Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Tame Topology and O-Minimal Structures

University of IllinoisUrbana-Champaign

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Structure

Definition

Given a set R a structure S on R is a sequence (Sn)n∈N such that

Sn ⊆ P(Rn).

Sn is closed under boolean operations.

A ∈ Sn implies A× R ∈ Sn+1 and R × A ∈ Sn+1.

A ∈ Sn+1 implies π(A) ∈ Sn.

{(x1, · · · , xn) : x1 = xn} ∈ Sn

We say that a set A is definable if it is in S, that is if it is in any ofthe Sn composing S

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Example

Example

For any set R let Sn = P(Rn).

Constructible sets of an algebraically closed field.

Semialgebraic sets of R.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Generating Structures

Given X a collection of subsets of powers of R, we can talk aboutthe Structure generated by X . That is, the smallest structure〈R,X〉 containing X . A way of constructing it is adding thediagonals, and closing under the operations in the definition (i.e.unions, intersections, complements, Cartesian products andprojections).

Example

Let X = {{m} : m ∈ R} and S = 〈R,X〉 then S1 is the set of allfinite and cofinite subsets of R

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Complexity Dichotomy

If you start with a collection of subsets X and then start closing bythe before mentioned operations either one of two things are goingto happen.

(Tame) You reach some form of stabilization. For exampletaking complements becomes superfluous, or takingprojections gives sets already definable.

(Wild) You come up with more complicated sets as you keepapplying the operations, such as Borel or Cantor-like sets.

Example

Let X = {{m} : m ∈ R} is tame.

Z, the graph of addition and multiplication gives rise to all theprojective sets, hence it is wild.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

O-minimal Structures

Definition

Let (R, <) be a dense linear ordered non-empty set with noendpoints. An o-minimal structure over R is a structure in whichall the definable subsets of R are finite unions of points andintervals, and the order is a definable subset of R2

We equip R with the order topology, and Rn with the producttopology. It is clear that not all open sets are definable (take aninfinite union of disjoint intervals). But If A is a definable set, thenthe closure and the interior of A are also definable.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Examples of O-minimal Structures

The following are o-minimal structures, and thus motivate thestudy of such elements.

1 R̄ generated by singletons in R, the graph of +, · and theorder. (i.e. semialgebraic sets in R).

2 Qalg ∩ R with the with the graph of + and ·.3 Ran same as R̄ with the graphs of restricted analytic functions.

4 Rexp same as R̄ with the graph of the exponential function.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

The Semialgebraic Case

Consider the structure R̄. This structure has the followingproperties:

Stratification,

Piecewise smoothness of semialgebraic maps,

Triangulation,

Finiteness of topological type.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Monotonicity Theorem

Theorem

Let f : (a, b)→ R be a definable function then there are pointsa1 < · · · < an with a = a0 and b = an+1 such that for all intervals(ai , ai+1) f is constant, or strictly monotone and continuous.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Cells

Definition

We construct k-cells inductively in the following way

The element in R0 is a 0-cell

If C is a k-cell in Rn and f is a definable continuous functionon C , then the graph of f in Rn+1 is a k-cell

If C is a k-cell in Rn and f and g are definable continuousfunction over C such that for all c ∈ C we have f (c) < g(c),then {(x , y) ∈ Rn + 1 : x ∈ C , y ∈ R f (x) < y < g(x)} is ak + 1-cell

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Decomposition

Definition

A decomposition of R consists of a finite set{(−∞, a1), (a1, a2), · · · , (ak ,∞), {a1}, · · · {ak}} wherea1 < a2 < · · · < an.

A decomposition of Rn+1 consists of a finite partition in Cellsof Rn+1 such that the set of projections of those Cells is adecomposition of Rn

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Cell Decomposition Theorem

Theorem

Let X1, · · · ,Xk be definable sets in Rn then there is adecomposition of Rn such that it forms a partition of each ofthe Xi ’s

Let f : X → R be a definable function with X ⊆ Rm thenthere is a decomposition of Rm such that for each cell C inthat decomposition, the restriction f �C is continuous.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Dimension

Definition

Let X be a definable set, then the dimension of X is defined to be

Dim(X ) = max {k : C ⊆ X and C is a k − cell}

This notion of dimension has very nice properties like if A is anonempty definable set,

if f : A→ Rn is a definable map, then Dim(A) ≥ Dim(f (A));

Dim(cl(A)− A) < Dim(A)

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Euler Characteristic

Definition

Let A be a definable set and let C = {Ci} be a finite partition of Ainto cells, then we define the Euler characteristic of A,EC(A) :=

∑i (−1)Dim(Ci)

This characteristic is in fact independent of the choice of thepartition, hence it is customary to drop the C from EC and it isalso preserved under definable injective maps.This Euler Characteristic has been previously used to prove ananalogue of Sylow’s theorems on groups definable in o-minimalstructures.In the topological setting, the Euler Characteristic is seen as afinitely additive measure and integration with respect to it isdeveloped.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Definable Families

It is often of interest to study collections or families of definablesets in a certain uniform way.

