for some cell $C’ \subseteq R^n$ and continuous definable functions $f$ and $g$ on $C$ with $f < g$ (or $f \equiv -\infty$ or $g \equiv +\infty$) of $C = \Gamma_{C’}(f)$ for some cell $C’ \subseteq R^n$ and deifnable continuous function $f:C’ \to R$. The arguments in both cases will be similar. Since $i < n$, we may overload notation and regard $\sigma$ as both a map $R^n \to R^n$ and as a map $R^{n+1} \to R^{n+1}$. By induction, there is a finite partition $\mathcal{C}’$ of $C’$ into cells so that $\sigma D$ is a cell for each $D \in \mathcal{C}$. Let $\mathcal{C} := \{ (f \upharpoonright D, g \upharpoonright D)_D : D \in \mathcal{C}’ \}$ or $\mathcal{C} := \{ \Gamma_D(f \upharpoonright D) : D \in \mathcal{C}’ \}$ depending on kind of $C$. The set $\mathcal{C}$ is a finite partition of $C$ into cells. Unwinding the definitions, we see that $\sigma \Gamma_{D}(f \upharpoonright D) = \Gamma_{\sigma(D)} ((f \circ \sigma) \upharpoonright \sigma D)$, which is a cell, and $\sigma (f \upharpoonright D, g \upharpoonright D)_D = ((f \circ \sigma) \upharpoonright \sigma D, (g \circ \sigma) \upharpoonright \sigma D)_{\sigma D}$, which is also a cell. In each of these computations we make use of the fact that $\sigma^2 = \text{id}$.
We now consider the case that $i = n$. This also breaks into subcases based on whether $\text{kind}(C) = \ast \smallfrown 00$, $\ast \smallfrown 01$, $\ast \smallfrown 10$, or $\ast \smallfrown 11$.
If $\text{kind}(C) = \ast \smallfrown 00$, then $C = \{ (x,y,z) \in C’ \times R^2 : y = f(x) \text{ and } z = g(x,y) \}$ for some cell $C’ \subseteq R^{n-1}$ and continuous definable functions $f:C’ \to R$ and $g:\Gamma_{C’}(f) \to R$. Since $y = f(x)$ on $\Gamma_{C’}(f)$, $g \upharpoonright \Gamma_{C’}(f) = g(x,f(x)) =: G(x)$, a continuous definable function on $C’$. Thus, $$\begin{align} \sigma C &= \{ (x,y,z) \in C’ \times R^2 : y = G(x) \text{ and } z = f(x) \} \\ &= \Gamma_{\Gamma_{C’}(G)}(f) \end{align} $$ is also a cell.
If $\text{kind}(C) = \ast \smallfrown 01$, then there is a cell $C’ \subseteq R^{n-1}$, a definable continuous function $f:C’ \to R$, and definable continuous functions $g:\Gamma_{C’}(f) \to R$ and $h:\Gamma_{C’} \to R$ with $-\infty \leq g < h \leq \infty$ so that $C = (g,h)_{\Gamma_{C’}(f)} = \{ (x,y,z) \in C’ \times R^2 : y = f(x) \text{ and } g(x,y) < z < h(x,y) \}$. As before, if $G := g(x,f(x))$ and $H := h(x,f(x))$, then we also have $C = \{ (x,y,z) \in C’ \times R^2 : y = f(x) \text{ and } G(x) < z < H(x) \}$ so that $\sigma C = \Gamma_{(G,H)_{C’}}(f)$ is also a cell.
If $\text{kind}(C) = \ast \smallfrown 10$, then there is a cell $C’ \subseteq R^{n-1}$, definable continuous functions $f:C’ \to R$ and $g:C’ \to R$ with $f < g$, and a definable continuous function $h:(f,g)_{C’} \to R$ so that $C = \Gamma_{(f,g)_{C’}}(h)$. By cell composition and the monotonicity theorem, we may cell decompose $C’$ into cells so that for each such $D$ for all $x \in D$ either $h_x:(f(x),g(x)) \to R$ given by $y \mapsto h(x,y)$ is constants, strictly increasing, or strictly decreasing. Let us write $\widetilde{D}$ for $D \times R^2 \cap C$. If $h_x$ is constantly equal to $c(x)$ on $D$, then $\widetilde{D} = \{(x,y,z) \in D \times R^2 : f(x) < y < g(x) \text{ and } z = c(x) \}$ and $\sigma \widetilde{D} = \{ (x,y,z) \in D \times R^2 : y = c(x) \text{ and } f(x) < z < g(x) \} = (f,g)_{\Gamma_D(\ast)}$ is a cell. If $h_x$ is increasing for all $x \in D$, then define $\alpha(x) := \lim_{y \to f(x)} h_x(y)$ and $\beta(x) := \lim_{y \to g(x)} h_x(y)$ and $\gamma(x,y) = h^{-1}_x(y)$. We have that $\sigma \widetilde{D} = \Gamma_{(\alpha,\beta)_D}(\gamma)$ is a cell. In the case that $f_x$ is decreasing for all $x \in D$, we have $\sigma \widetilde{D} = \Gamma_{(\beta,\alpha)_D}(\gamma)$.
Finally, if $\text{kind}(C) = \ast \smallfrown 11$, then modify the construction from the last case. There is a cell $C’ \subseteq R^{n-1}$, definable continuous functions $f:C’ \to R$ and $g:C’ \to R$ with $-\infty \leq f < g \leq \infty$, and definable continuous functions $h:(f,g)_{C’}\to R$ and $k:(f,g)_{C’} \to R$ with $-\infty \leq h < k \leq \infty$ so that $C = (h,k)_{(f,g)_{C’}}$. As before, we cell decompose $C’$ so that on each cell $D$ in the decomposition each of $h_x$ and $k_x$ is constant, strictly increasing, or strictly decreasing (so that there are nine such cases). As above, $\widetilde{D} := D \times R^2 \cap C$ is a cell and $C$ is decomposed by these $\widetilde{D}$. Let us treat the case where both $h_x$ and $k_x$ are increasing on $D$. The other cases are similar. Let $\alpha(x) := \lim_{y \to f(x)^+} h_x(y)$, $\beta(x) := \lim_{y \to f(x)^+} k_x(y)$, $\gamma(x) := \lim_{y \to g(x)^-} h_x(y)$, and $\delta(x) := \lim_{y \to g(x)^-} k_x(y)$. Then $h_x:(f(x),g(x)) \to (\alpha(x),\gamma(x))$ and $k_x:(f(x),g(x)) \to (\beta(x),\delta(x))$ are order preserving bijections. Define functions $$\lambda(x,z) := \begin{cases} f(x) & \text{ if } \alpha(x) < z \leq \beta(x) \\ k^{-1}_x(z) & \text{ if } \beta(x) < z < \delta(x) \end{cases}$$ and $$\mu(x,z) := \begin{cases} g(x) & \text{ if } \gamma(x) \leq z < \delta(x) \\ h^{-1}_x(z) & \text{ if } \alpha(x) < z < \gamma(x) \end{cases}$$ Then $\sigma \widetilde{D} = (\lambda,\mu)_{(\alpha,\delta)_D}$ is a cell. The other cases are similar. $\Box$