{"id":215,"date":"2024-12-09T22:26:37","date_gmt":"2024-12-09T22:26:37","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/model-theory\/?p=215"},"modified":"2024-12-09T22:30:37","modified_gmt":"2024-12-09T22:30:37","slug":"o-minimal-euler-characteristic","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/model-theory\/2024\/12\/09\/o-minimal-euler-characteristic\/","title":{"rendered":"O-minimal Euler characteristic"},"content":{"rendered":"

Every o-minimal structure admits an Euler characteristic with values in $\\mathbb{Z}$.\u00a0 Our computation showing that $K_0(\\mathfrak{R})$ is a quotient of $\\mathbb{Z}$\u00a0 when $\\mathfrak{R}$ is an o-minimal expansion of an ordered field may be reversed to define the o-minimal Euler characteristic.<\/p>\n

<\/p>\n

Throughout this post, $\\mathfrak{R}$ is an o-minimal structure.<\/p>\n

Definition 1:<\/strong>\u00a0 We define by recursion on $n$ what it means for $\\mathcal{C}$ to be a cylindrical decomposition of a definable set $X \\subseteq R^n$.\u00a0 For $n = 0$, either $\\mathcal{C} = \\varnothing = X$ or $\\mathcal{C} = \\{\u00a0 R^0 \\}$ and $X = R^0$.\u00a0 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.<\/p>\n

Remark 2:\u00a0<\/strong> From our proof of the cell decomposition theorem, every definable set admits a cylindrical decomposition.\u00a0 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 $X_i$.<\/p>\n

Definition 2:<\/strong>\u00a0 For a definable set $X \\subseteq R^n$ and a cylindrical decomposition $\\mathcal{C}$ of $X$, we define $\\chi_{\\mathcal{C}}(X) := \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim C}$.<\/p>\n

Our aim is to define $\\chi(X) := \\chi_{\\mathcal{C}}(X)$ for any cylindrical decomposition $\\mathcal{C}$ of $X$.\u00a0 For this to give a well-defined value we need to show that it is independent of the choice of decomposition.\u00a0 \u00a0We do so in stages.\u00a0 We start by showing the independence of this value when $X$ is itself a cell.<\/p>\n

Proposition 3:\u00a0<\/strong> 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\u00a0 $(-1)^{\\dim X}$.<\/p>\n

Proof:<\/strong>\u00a0 We work by induction on $n$.<\/p>\n

When $n = 0$, we have $X = \\{ \\ast \\} = R^0$ and there is only one decomposition of $X$, namely $\\mathcal{C} = \\{ X \\}$.\u00a0 Hence, the claimed equality holds as the two expressions are identical.<\/p>\n

In the inductive case of $n+1$ we have two cases to consider.<\/p>\n

Suppose that $X = \\Gamma_Y(f)$ is the graph of the continuous definable function $f:Y \\to R$ on the cell $Y \\subseteq R^n$.\u00a0 The map $\\pi \\upharpoonright X: X \\to Y$ is a definable homeomorphism taking cells in $X$ to cells $Y$.\u00a0 In particular, the association $C \\mapsto \\pi C$ from $\\mathcal{C}$ to $\\pi \\mathcal{C}$ is a bijection and it preserves dimensions.\u00a0 \u00a0Thus, $$\\begin{align} \\chi_{\\mathcal{C}} (X) &= \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim C} \\\\ &= \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim \\pi C} \\\\ &= \\sum_{C’ \\in \\pi \\mathcal{C}} (-1)^{\\dim C’} \\\\ &= \\chi_{\\pi \\mathcal{C}} (Y) \\\\ &= (-1)^{\\dim Y} \\\\ &= (-1)^{\\dim X} \\end{align}$$<\/p>\n

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 $f < g$.\u00a0 \u00a0For 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 $\\mathcal{C}_{C’} := \\{ C \\in \\mathcal{C} : \\pi C = C’ \\}$.\u00a0 \u00a0As these cells partition $X \\cap (C’ \\times R) = (f \\upharpoonright C’, g \\upharpoonright C’)_{C’}$ we must have definable continuous functions $f \\upharpoonright C’ = \\alpha_1 < \\alpha_2 < \\ldots <\u00a0 \\alpha_{m_{C’}} = g \\upharpoonright C’$ so that<\/p>\n

$$\\mathcal{C}_{C’} = \\{ \\Gamma_{C’}(\\alpha_i) : 1 < i < m_{C’} \\} \\cup \\{(\\alpha_i,\\alpha_{i+1})_{C’} : 1 \\leq i < m_{C’} \\}$$<\/p>\n