Definition

Given S ⊆ Rm+n we consider the definable family (Sx)x∈Rm whereSx = {y ∈ Rn | (x , y) ∈ S}.

We are often interested in the properties of Sx that depend on x .We get results such as

For each d ∈ {1, · · · , n} the setS(d) := {x ∈ Rm | Dim(Sx) = d} is definable andDim({(x , y) ∈ S | Dim(Sx) = d} = Dim(S(d)) + d ,

E (Sx) takes only finitely many values as x runs through Rm,and for each e the set A(e) := {x ∈ Rm | E (Sx) = e} isdefinable and E{(x , y) ∈ S : x ∈ A(e)} = eE (A(e)).

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

VC-Minimalty

Consider the collection C = {Sx : x ∈ Rm} ⊆ P(Rn). Now for eachfinite F ⊆ Rn we can consider F ∩ C := {F ∩ Sx : x ∈ Rm} then0 ≤ |F ∩ C| ≤ 2|F |. We define the function gS : N→ N given bygS(k) := max{|F ∩ C| : F is a k-element set}This function has polynomial growth with respect to k .

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Definable Choice

So far everything holds in arbitrary o-minimal structures, that is weonly require the order to be in the structure. If we further on askfor an additive group operation to be part of the structure we geteven nicer results such as

Theorem (Definable Choice)

1 If S ⊆ Rm+n is definable and π : Rm+n → Rm the projectionon the first m coordinates, then there is a definable mapf : πS → Rn such that Γ(f ) ⊆ S.

2 Each definable equivalence relation on a definable set X has adefinable set of representatives.

In Model Theoretic terms, these two conditions are having skolemfunctions and elimination of imaginaries.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Curve Selection

We cannot have infinite definable sequences of (distinct) points, asthe structure generated by such a set would not be o-minimal, butwe can find a suitable substitute for convergent sequences.

Lemma (Curve Selection)

Let X be a definable set a ∈ cl(X )− X then there is a continuousdefinable map γ : (0, ε)→ X for some ε > 0 such thatlimt→0 γ(t) = 0.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Differentiability

If we further require the structure to have multiplication we candefine the notion of differentiability of a function. In particular ifwe have an o-minimal structure with an addition and a product, wecan define the following.

Definition

Let f = (f1, · · · fn) : U → Rn be a definable map from an opensubset U of Rm. We call f a C k -map if all the partial derivativesof order k of f are defined as R-valued functions on U and arecontinuous.

We actually extend the notion of C k maps to arbitrary definablesets.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

C k-cells

Definition

We construct C k − l-cells inductively in the following way

The element in R0 is a 0-cell

If A is a C k − l-cell in Rn and f is a definable C k functionover A then the graph of f in Rn+1 is a C k − l-cell

If C is a k-cell in Rn and f and g are definable C k functionover C such that for all c ∈ C we have f (c) < g(c) then{(x , y) ∈ Rn + 1 : x ∈ C , y ∈ R f (x) < y < g(x)} is aC k − (l + 1)-cell.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

C k- cell decomposition

Theorem (C k -cell decomposition)

Let k ∈ N1 Let m ∈ N. For any definable sets A1, · · · ,An ⊆ Rm there is a

decomposition of Rm into C k -cells partitioning A1, · · ·An

2 Let m ∈ N. For every definable function f : A→ R,A ⊆ Rm ,there is a decomposition of Rm into C k -cells partitioning Asuch that each restriction f �C : C → R is C k -for each cellC ⊆ A of the decomposition.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Good Directions Lemma

Theorem

Let A ⊆ Rm+1 be definable of dimension < m + 1. Let B ⊆ Rm bea box contained in the disc ‖x‖ < 1. Then there is x ∈ B such thatfor each point p ∈ Rm+1 the set {t ∈ R : p + tv̇(x) ∈ A} is finite.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Triangulation Theorem

Theorem

Each definable set is definably homeomorphic to a boundedsemilinear set.

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Trivial Maps

Definition

A definable map f : E → B between definable sets is definablytrivial if there are a definable set F and a definablehomeomorphism E → B × F such that the diagram

E

f ��???

????

∼ // B × F

{{xxxx

xxxx

x

B

commutes.

Notice that in such a case all the fibers are definablyhomeomorphic to F .

2013

Tame Topology and O-Minimal Structures

Structures Cell Decomposition Dimension and Euler Characteristic Definable Families and Collections Adding more Structure

Trivialization Theorem

Theorem (Trivialization Theorem)

Let f : E → B be a definable continuous map between definablesets E and B. Then B can be partitioned into definable setsB1, · · ·Bk such that the restrictions

f � f −1(Bi ) : f −1(Bi )→ Bi , i = 1, · · · k

are definably trivial.

As an easy consequence we get that definable collections of adefinable set have only finitely many homeomorphism types.

2013

Tame Topology and O-Minimal Structures