EQUIVARIANT DEFINALE MORSE FUNCTIONS IN DEFINABLY COMPLETE STRUCTURES

TOMOHIRO KAWAKAMI

Abstract. Let G be a compact definable C^{r} group, X a compact affine definable C^{r}G manifold and 2 ≤ r < ∞. We prove that the set of equivariant definable Morse functions on X whose loci are finite unions of nondegenerate critical orbits is open and dense in the set of G invariant definable C^{r} functions with respect to the definable C^{r} topology.

1. Introduction

Let N = (R, +, ·, <, …) be an expansion of a real closed field R. We say that N is definably complete if every nonempty definable subset A of R, sup A, inf A ∈ R ∪ {∞, −∞}. Every o-minimal expansion of R is definably complete. Definably complete structures are studied in [1], [2]. A weakly o-minimal structure is not always definably complete. For example (ℝ_{alg}, +, ·, <, (−π, π)∩ℝ_{alg}) is weakly o-minimal but not definably complete.

If R is the field ℝ of real numbers, then an expansion M of the standard structure R = (ℝ, +, ·, <) is definably complete.

In this paper we consider its equivariant definable C^{r} version of Morse theory on M. It is a generalization of [8], [12]. Everything is considered in M, r ≥ 2 and every definable map is continuous unless otherwise stated. Remark that the condition that r ≥ 2 is necessary to define Morse functions. Definable C^{r}G manifolds in o-minimal structures are studied in [10], [9]. Their definitions work in M.

Let X be an n-dimensional definable C^{r} manifold and f : X → ℝ a definable C^{r} function. We say that a point p ∈ X is a critical point of f if the differential of f at p is zero. We say that f(p) is called a critical value of f if p is a critical point of f. Let p be a critical point of f and (U, u) a definable C^{r} coordinate system on X at p. The critical point p is nondegenerate if the Hessian matrix of f ∘ u^{−1} at 0 is nonsingular. Direct computations show this definition is well-defined.

In the non-equivariant setting, Y. Peterzil and S. Starchenko [15] introduced definable C^{r} Morse functions in an o-minimal expansion of the standard structure of a real closed field.

Let G be a definable C^{r} group, X a definable C^{r}G manifold and f : X → ℝ a G invariant definable C^{r} function on X. A closed definable C^{r}G submanifold Y of X is a critical manifold (resp. a nondegenerate critical manifold) of f if each p ∈ Y is a critical point (resp. a nondegenerate critical point) of f. We say that f is an equivariant definable Morse function if the critical locus of f is a finite union of nondegenerate critical manifolds of f without interior.

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

Keywords and P hrases. Definably complete structures, Morse theory, equivariant definable Morse functions.

Theorem 1.1. Let G be a compact definable C^{r} group, f an equivariant definable Morse function on a compact affine definable C^{r}G manifold X and 2 ≤ r < ∞. If f has no critical value in [a, b], then f^{a} := f^{−1}((−∞, a]) is definably C^{r}G diffeomorphic to f^{b} := f^{−1}((−∞, b]). If M is exponential, then we can take r = ∞.

Theorem 1.1 is an equivariant definable C^{r} version of Theorem 4.3 [17]. An O-minimal version of Theorem 1.1 is considered in [8], [12].

Note that the method of the proof Theorem 4.3 [17] is the integration of a G invariant C^{∞} vector field. This method does not work in the definable category because the integration of a G invariant definable C^{r} vector field is not always definable.

In the non-equivariant o-minimal case, T.L. Loi [13] proved density and openness of definable Morse functions.

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 1.2 ([13]). Let L be an o-minimal expansion of R and X a definable C^{r} submanifold of ℝ^{n}. Then the set of definable C^{r} functions on ℝ^{n} which are Morse functions on X and have distinct critical values are open and dense in Def^{r}(ℝ^{n}) with respect to the definable C^{r} topology.

