DEFINABLE MORSE FUNCTIONS IN A REAL CLOSED FIELD

TOMOHIRO KAWAKAMI

Abstract. Let X be a definably compact definable C^{r} manifold and 2 ≤ r < ∞. We prove that the set of definable Morse functions is open and dense in the set of definable C^{r} functions on X with respect to the definable C^{2} topology.

1. Introduction

In Morse theory the topological data of a given space can be described by Morse functions defined on the space. We refer the reader to the book by J. Milnor [10] for Morse theory on compact C^{∞} manifolds.

Let N = (R, +, ·, <, …) be an o-minimal expansion of a real closed field R. Everything is considered in N, the term “definable” is used throughout in the sense of “definable with parameters in N ”, each definable map is assumed to be continuous and 2 ≤ r < ∞.

General references on o-minimal structures are [2], [3], also see [13].

Definable C^{r} Morse functions in an o-minimal expansion of the standard structure of a real closed field are considered in [11].

In this paper we consider a definable C^{r} version of Morse theory in a real closed field R when 2 ≤ r < ∞.

Definable C^{r} manifolds are studied in [11], [1], and definable C^{r}G manifolds are studied in [4]. If R is the field ℝ of real numbers, then definable C^{r}G manifolds are considered in [8], [7], [6] [5].

Let Def^{r}(R^{n}) denote the set of definable C^{r} functions on R^{n}. For each f ∈ Def^{r}(R^{n}) and for each positive definable function ϵ : R^{n} → R, the ϵ-neighborhood N(f; ϵ) of f in Def^{r}(R^{n}) is defined by {h ∈ Def^{r}(R^{n})||∂^{α}(h − f)| < ϵ, ∀α ∈ (ℕ ∪ {0})^{n}, |α| ≤ r}, where α = (α_{1}, …, α_{n}) ∈ (ℕ∪\{0\})^{n}, |α| = α_{1} +⋯+α_{n}, ∂^{α}F = \frac{∂^{|α|}F}{∂x_{1}^{α_{1}}...∂x_{n}^{αn}}. We call the topology defined by these ϵ-neighborhoods the definable C^{r} topology.

Theorem 1.1 (10.7 [1]). Every definably compact definable C^{r} manifold X is definably C^{r} diffeomorphic to a definable C^{r} submanifold of some R^{n}.

By Theorem 1.1, we can consider the set Def^{r}(X) of definable C^{r} functions on X as a subspace of Def^{r}(R^{n}).

Theorem 1.2. Let X be a definably compact definable C^{r} manifold. Then the set of definable Morse functions Def_{Morse}^{r}(X) is open and dense in the set Def^{r}(X) of definable C^{r} functions on X with respect to the definable C^{2} topology.

Theorem 1.1 is a generalization of [9].

2010 Mathematics Subject Classification. 14P10, 14P20, 57R35, 58A05, 03C64.

Keywords and P hrases. O-minimal, real closed fields, Morse theory, definable C^{r} functions, critical points, critical values, definably compact.

2. Preliminaries.

Let W_{1} ⊂ R^{n}, W_{2} ⊂ R^{m} be definable open sets and f : W_{1} → W_{2} a definable map. We say that f is a definable C^{r} map if f is of class C^{r}. A definable C^{r} map is a definable C^{r} diffeomorphism if f is a C^{r} diffeomorphism.

Definition 2.1. A Hausdorff space X is an n-dimensional definable C^{r} manifold if there exist a finite open cover {U_{i}}_{i=1}^{k} of X, finite open sets {V_{i}}_{i=1}^{k} of R^{n}, and a finite collection of homeomorphisms {ϕ_{i} : U_{i} → V_{i}}_{i=1}^{k} such that for any i, j with U_{i} ∩ U_{j} ≠ ∅, ϕ_{i}(U_{i} ∩ U_{j}) is definable and ϕ_{j} ∘ ϕ_{i}^{−1} : ϕ_{i}(U_{i} ∩ U_{j}) → ϕ_{j}(U_{i} ∩ U_{j}) is a definable C^{r} diffeomorphism. This pair ({U_{i}}_{i=1}^{k}, {ϕ_{i} : U_{i} → V_{i}}_{i=1}^{k}) of sets and homeomorphisms is called a definable C^{r} coordinate system.

A definable C^{r} manifold X is definably compact if for every a, b ∈ R ∪ {∞} ∪ {−∞} with a < b and for every definable map f : (a, b) → X, lim_{x→a+0} f(x) and lim_{x→b−0} f(x) exist in X.

If R = ℝ, then for any definable C^{r} manifold X of ℝ^{n}, X is compact if and only if it is definably compact. In general a definably compact set is not necessarily compact. For example, if R = ℝ_{alg}, then [0, 1]_{ℝ_{alg}} = {x ∈ ℝ_{alg}|0 ≤ x ≤ 1} is definably compact but not compact.

