デファイナブルC^r群上の代表デファイナブルC^r関数について

(1)

Representative deﬁnable C

r

functions on deﬁnable C

r

groups

Tomohiro Kawakami

Department of Mathematics, Faculty of Education, Wakayama University,

Sakaedani Wakayama 640-8510, Japan

kawa@center.wakayama-u.ac.jp

Partially supported by Kakenhi (23540101)

Abstract

Let G be a compact aﬃne deﬁnable Cr _{group and let r be} _{∞ or ω. We prove that the}

representative deﬁnable Cr _{functions on G is dense in the space of continuous functions on G.}

2010 M athematics Subject Classif ication. 57S15, 03C64.

Keywords and P hrases. O-minimal, deﬁnable Cr _{groups, deﬁnable C}r_{G manifolds.}

1 . Introduction.

Let _{M = (R, +, ·, <, . . . ) be an} o-mini-mal expansion of the standard structureR = (R, +, ·, <) of the field R of real numbers. Everything is considered inM, every defin-able map is assumed to be continuous and the term “definable” is used throughout in the sense of “definable with parameters in M”unless otherwise stated. We assume that r denotes ∞ or ω.

General references on o-minimal struc-tures are [1], [2], also see [13].

Deﬁnable Cr_{G manifolds and deﬁnable}

G sets inM are studied in [8], [7], [6]. Let G be a deﬁnable Cr_{group and Def}r₍

G) denote the space of deﬁnable Cr

func-tions. Left translations in G induce an ac-tion of G deﬁned by f : G→ R �→ L(g, f) = f (g−1_{x) : G} _{→ R. A function f on G is}

representative if the functions {L(g, f)|g ∈ G_{} generate a ﬁnite dimensional subspace of}

Defr_(G).

Theorem 1.1.

Let G be a compact aﬃne deﬁnable Cr _{group. Then the representative}

deﬁnable Cr _{functions on G is dense in the}

strong topology in the space of continuous functions on G.

Let X be a deﬁnable Cr_{G manifold. We}

say that the action of G on X is def inably Cr _{linearizable (resp. C}r _{linearizable) if}

there exist a deﬁnable Cr _{representation of}

G whose representation space is Rn_{, a}

deﬁn-able Cr_{G submanifold Y of} _Rn _{and a}

deﬁn-able Cr_{G diﬀeomorphism (resp. C}r

diﬀeo-morphism) from X to Y .

Theorem 1.2.

Let G be a compact aﬃne deﬁnable Cr _{group and X a compact}

deﬁn-able Cr_{G manifold. Then the action is C}r

linearizable.

Remark that if _{M = R, then for any} positive dimensional compact connected C∞

Representative definable Cr_{functions on definable}_Cr _groups

−31−

(2)

和歌山大学教育学部紀要　自然科学　第62集（2012）

−32−

G manifold, it admits uncountably many nonaﬃne deﬁnable C∞_{G manifold structures}

([10]). In Theorem 1.2, we cannot replace Cr

linearizable by deﬁnably Cr _{linearizable.}

Locally deﬁnable Cr _{manifolds are}

de-ﬁned in [9].

Theorem 1.3.

Let G be a connected lo-cally deﬁnable Cr _{group and ( ˜}_{G, π) the}

uni-versal cover of G. Then ˜G can be equipped uniquely with the structure of a locally deﬁn-able Cr _{group such that π is a locally}

deﬁn-able Cr _{group homomorphism.}

A locally Nash case of Theorem 1.3 is proved in [5].

2 Preliminaries and proof

of results

Let X _{⊂ R}n _{and Y} _{⊂ R}m _{be deﬁnable sets.}

A continuous map f : X → Y is definable if the graph of f (_{⊂ X × Y ⊂ R}n_{× R}m_{) is a}

deﬁnable set.

We say that a group G is a def inable group if G is a deﬁnable set and the group

operations G _{× G → G and G → G are}

deﬁnable.