Theorem 1.2 is generalized in o-minimal expansions of real closed fields ([6]).

Remark that the definable C^{r} topology and the C^{r} Whitney topology do not coincide in general. If X is compact, then these topologies of the set Def^{r}(X) of definable C^{r} functions on X are the same (P156 [16]).

A nondegenerate critical manifold of an equivariant Morse function on a definable C^{r}G manifold is called a nondegenerate critical orbit if it is an orbit.

Theorem 1.3. Let G be a compact definable C^{r} group, X a compact affine definable C^{r}G manifold and 2 ≤ r < ∞.

(1) The set Def_{equi−Morse,o}(X) of equivariant definable Morse functions on X whose critical loci are finite unions of nondegenerate critical orbits is dense in the set C_{inv}^{r}(X) of G invariant C^{r} functions on X with respect to the C^{r} Whitney topology. Moreover Def_{equi−Morse,o}(X) is open and dense in the set Def_{inv}^{r}(X) of G invariant definable C^{r} functions with respect to the definable C^{r} topology.

(2) If M is exponential, then the set Def_{equi−Morse,o}(X) of equivariant definable Morse functions on X whose critical loci are finite unions of nondegenerate critical orbits is dense in the set C_{inv}^{∞}(X) of G invariant C^{∞} functions on X with respect to the C^{r} Whitney topology. Moreover Def_{equi−Morse,o}(X) is open and dense in the set Def_{inv}^{∞}(X) of G invariant definable C^{∞} functions with respect to the definable C^{r} topology.

Definable G CW complexes are introduced in [5]. They are generalized in o-minimal expansions of real closed fields ([4]). In the o-minimal setting L, the following result holds.

Theorem 1.4 ([8]). Let L be an o-minimal expansion of R, G a compact definable group and X a definable G manifold.

(1) X is definably G homeomorphic to a finite union of open G cells of a definable G CW complex.

(2) If X is compact, then X is definably G homeomorphic to a definable G CW complex. In particular, X is G homeomorphic to a finite G CW complex.

However if M = (ℝ, +, ·, <, ℤ), then Theorem 1.4 does not hold even when G is the trivial group because a definable set ℤ is not homeomorhic to a finite union of open cells.

By a way similar to the proof of 1.6 [8], we have the following result. It is a definable version of a well-known topological result (e.g. 6.2.4 [3]).

Theorem 1.5. Let X be an n-dimensional compact definable C^{r} manifold having a definable Morse function f : X → ℝ with only two critical points and 2 ≤ r < ∞. Then X is definably homeomorphic to the n-dimensional unit sphere S^{n}. If n ≤ 6, then X is definably C^{r} diffeomorphic to S^{n}. If M is exponential, then we can take r = ∞.

Remark that if n = 7, then there exsits a C^{∞} manifold which is homeomorphic to S^{7}, but not C^{∞} diffeomorphic to S^{7} ([14]).

2. Preliminaries and proof of Theorem 1.1.

A group G is a definable C^{r} group if G is a definable C^{r} manifold such that the group operations G × G → G and G → G are definable C^{r} maps. Let G be a definable C^{r} group. A definable C^{r}G manifold is a pair (X, ϕ) consisting of a definable C^{r} manifold X and a group action ϕ : G × X → X such that ϕ is a definable C^{r} map. For simplicity, we write X instead of (X, ϕ).

Let G be a definable C^{r} group. A representation map of G means a group homomorphism from G to some O_{n}(ℝ) which is of class definable C^{r} and the representation of this representation map is ℝ^{n} with the orthogonal action induced by the representation map. In this paper, we always assume that every representation is orthogonal. A definable C^{r}G submanifold of a representation Ω of G is a G invariant definable C^{r} submanifold of Ω. We say that a definable C^{r}G manifold is affine if it is definably C^{r}G diffeomorphic to a definable C^{r}G submanifold of some representation of G.