We compute\u00a0 $$\\begin{align} \\chi_{\\mathcal{C}}(X) &= \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim C} \\\\ &= \\sum_{C’ \\in \\pi \\mathcal{C}} \\sum_{C \\in \\mathcal{C}_{C’}} (-1)^{\\dim C} \\\\ &= \\sum_{C’ \\in \\pi \\mathcal{C}} \\left( \\sum_{i=2}^{m_{C’}}\u00a0 (-1)^{\\dim \\Gamma_{C’}(\\alpha_i)} + \\sum_{i=1}^{m_{C’} – 1} (-1)^{\\dim (\\alpha_i,\\alpha_{i+1})_{C’} } \\right) \\\\ &= \\sum_{C’ \\in \\pi \\mathcal{C}} \\left( \\sum_{i=2}^{m_{C’}}\u00a0 (-1)^{\\dim C’} + \\sum_{i=1}^{m_{C’} – 1} (-1)^{\\dim C’ + 1 } \\right) \\\\ &= \\sum_{C’ \\in \\pi \\mathcal{C}} (-1)^{\\dim C’} \\left( (m_{C’} – 2)\u00a0 – (m_{C’} – 1) \\right) \\\\ &= \\sum_{C’ \\in \\pi \\mathcal{C}} (-1)^{\\dim C’}\u00a0 (-1)\u00a0 \\\\ &= \\chi_{\\pi \\mathcal{C}} (\\pi X) (-1)\u00a0 \\\\ &= (-1)^{\\dim \\pi X} (-1) \\\\ &= (-1)^{\\dim \\pi X + 1} \\\\ &= (-1)^{\\dim X} \\end{align} $$ as required. $\\Box$<\/p>\n

Proposition 5:<\/strong>\u00a0 If $\\mathcal{C}$ and $\\mathcal{D}$ are cylindrical decompositions of\u00a0 $X \\subseteq R^n$ and $\\mathcal{C} \\subseteq \\mathcal{D}$, then $\\chi_{\\mathcal{C}})(X) = \\chi_\\mathcal{D}(X)$.<\/p>\n

Proof:\u00a0<\/strong> For any $C \\in \\mathcal{C}$, let us set $\\mathcal{D}_C := \\{ D \\in \\mathcal{D} : D \\subseteq C \\}$.\u00a0 This set $\\mathcal{D}_C$ is a cylindrical decomposition of $C$.<\/p>\n

Let us compute $\\chi_\\mathcal{D}(X)$.<\/p>\n

$$\\begin{align} \\chi_{\\mathcal{D}}(X) &= \\sum_{D \\in \\mathcal{D}} (-1)^{\\dim D} \\\\ &= \\sum_{C \\in \\mathcal{C}} \\sum_{D \\in \\mathcal{D}_C} (-1)^{\\dim D} \\\\ &= \\sum_{C \\in \\mathcal{C}} \\chi_{\\mathcal{D}}(C) \\\\ &= \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim C} \\\\ &= \\chi_\\mathcal{C}(X) \\end{align} $$ as required. $\\Box$<\/p>\n

Lemma 6:<\/strong>\u00a0 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 $\\mathcal{C}_i$.<\/p>\n

Proof:\u00a0<\/strong> Any cell decomposition of $R^n$ subjacent to $\\mathcal{C}_1 \\cup \\mathcal{C}_2$ constructed from our proof of cell decomposition suffices. $\\Box$<\/p>\n

Proposition 7:\u00a0<\/strong> For any definable set $X$ and pair of cylindrical decompositions $\\mathcal{C}$ and $\\mathcal{D}$ of $X$ we have $\\chi_{\\mathcal{C}}(X) = \\chi_{\\mathcal{D}}(X)$.<\/p>\n

Proof:\u00a0 By Lemma 6 we can find a cylindrical decomposition $\\mathcal{E}$ of $X$ refining both $\\mathcal{C}$ and $\\mathcal{D}$.\u00a0 Applying Proposition 5 twice we have $\\chi_{\\mathcal{C}}(X) = \\chi_{\\mathcal{E}}(X) = \\chi_{\\mathcal{D}}(X)$, as required. $\\Box$<\/p>\n

Definition 8:\u00a0<\/strong> For a definable set $X$ we define $\\chi(X) := \\chi_\\mathcal{C}(X)$ for any choice of a cylindrical decomposition $\\mathcal{C}$ of $X$.<\/p>\n

By Proposition 7, the value of $\\chi(X)$ does not depend on the choice of the decomposition $\\mathcal{C}$.<\/p>\n