A Hausdorﬀ space X is an n-dimensional def inable Cr _{manif old if there exist a}

ﬁ-nite open cover_{Ui}ki=1of X, ﬁnite open sets

{Vi}ki=1ofRn, and a ﬁnite collection of

home-omorphisms {φi : Ui → Vi}ki=1 such that for

any i, j with Ui∩Uj �= ∅, φi(Ui∩Uj) is

deﬁn-able and φj◦φ−1i : φi(Ui∩Uj)→ φj(Ui∩Uj) is

a definable Cr _{diffeomorphism. A definable}

Cr_{manifold X is af f ine if X is deﬁnably C}r

diﬀeomorphic to a deﬁnable Cr_submanifold

of some Rn_.

A deﬁnable Cr _{manifold (resp. An aﬃne}

deﬁnable Cr _{manifold) G is a def inable C}r

group (resp. an af f ine def inable Cr_group)

if G is a group and the group operations G× G_{→ G, G → G are deﬁnable C}r _maps.

A subgroup of a deﬁnable Cr _{group is a}

def inable subgroup of it if it is a deﬁnable Cr _{submanifold of it. Note that every}

deﬁn-able Cr _{subgroup of a deﬁnable C}r _{group is}

closed ([12]) and a closed subgroup of a de-ﬁnable Cr _{group is not necessarily deﬁnable.}

Let G be a deﬁnable Cr _{group. A group}

homomorphism from G to some On(R) is a

def inable Cr _{representation if it is a}

de-ﬁnable Cr _{map. A def inable C}r

represen-tation space of G is Rn _{with the orthogonal}

action induced from a deﬁnable Cr

represen-tation of G. A def inable Cr_{G submanif old}

means a G invariant deﬁnable Cr

submani-fold of some deﬁnable Cr _{representation}

space of G.

Let G be a deﬁnable Cr _{group. A def}

in-able Cr_{G manif old is a pair (X, φ)}

consist-ing of a deﬁnable Cr _{manifold X and a}

de-ﬁnable Cr _{action φ : G}_{× X → X on X of}

G. For abbreviation, we write X instead of (X, φ). A deﬁnable Cr_{G manifold is af f ine}

if it is deﬁnably Cr_{G diﬀeomorphic to a}

de-ﬁnable Cr_{G submanifold of some deﬁnable}

Cr _{representation space of G.}

P roof of T heorem 1.1. Since G is compact and aﬃne, there exists a deﬁnable Cr_G

dif-feomorphism f from G to a deﬁnable Cr_G

submanifold G� _{of some deﬁnable C}r

repre-sentation space Ω of G.

Let r : G _{→ R be a continuous} func-tion. Applying Polynomial Approximation Theorem to r _{◦ f}−1 _{: G}� _{→ R, we have a}

polynomial function q : G� → R approxi-mating r_{◦ f}−1_{. Since f is equivariant and}

G acts orthogonally on Ω and by P107 [11], q _{◦ f : G → R is a representative on G} which is a deﬁnable Cr_function

approximat-ing r.

By a way similar to the proof of results of [10], we have the following result.

Theorem 2.1.

Let G be a compact aﬃne deﬁnable Cr _{group and X a compact C}∞_G

manifold. Then X is C∞_{G diﬀeomorphic}

to a deﬁnable Cr_{G submanifold Y of some}

representation space of G.