Let X be an m-dimensional definable C^{r} manifold and f : X → R a definable C^{r} function. A point p ∈ X is a critical point of f if the differential of f at p is zero. If p is a critical point of f, then f(p) is called a critical value of f. Let p be a critical point of f and (U, ϕ : (U, p) → (V, 0)) a definable C^{r} neighborhood around p. The critical point p is nondegenerate if the Hessian of f ∘ ϕ^{−1} at 0 is nonsingular. Direct computations show that the notion of nondegeniricity does not depend on the choice of a local coordinate neighborhood. We say that f is a definable Morse function if every critical point of f is nondegenerate.

3. Proof of Theorem 1.2

To prove Theorem 1.2, we need the following results.

Lemma 3.1 (6.3.6 [2]). Let A ⊂ R^{n} be a definable set which is the union of definable open subsets U_{1}, …, U_{n} of A. Then A is the union of definable open subsets W_{1}, …, W_{n} of A with cl_{A}(W_{i}) ⊂ U_{i} for i = 1, …, n, where cl_{A}(W_{i}) denotes the closure of W_{i} in A.

Theorem 3.2 ([12]). For a definable subset of R^{n}, it is definably compact if and only if it is closed and bounded.

Theorem 3.3 (5.8 [1]). Let X ⊂ R^{l} be a definable C^{r} manifold. Given two disjoint definable sets F_{0}, F_{1} ⊂ X closed in X, there exists a definable C^{p} function δ : X → R which is 0 exactly on F_{0}, 1 exactly on F_{2} and 0 ≤ δ ≤ 1.

The following result is a definable version of Sard’s Theorem.

Theorem 3.4 (3.5 [1]). Let X_{1} ⊂ R^{s} and X_{2} ⊂ R^{t} be definable C^{r} manifolds of dimension m and n, respectively. Let f : X_{1} → X_{2} be a definable C^{r} map. Then the set of critical values of f has dimension less than n.

By Theorem 3.4, we have the following lemma.

Lemma 3.5. Let U be a definable open subset of R^{m} and f : U → R a definable C^{r} function. There exist a_{1}, …, a_{m} ∈ R such that F (x_{1}, …, x_{m}) = f(x_{1}, …, x_{m}) − (a_{1}x_{1} + ⋯ + a_{m}x_{m}) is a definable Morse function on U and |a_{1}|, …, |a_{m}| are sufficiently small.

Let {ϕ_{i} : U_{i} → V_{i}}_{i=1}^{k} be a definable C^{r} coordinate system of X. By Lemma 3.1, Theorem 3.2, Theorem 1.1 and X is definably compact, shrinking {U_{i}}_{i=1}^{k}, if necessary, there exists a finite collection {K_{i}}_{i=i}^{k} of definably compact subsets with K_{i} ⊂ U_{i} such that X = ∪_{i=1}^{k}K_{i}. From now on we fix {U_{i}}_{i=1}^{k} and {K_{i}}_{i=1}^{k}.

Let f, g : X → R be definable C^{r} functions and ϵ > 0. We say that g is a (C^{2}, ϵ) approximation of f on a definably compact subset K of X if the following three inequalities hold for any point p ∈ K.

\{ \begin{matrix}|f(p) − g(p)| < ϵ, \\ |\frac{∂f}{∂x_{i}} (p) − \frac{∂g}{∂x_{i}} (p)| < ϵ, 1 ≤ i ≤ n, \\ |\frac{∂^{2}f}{∂x_{i}∂x_{j}} (p) − \frac{∂^{2}g}{∂x_{i}∂x_{j}} (p)| < ϵ, 1 ≤ i, j ≤ n. \end{matrix}

Definition 3.6. Let f : X → R be a definable C^{r} function and ϵ > 0. A definable C^{r} function g : X → R is a (C^{2}, ϵ) approximation of f if g is a (C^{2}, ϵ) approximation of f on any K_{i}.

Proposition 3.7. Let C be a definably compact subset of X, h : X → R a definable C^{r} function and ϵ > 0 is sufficiently small. If there are no degenerate critical points of h in C, then for every definable C^{r} function h′ : X → R which is a (C^{2}, ϵ) approximation of h, C does not contain a degenerate critical point of h′. In particular Def_{Morse}^{r}(X) is open in Def^{r}(X) with respect to the definable C^{2} topology.

P roof. We consider in a definable C^{r} coordinate neighborhood (U_{l}, (x_{1}, …, x_{m})). Let the Hessian of h with respect to (U_{l}, (x_{1}, …, x_{m})) be (\frac{∂^{2}h}{∂x_{i}∂x_{j}}). Then h has no degenerate

critical points in C ∩ K_{l} if and only if |\frac{∂h}{∂x_{1}} | + ⋯ + |\frac{∂h}{∂x_{n}} | + |det(\frac{∂^{2}h}{∂x_{i}∂x_{j}})| > 0 holds in C ∩ K_{l}. If ϵ > 0 is sufficiently small, then for any h′ which is a (C^{2}, ϵ) approximation of h, |\frac{∂h′}{∂x_{1}} | + ⋯ + |\frac{∂h′}{∂x_{n}} | + |det(\frac{∂^{2}h′}{∂x_{i}∂x_{j}})| > 0 holds in C ∩ K_{l}. Thus h′ has no degenerate critical points in C ∩ K_{l}. By a similar argument, h′ has no degenerate critical points in C = ∪_{i=1}^{k}C ∩ K_{l}. □

P roof of T heorem 1.2. Proposition 3.7 proves that Def_{Morse}^{r}(X) is open in Def^{r}(X).