Proposition 9:<\/strong>\u00a0 The function $\\chi:\\text{Def}(\\mathfrak{R}) \\to \\mathbb{N}$ satisfies the following.<\/p>\n

    \n
  1. $\\chi(\\varnothing) = 0$<\/li>\n
  2. $\\chi(\\{ \\ast \\}) = 1$<\/li>\n
  3. $\\chi(X \\times Y) = \\chi(X) \\cdot \\chi(Y)$ for any pair of definable sets $X$ and $Y$, and<\/li>\n
  4. $\\chi(X \\dot{\\cup} Y) = \\chi(X) + \\chi(Y)$ for any pair of disjoint definable subsets $X$ and $Y$ of $R^n$.<\/li>\n<\/ol>\n

    Proof:\u00a0<\/strong> \u00a0For the first point, any cylindrical decomposition $\\mathcal{C}$ of the empty set is itself empty so that $$ \\begin{align} \\chi(\\varnothing) &= \\chi_{\\mathcal{C}}(\\varnothing) \\\\ &= \\chi_{\\varnothing}(\\varnothing) \\\\ &= \\sum_{C \\in \\varnothing} (-1)^{\\dim C} \\\\ &= 0 \\end{align} $$<\/p>\n

    For the second point, the only cylindrical decomposition $\\mathcal{C}$ of the singleton $\\{ \\ast \\}$ is $\\mathcal{C} = \\{ \\{ \\ast \\} \\}$ so that $$ \\begin{align} \\chi(\\{ \\ast \\}) &= \\chi_{\\mathcal{C}}(\\{ \\ast \\}) \\\\ &= \\chi_{\\{ \\{ \\ast \\} \\} }(\\varnothing) \\\\ &= \\sum_{C \\in \\{ \\{ \\ast \\} \\} } (-1)^{\\dim C} \\\\ &= (-1)^{\\dim \\{ \\ast \\}} \\\\ &= (-1)^0 \\\\ &= 1 \\end{align} $$<\/p>\n

    For the third point, let $\\mathcal{C}$ be a cylindrical decomposition of $X$ and $\\mathcal{D}$ be a cylindrical decomposition of $Y$.\u00a0 Then $\\mathcal{C} \\times \\mathcal{D} := \\{ C \\times D : C \\in \\mathcal{C} \\text{ and } D \\in \\mathcal{D} \\}$ is a cylindrical decomposition of $X \\times Y$.\u00a0 \u00a0We compute $$\\begin{align} \\chi(X \\times Y) &= \\chi_{\\mathcal{C} \\times \\mathcal{D}} (X \\times Y) \\\\ &= \\sum_{E \\in \\mathcal{C} \\times \\mathcal{D}} (-1)^{\\dim E} \\\\ &= \\sum_{C \\in \\mathcal{C}} \\sum_{D \\in \\mathcal{D}} (-1)^{\\dim (C \\times D)} \\\\\u00a0 &= \\sum_{C \\in \\mathcal{C}} \\sum_{D \\in \\mathcal{D}} (-1)^{\\dim C + \\dim\u00a0 D} \\\\ &= \\sum_{C \\in \\mathcal{C}} \\sum_{D \\in \\mathcal{D}} (-1)^{\\dim C} \\cdot (-1)^{\\dim D} \\\\ &= \\left( \\sum_{C \\in \\mathcal{C}}\u00a0 (-1)^{\\dim C} \\right) \\left( \\sum_{D \\in \\mathcal{D}} (-1)^{\\dim D}\u00a0 \\right) \\\\ &= \\chi_{\\mathcal{C}}(X) \\cdot \\chi_{\\mathcal{D}}(Y) \\\\ &= \\chi(X) \\cdot \\chi(Y) \\end{align}$$<\/p>\n

    Finally, consider the additivity condition.\u00a0 \u00a0By 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.\u00a0 Thus, $$\\begin{align} \\chi(X \\cup Y) &= \\chi_{\\mathcal{C}}(X \\cup Y) \\\\ &= \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim C} \\\\ &= \\sum_{C \\in \\mathcal{C} : C \\subseteq X } (-1)^{\\dim C} + \\sum_{C \\in \\mathcal{C} : C \\subseteq Y } (-1)^{\\dim C}\u00a0 \\\\ &= \\sum_{C \\in \\mathcal{C}_X\u00a0 } (-1)^{\\dim C} + \\sum_{C \\in \\mathcal{C}_Y } (-1)^{\\dim C}\u00a0 \\\\ &= \\chi_{\\mathcal{C}_X}(X) + \\chi_{\\mathcal{C}_Y}(Y) \\\\ &= \\chi(X) + \\chi(Y) \\end{align} $$\u00a0 $\\Box$<\/p>\n