P roof of T heorem 1.2. We only have to prove the case where r = ω. By Theorem 2.1, there exist a representation space Ω of a deﬁnable Cr _{representation of G, a deﬁnable}

Cr_{G submanifold Y of Ω and a C}∞_G

diﬀeo-morphism f : X _{→ Y . By [P 233 [4]], any} Whitney neighborhood of a C∞_{G map to a}

representation space contains a Cω_{G map.}

Thus we can approximate f by a Cω_{G map}

h : X _{→ Ω. Therefore we have a required} Cω_{G imbedding.}

A Hausdorﬀ space X is an n-dimensional locally def inable Cr _{manif old if there exist}

a countable open cover_{Ui}∞i=1of X,

count-able open sets{Vi}∞i=1ofRn, and a countable

collection of homeomorphisms _{φi : Ui →

Vi}∞i=1 such that for any i, j with Ui∩ Uj �=

∅, φi(Ui ∩ Uj) is deﬁnable and φj ◦ φ−1i :

φi(Ui∩ Uj) → φj(Ui∩ Uj) is a deﬁnable Cr

diﬀeomorphism. We call the (Ui, φi)�s the

def inable charts of X.

Note that locally deﬁnable (C0₎

mani-folds are considered in [3].

Let X, Y be locally deﬁnable Cr

mani-folds with deﬁnable charts (Ui, φi)i∈I, (Wj,

ψj)j∈J respectively. A continuous map f :

X _{→ Y is a locally definable C}r _{map if for}

every ﬁnite subset I� of I, there exists a ﬁ-nite subset J� _{of J such that f (}_∪

i∈IU_i) ⊂

∪j∈JVj and that f| ∪i∈I Ui : ∪i∈IUi →

∪j∈JVj is a deﬁnable Cr map.

A bijective locally deﬁnable Cr _{map f}

between locally deﬁnable Cr _{manifolds is a}

locally def inable Cr_{dif f eomorphism if f}−1

is a locally deﬁnable Cr _map.

A locally deﬁnable Cr _{manifold X is}

af f ine if X is locally deﬁnably Cr

diﬀeo-morphic to a locally deﬁnable Cr

submani-fold of someRn_. Note that for any positive

integer s, a locally deﬁnable Cr _{manifold is}

locally deﬁnably Cs _{imbeddable into some}

Rl _{(1.3 [9]).}

A locally deﬁnable Cr_{manifold (resp. An}

aﬃne locally deﬁnable Cr _{manifold) G is a}

locally def inable Cr_{group (resp. an af f ine}

locally def inable Cr _{group) if G is a group}

and the group operations G× G → G, G → G are locally deﬁnable Cr _maps.

P roof of T heorem 1.3. By the construction of the universal cover ˜G of G, ˜G is a Cr_group

whose charts are countable and π is a Cr

map. Since G is a locally deﬁnable Cr_group,

every transition function is deﬁnable.

References

[1] L. van den Dries, Tame topology and o-minimal structures, Lecture notes series 248, London Math. Soc. Cambridge Univ. Press (1998).

[2] L. van den Dries and C. Miller, Geomet-ric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540.

[3] M.J. Edmundo, G.O. Jones and

N.P. Peatﬁeld, Invariance results for deﬁnable extension of groups, Arch. Math. Logic 50 (2011), 19–31.

[4] P. Heinzner, A.T. Huckleberry, F. Kutzschebauch, A real analytic version of Abels’ theorem and com-plexiﬁcations of proper Lie group actions. Complex analysis and geome-try, Lecture Notes in Pure and Appl. Math., 173, Dekker, New York, (1996), 229–273.

[5] E. Hrushovski and A. Pillay, Groups de-finable in local fields and pseudo-finite fields, Israel J. Math. 85 (1994), 203– 262.

[6] T. Kawakami, Deﬁnable Cr _{groups and}

proper deﬁnable actions, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 58 (2008), 9–18.

[7] T. Kawakami, Deﬁnable G CW com-plex structures of deﬁnable G sets and their applications, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 54 (2004), 1–15.

[8] T. Kawakami, Equivariant diﬀerential topology in an o-minimal expansion of the ﬁeld of real numbers, Topology Appl. 123 (2002), 323–349.

[9] T. Kawakami, Locally deﬁnable Cs_G

manifold structures of locally deﬁn-able Cr_{G manifolds, Bull. Fac. Ed.}

Wakayama Univ. Natur. Sci. 56 (2006), 1–12.