Theorem 2.1. Let X and Y be compact affine definable C^{r}G manifolds possibly with boundary and 2 ≤ r < ∞. Then X and Y are C^{1}G diffeomorphic if and only if they are definably C^{r}G diffeomorphic. If M is exponential, then we can take r = ∞.

Let G be a compact group, f a map from a C^{r}G manifold X to a representation Ω of G and 0 ≤ r ≤ ∞. Denote the Haar measure of G by dg, and let x be a point in X. Recall the averaging operator A defined by

A(f)(x) = ∫_{G} g^{−1}f(gx)dg.

Proposition 2.2 (e.g. 2.11 [7]). Let G be a compact group and 0 ≤ r ≤ ∞. Suppose that C^{r}(X, Ω) denotes the set of C^{r} maps from a C^{r}G submanifold X of a representation of G to a representation Ω of G.

(1) The averaged map A(f) of f is equivariant, and A(f) = f if f is equivariant.

(2) If f ∈ C^{r}(X, Ω), then A(f) ∈ C^{r}(X, Ω).

(3) If f is a polynomial map, then so is A(f).

(4) If X is compact and r < ∞, then A : C^{r}(X, Ω) → C^{r}(X, Ω) is continuous in the C^{r} Whitney topology.

By a way similar to the proofs of 4.5, 4.6 [7], we have the following two propositions.

Proposition 2.3. Let X be a compact definable C^{r}G submanifold possibly with boundary of a representation Ω of G and 1 ≤ r < ∞. Then there exists a definable C^{r}G tubular neighborhood (U, θ) of X in Ω. If M is exponential, then we can take r = ∞.

Proposition 2.4. Let X be a compact affine definable C^{r}G manifold with boundary and 2 ≤ r < ∞. Then X admits a definable C^{r}G collar, namely there exists a definable C^{r}G imbedding ϕ : ∂X × [0, 1] → X such that ϕ|(∂X × {0}) is the inclusion ∂X → X, where the action on [0, 1] is trivial. If M is exponential, then we can take r = ∞.

Theorem 2.5 (P 38 [3]). (1) Let X, Y be C^{1} manifolds. Then the set of C^{1} diffeomorphisms from X onto Y is open in the set C^{1}(X, Y) of C^{1} maps from X to Y with respect to the C^{1} Whitney topology.

(2) Let X, Y be C^{1} manifolds with boundary ∂X, ∂Y, respectively. Then the set of C^{1} diffeomorphisms from X onto Y is open in {f ∈ C^{1}(X, Y)|f(∂X) ⊂ f(∂Y)} with respect to the C^{1} Whitney topology.

By a way similar to the proof of 2.5 [11], we have the following theorem.

Theorem 2.6. Let G be a compact definable C^{r} group and X a compact affine definable C^{r}G manifold and 1 ≤ r < ∞. Suppose that A, B are G invariant definable disjoint closed subsets of X. Then there exists a G invariant definable C^{r} function f : X → ℝ such that f|A = 1 and f|B = 0. If M is exponential, we can take r = ∞.

P roof of T heorem 2.1. Let Ω (resp. Ξ) be a representation of G containing X (resp. Y) as a definable C^{r}G submanifold of Ω (resp. Ξ). We first assume that ∂X = ∂Y = ∅. By Polynomial Approximation Theorem, Proposition 2.2, Proposition 2.3 and Theorem 2.5, X and Y are C^{1}G diffeomorphic if and only if they are definably C^{r}G diffeomorphic. If M is exponential, then we can take r = ∞.

We assume that ∂X ≠ ∅ and ∂Y ≠ ∅. Let f : X → Y be a C^{1}G diffeomorphism. Since f|∂X : ∂X → ∂Y is a C^{1}G diffeomorphism and ∂X is compact, one can find a definable C^{r}G diffeomorphism f′ : ∂X → ∂Y as an approximation of f|∂X : ∂X → ∂Y in the C^{1} Whitney topology. Using definable C^{r}G collars of ∂X and ∂Y in X and Y, respectively, we have a G invariant definable open neighborhoods U and V of ∂X and ∂Y in X and Y, respectively, and a definable C^{r}G diffeomorphism f_{1} : U → V with f_{1}|∂X = f′.