    It follows from Proposition that we may weaken the requirement that $\\mathcal{C}$ be a cylindrical decompostion in the definition of $\\chi(X)$.<\/p>\n

    Corollary 10:\u00a0<\/strong> If $X$ is a definable set and $\\mathcal{C}$ is a finite partition of $X$ into cells, then $\\chi(X) = \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim C}$.<\/p>\n

    Proof:\u00a0<\/strong> Using the additivity in Proposition 9 and the calculation of $\\chi(C)$ for $C$ a cell of Proposition 3, we have $$\\begin{align} \\chi(X) &= \\chi( \\dot{\\bigcup}_{C \\in \\mathcal{C}} C) \\\\ &= \\sum_{C \\in \\mathcal{C}} \\chi(C) \\\\ &= \\sum_{C \\in \\mathcal{C}} (-1)^{\\dim C} \\end{align} $$ $\\Box$<\/p>\n

    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.<\/p>\n

    Our definition of cells and of cylindrical decompositions depends on a choice of the ordering of the coordinates.\u00a0 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.<\/p>\n

    Definition 11:<\/strong>\u00a0 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 $(x_1, \\ldots, x_n) \\mapsto (x_{\\sigma(1)}, \\ldots, x_{\\sigma(n)})$.<\/p>\n

    Lemma 12:<\/strong> If $n \\in \\mathbb{N}$, $1 \\leq i < n$,\u00a0 $\\sigma$ 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.<\/p>\n

    Before we prove Lemma 12, let us recall the definition of the kind of a cell.<\/p>\n

    Definition 13:\u00a0<\/strong> We defined the kind of a cell $C \\subseteq R^n$ by recursion on $n$.\u00a0 For $n = 0$, $\\text{kind}(C) = \\varnothing$.\u00a0 For $n+1$, if $C = \\Gamma_{C’}(f)$ is the graph of a continuous definable function on the cell $C’ \\subseteq R^n$, then $\\text{kind}(C) := \\text{kind}(C’) \\smallfrown 0$.\u00a0 If $C = (f,g)_{C’}$ is the interval between two continuous definable functions on the cell $C’ \\subseteq R^n$, then $\\text{kind}(C) := \\text{kind}(C’) \\smallfrown 1$.<\/p>\n

    Proof of Lemma 12: We argue by induction on $n$.\u00a0 If $n \\leq 1$, then the lemma is trivial.\u00a0 Consider now the case of $n+1$ (with $n > 0$).<\/p>\n

    Let us consider first the case where $i < n$.\u00a0 \u00a0We 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 $g \\equiv +\\infty$)\u00a0 of $C = \\Gamma_{C’}(f)$ for some cell $C’ \\subseteq R^n$ and deifnable continuous function $f:C’ \\to R$.\u00a0 The arguments in both cases will be similar.\u00a0 Since $i < n$, we may overload notation and regard $\\sigma$ as both a map $R^n \\to R^n$ and as a map $R^{n+1} \\to R^{n+1}$.\u00a0 \u00a0By induction, there is a finite partition $\\mathcal{C}’$ of $C’$ into cells so that $\\sigma D$ is a cell for each $D \\in \\mathcal{C}$.\u00a0 Let $\\mathcal{C} :=\u00a0 \\{ (f \\upharpoonright D, g \\upharpoonright D)_D : D \\in \\mathcal{C}’ \\}$ or\u00a0 $\\mathcal{C} := \\{ \\Gamma_D(f \\upharpoonright D) : D \\in \\mathcal{C}’ \\}$\u00a0 depending on kind of $C$.\u00a0 The set $\\mathcal{C}$ is a finite partition of $C$ into cells.\u00a0 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 $\\sigma (f \\upharpoonright D, g \\upharpoonright D)_D = ((f \\circ \\sigma) \\upharpoonright \\sigma\u00a0 D, (g \\circ \\sigma) \\upharpoonright \\sigma D)_{\\sigma D}$, which is also a cell.\u00a0 In each of these computations we make use of the fact that $\\sigma^2 = \\text{id}$.<\/p>\n

    We now consider the case that $i = n$.\u00a0 This also breaks into subcases based on whether $\\text{kind}(C) = \\ast \\smallfrown 00$, $\\ast \\smallfrown 01$, $\\ast \\smallfrown 10$, or $\\ast \\smallfrown 11$.<\/p>\n