(3)

Representative definable Cr_{functions on definable}_Cr _groups

−33−

G manifold, it admits uncountably many nonaﬃne deﬁnable C∞_{G manifold structures}

([10]). In Theorem 1.2, we cannot replace Cr

linearizable by deﬁnably Cr _{linearizable.}

Locally deﬁnable Cr _{manifolds are}

de-ﬁned in [9].

Theorem 1.3.

Let G be a connected lo-cally deﬁnable Cr _{group and ( ˜}_{G, π) the}

uni-versal cover of G. Then ˜G can be equipped uniquely with the structure of a locally deﬁn-able Cr _{group such that π is a locally}

deﬁn-able Cr _{group homomorphism.}

A locally Nash case of Theorem 1.3 is proved in [5].

2 Preliminaries and proof

of results

Let X _{⊂ R}n _{and Y} _{⊂ R}m _{be deﬁnable sets.}

A continuous map f : X → Y is definable if the graph of f (_{⊂ X × Y ⊂ R}n_{× R}m_{) is a}

deﬁnable set.

We say that a group G is a def inable group if G is a deﬁnable set and the group

operations G _{× G → G and G → G are}

deﬁnable.

A Hausdorﬀ space X is an n-dimensional def inable Cr _{manif old if there exist a}

ﬁ-nite open cover_{Ui}ki=1of X, ﬁnite open sets

{Vi}ki=1ofRn, and a ﬁnite collection of

home-omorphisms {φi : Ui → Vi}ki=1 such that for

any i, j with Ui∩Uj �= ∅, φi(Ui∩Uj) is

deﬁn-able and φj◦φ−1i : φi(Ui∩Uj)→ φj(Ui∩Uj) is

a definable Cr _{diffeomorphism. A definable}

Cr_{manifold X is af f ine if X is deﬁnably C}r

diﬀeomorphic to a deﬁnable Cr _submanifold

of some Rn_.

A deﬁnable Cr _{manifold (resp. An aﬃne}

deﬁnable Cr _{manifold) G is a def inable C}r

group (resp. an af f ine def inable Cr_group)

if G is a group and the group operations G× G_{→ G, G → G are deﬁnable C}r _maps.

A subgroup of a deﬁnable Cr _{group is a}

def inable subgroup of it if it is a deﬁnable Cr _{submanifold of it. Note that every}

deﬁn-able Cr _{subgroup of a deﬁnable C}r _{group is}

closed ([12]) and a closed subgroup of a de-ﬁnable Cr _{group is not necessarily deﬁnable.}

Let G be a deﬁnable Cr _{group. A group}

homomorphism from G to some On(R) is a

def inable Cr _{representation if it is a}

de-ﬁnable Cr _{map. A def inable C}r

represen-tation space of G is Rn _{with the orthogonal}

action induced from a deﬁnable Cr

represen-tation of G. A def inable Cr_{G submanif old}

means a G invariant deﬁnable Cr

submani-fold of some deﬁnable Cr _{representation}

space of G.

Let G be a deﬁnable Cr _{group. A def}

in-able Cr_{G manif old is a pair (X, φ)}

consist-ing of a deﬁnable Cr _{manifold X and a}

de-ﬁnable Cr _{action φ : G}_{× X → X on X of}

G. For abbreviation, we write X instead of (X, φ). A deﬁnable Cr_{G manifold is af f ine}

if it is deﬁnably Cr_{G diﬀeomorphic to a}

de-ﬁnable Cr_{G submanifold of some deﬁnable}

Cr _{representation space of G.}

P roof of T heorem 1.1. Since G is compact and aﬃne, there exists a deﬁnable Cr_G

dif-feomorphism f from G to a deﬁnable Cr_G

submanifold G� _{of some deﬁnable C}r

repre-sentation space Ω of G.