To prove density of Def_{Morse}^{r}(X), we proceed by induction on l. Let g : X → R be a definable C^{r} function and ϵ > 0. Assume that we have a definable C^{r} function f_{l−1} : X → R such that f_{l−1} has no degenerate critical points in C_{l−1} := ∪_{i=1}^{l−1}K_{i} and it is a (C^{2}, δ_{l−1}) approximation of g, where δ_{l−1} > 0 is sufficiently smaller than ϵ.

We consider a definable C^{r} coordinate neighborhood (U_{l}, (x_{1}, …, x_{m})). By Lemma 3.5, there exist a_{1}, …, a_{m} ∈ R such that f(x_{1}, …, x_{m}) − (a_{1}x_{1} + ⋯ + a_{m}x_{m}) is a definable Morse function on U_{l} and |a_{1}|, …, |a_{m}| are sufficiently small. By Theorem 3.3, we have a definable C^{r} function h_{l} : X → R such that h_{l} is identically 1 on some definable open neighborhood V_{l} of K_{l} in U_{l}, h_{l} is identically 0 outside of some definably compact set L_{l} with V_{l} ⊂ L_{l} ⊂ U_{l} and 0 ≤ h_{l} ≤ 1. We define f_{l} : X → R, f_{l} = f_{l−1}(x_{1}, …, x_{m}) − (a_{1}x_{1} + ⋯ + a_{m}x_{m})h_{l}(x_{1}, …, x_{m}) on U_{l} and f_{l} = f_{l−1}(x_{1}, …, x_{m}) outside of L_{l}. By the definition of f_{l}, f_{l} is a definable C^{r} function on X.

Calculating on U_{l}, |f_{l−1}(p) − f_{l}(p)| = |a_{1}x_{1} + ⋯ + a_{m}x_{m}|h_{l}(p), |\frac{∂f_{l−1}}{∂x_{i}} (p) − \frac{∂f_{l}}{∂x_{i}} (p)| =

|a_{i}h_{l}(p) + (a_{1}x_{1} + ⋯ + a_{m}x_{m})\frac{∂h_{l}}{∂x_{i}} (p)|, 1 ≤ i ≤ m, |\frac{∂^{2}f_{l−1}}{∂x_{i}∂x_{j}} (p) − \frac{∂^{2}f_{l}}{∂x_{i}∂x_{j}} (p)| = |a_{i} \frac{∂h_{l}}{∂x_{j}} (p) +

a_{j} \frac{∂h_{l}}{∂x_{i}} (p) + (a_{1}x_{1} + ⋯ + a_{m}x_{m})\frac{∂^{2}h_{l}}{∂x_{i}∂x_{j}} (p)|, 1 ≤ i, j ≤ m, where p = (x_{1}, …, x_{m}).

By the construction of h_{l} and since X is definably compact, |h_{l}|, |\frac{∂h_{l}}{∂x_{i}} |, |\frac{∂^{2}h_{l}}{∂x_{i}∂x_{j}} | are bounded. Thus f_{l} is a (C^{2}, δ′_{l}) approximation of f_{l−1} on K_{l} if |a_{1}|, …, |a_{m}| > 0 are sufficiently small.

We now consider on K_{j} when j ≠ l. Since f_{l−1} = f_{l} outside of L_{l}, we only have to evaluate them on K_{j} ∩ L_{l}. Since K_{j} ∩ L_{l} ⊂ U_{j} ∩ U_{l}, they are evaluated by the Jacobian of (U_{j}, (y_{1}, …, y_{m})) between (U_{l}, (x_{1}, …, x_{m})). It is bounded on K_{j} ∩L_{l} because K_{j} ∩ L_{l} is definably compact. Thus they are sufficiently small if |a_{1}|, …, |a_{m}| > 0 are sufficiently small. Hence f_{l} is a (C^{2}, δ_{l}) approximation of f_{l−1}. By Proposition 3.7, f_{l} has no degenerate critical points in C_{l−1}. By the construction of f_{l}, f_{l} has no degenerate critical points in K_{l}. Thus there are no degenerate critical points of f_{l} in C_{l} := ∪_{i=1}^{l}K_{i}. Therefore f_{k} : X → R is the required definable Morse function on X. □

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Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani Wakayama 640-8510, Japan

E-mail address: kawa@center.wakayama-u.ac.jp