    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 $g:\\Gamma_{C’}(f) \\to R$.\u00a0 Since $y = f(x)$ on $\\Gamma_{C’}(f)$, $g \\upharpoonright \\Gamma_{C’}(f) = g(x,f(x)) =: G(x)$, a continuous definable function on $C’$.\u00a0 Thus, $$\\begin{align} \\sigma C &= \\{ (x,y,z) \\in C’ \\times R^2 : y = G(x) \\text{ and } z = f(x) \\} \\\\ &= \\Gamma_{\\Gamma_{C’}(G)}(f) \\end{align}\u00a0 $$\u00a0 is also a cell.<\/p>\n

    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 $C = (g,h)_{\\Gamma_{C’}(f)} = \\{ (x,y,z) \\in C’ \\times R^2 : y = f(x) \\text{ and } g(x,y) < z < h(x,y) \\}$.\u00a0 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.<\/p>\n

    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 $C = \\Gamma_{(f,g)_{C’}}(h)$.\u00a0 \u00a0By cell composition and the monotonicity theorem, we may cell decompose $C’$\u00a0 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.\u00a0 \u00a0Let us write $\\widetilde{D}$ for $D \\times R^2 \\cap C$.\u00a0 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 $\\sigma \\widetilde{D}\u00a0 = \\{ (x,y,z) \\in D \\times R^2 : y = c(x) \\text{ and } f(x) < z < g(x) \\} = (f,g)_{\\Gamma_D(\\ast)}$ is a cell.\u00a0 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 $\\gamma(x,y) = h^{-1}_x(y)$.\u00a0 We have that $\\sigma \\widetilde{D} = \\Gamma_{(\\alpha,\\beta)_D}(\\gamma)$ is a cell.\u00a0 \u00a0 In the case that $f_x$ is decreasing for all $x \\in D$, we have $\\sigma \\widetilde{D} = \\Gamma_{(\\beta,\\alpha)_D}(\\gamma)$.<\/p>\n

    Finally, if $\\text{kind}(C) = \\ast \\smallfrown 11$, then modify the construction from the last case.\u00a0 \u00a0There 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 $C = (h,k)_{(f,g)_{C’}}$.\u00a0 \u00a0As 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).\u00a0 \u00a0As above, $\\widetilde{D} := D \\times R^2 \\cap C$ is a cell and $C$ is decomposed by these $\\widetilde{D}$.\u00a0 Let us treat the case where both $h_x$ and $k_x$ are increasing on $D$.\u00a0 The other cases are similar.\u00a0 \u00a0Let $\\alpha(x) := \\lim_{y \\to f(x)^+} h_x(y)$, $\\beta(x) := \\lim_{y \\to f(x)^+} k_x(y)$,\u00a0 $\\gamma(x) := \\lim_{y \\to g(x)^-} h_x(y)$, and $\\delta(x) := \\lim_{y \\to g(x)^-} k_x(y)$.\u00a0 \u00a0Then $h_x:(f(x),g(x)) \\to (\\alpha(x),\\gamma(x))$ and $k_x:(f(x),g(x)) \\to (\\beta(x),\\delta(x))$ are order preserving bijections.\u00a0 \u00a0Define functions $$\\lambda(x,z) := \\begin{cases} f(x) & \\text{ if } \\alpha(x) < z \\leq \\beta(x) \\\\\u00a0 \u00a0k^{-1}_x(z) & \\text{ if } \\beta(x) < z < \\delta(x) \\end{cases}$$ and $$\\mu(x,z) := \\begin{cases} g(x) & \\text{ if } \\gamma(x) \\leq z < \\delta(x) \\\\\u00a0 \u00a0h^{-1}_x(z) & \\text{ if } \\alpha(x) < z < \\gamma(x) \\end{cases}$$\u00a0 Then $\\sigma \\widetilde{D} = (\\lambda,\\mu)_{(\\alpha,\\delta)_D}$ is a cell.\u00a0 The other cases are similar. $\\Box$<\/p>\n

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     <\/p>\n","protected":false},"excerpt":{"rendered":"

    Every o-minimal structure admits an Euler characteristic with values in $\\mathbb{Z}$.\u00a0 Our computation showing that $K_0(\\mathfrak{R})$ is a quotient of $\\mathbb{Z}$\u00a0 when $\\mathfrak{R}$ is an o-minimal expansion of an ordered field may be reversed to define the o-minimal Euler characteristic.<\/p>\n","protected":false},"author":99,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16],"tags":[],"class_list":["post-215","post","type-post","status-publish","format-standard","hentry","category-o-minimality-learning-seminar"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/215","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/users\/99"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/comments?post=215"}],"version-history":[{"count":3,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/215\/revisions"}],"predecessor-version":[{"id":233,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/215\/revisions\/233"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/media?parent=215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/categories?post=215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/tags?post=215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}