Take a G invariant definable open neighborhood U′ of ∂X in X with U′ ⊊ U. By Theorem 2.6, there exists a G invariant definable C^{r} function λ : X → ℝ such that λ = 1 on U′ and the support lies in U. By Proposition 2.3 and since Y is compact, there exists a definable C^{r}G tubular neighborhood (V, θ) of Y in Ξ. By Polynomial Approximation Theorem, Proposition 2.2 and since X is compact, there exists a polynomial G map f_{2} : X → Ξ which is an approximation of i ∘ f in the C^{1} Whitney topology, where i : Y → Ξ denotes the inclusion. If our approximation is sufficiently close, then H : X → Y, H(x) = θ(λ(x)f_{1}(x) + (1 − λ(x))f_{2}(x)) is a definable C^{r}G map such that it is an approximation of f in the C^{1} Whitney topology and H(∂X) ⊂ ∂Y. Therefore by Theorem 2.5 and the inverse function theorem, H is the required definable C^{r}G diffeomorphism. If M is exponential, then we can take r = ∞ in the general case. □

P roof of T heorem 1.1. By the proof of Theorem 4.3 [17], f^{a} = f^{−1}((−∞, a]) is C^{r−1}G diffeomorphic to f^{b} = f^{−1}((−∞, b]). Since X is compact and affine, these two manifolds are compact affine definable C^{r}G manifolds with boundary. Thus Theorem 1.1 follows from Theorem 2.1. □

3. Proof of Theorem 1.3.

By the proof of Lemma 4.8 [17] proves the following.

Theorem 3.1 ([17]). Let G be a compact C^{r} group, X a compact C^{r}G manifold and 2 ≤ r ≤ ∞. Then the set C_{equi−Morse,o}^{r}(X) of equivariant Morse functions on X whose critical loci are finite unions of nondegenerate critical orbits is open and dense in the set C_{inv}^{r}(X) of G invariant C^{r} functions on X with respect to the C^{r} Whitney topology.

P roof of T heorem 1.3. Let f ∈ C_{inv}^{r}(X) and N ⊂ C_{inv}^{r}(X) an open neighborhood of f in C_{inv}^{r}(X). By Theorem 3.1, there exists an open subset N′ ⊂ N such that each h ∈ N′ is an equivariant Morse function whose critical locus is a finite union of nondegenerate critical orbits. Let C^{r}(X) denote the set of C^{r} functions on X. Since A : C^{r}(X) → C^{r}(X) is continuous and A(C^{r}(X)) = C_{inv}^{r}(X), A : C^{r}(X) → C_{inv}^{r}(X) is continuous. Fix h ∈ N′. Since A(h) = h, A^{−1}(N′) is an open neighborhood of h in C^{r}(X). Applying Polynomial Approximation Theorem, we have a polynomial function h′ lies in A^{−1}(N′). Applying the averaging function, we have a G invariant polynomial function F := A(h′) lies in N′. Since F is a G invariant polynomial function, it is a G invariant definable C^{r} function. Thus F is an equivariant definable Morse function lies in N.

We now prove the second part. By the first part, Def_{equi−Morse,o}(X) is dense in C_{inv}^{r} (X). Thus it is dense in Def_{inv}^{r}(X). Let h ∈ Def_{equi−Morse,o}(X). By Theorem 3.1, there exists an open neighborhood V of h in C_{inv}^{r}(X) such that each h ∈ V is an equivariant Morse function whose critical locus is a finite union of nondegenerate critical orbits. Thus V ∩ Def_{inv}^{r}(X) is the required open neighborhood of h in Def_{inv}^{r}(X).

If M is exponential, then the above argument works when r = ∞. □

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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