Let r : G _{→ R be a continuous} func-tion. Applying Polynomial Approximation Theorem to r _{◦ f}−1 _{: G}� _{→ R, we have a}

polynomial function q : G� → R approxi-mating r_{◦ f}−1_{. Since f is equivariant and}

G acts orthogonally on Ω and by P107 [11], q _{◦ f : G → R is a representative on G} which is a deﬁnable Cr_function

approximat-ing r.

By a way similar to the proof of results of [10], we have the following result.

Theorem 2.1.

Let G be a compact aﬃne deﬁnable Cr _{group and X a compact C}∞_G

manifold. Then X is C∞_{G diﬀeomorphic}

to a deﬁnable Cr_{G submanifold Y of some}

representation space of G.

P roof of T heorem 1.2. We only have to prove the case where r = ω. By Theorem 2.1, there exist a representation space Ω of a deﬁnable Cr _{representation of G, a deﬁnable}

Cr_{G submanifold Y of Ω and a C}∞_G

diﬀeo-morphism f : X _{→ Y . By [P 233 [4]], any} Whitney neighborhood of a C∞_{G map to a}

representation space contains a Cω_{G map.}

Thus we can approximate f by a Cω_{G map}

h : X _{→ Ω. Therefore we have a required} Cω_{G imbedding.}

A Hausdorﬀ space X is an n-dimensional locally def inable Cr_{manif old if there exist}

a countable open cover_{Ui}∞i=1 of X,

count-able open sets{Vi}∞i=1ofRn, and a countable

collection of homeomorphisms _{φi : Ui →

Vi}∞i=1 such that for any i, j with Ui ∩ Uj �=

∅, φi(Ui ∩ Uj) is deﬁnable and φj ◦ φ−1i :

φi(Ui∩ Uj) → φj(Ui∩ Uj) is a deﬁnable Cr

diﬀeomorphism. We call the (Ui, φi)�s the

def inable charts of X.

Note that locally deﬁnable (C0₎

mani-folds are considered in [3].

Let X, Y be locally deﬁnable Cr

mani-folds with deﬁnable charts (Ui, φi)i∈I, (Wj,

ψj)j∈J respectively. A continuous map f :

X _{→ Y is a locally definable C}r _{map if for}

every ﬁnite subset I� of I, there exists a ﬁ-nite subset J� _{of J such that f (}_∪

i∈IU_i) ⊂

∪j∈JVj and that f| ∪i∈I Ui : ∪i∈IUi →

∪j∈JVj is a deﬁnable Cr map.

A bijective locally deﬁnable Cr _{map f}

between locally deﬁnable Cr _{manifolds is a}

locally def inable Cr_{dif f eomorphism if f}−1

is a locally deﬁnable Cr _map.

A locally deﬁnable Cr _{manifold X is}

af f ine if X is locally deﬁnably Cr

diﬀeo-morphic to a locally deﬁnable Cr

submani-fold of someRn_. Note that for any positive

integer s, a locally deﬁnable Cr _{manifold is}

locally deﬁnably Cs _{imbeddable into some}

Rl _{(1.3 [9]).}

A locally deﬁnable Cr_{manifold (resp. An}

aﬃne locally deﬁnable Cr _{manifold) G is a}

locally def inable Cr_{group (resp. an af f ine}

locally def inable Cr _{group) if G is a group}

and the group operations G× G → G, G → G are locally deﬁnable Cr _maps.

P roof of T heorem 1.3. By the construction of the universal cover ˜G of G, ˜G is a Cr_group

whose charts are countable and π is a Cr

map. Since G is a locally deﬁnable Cr_group,

every transition function is deﬁnable.

References

[1] L. van den Dries, Tame topology and o-minimal structures, Lecture notes series 248, London Math. Soc. Cambridge Univ. Press (1998).

[2] L. van den Dries and C. Miller, Geomet-ric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540.

[3] M.J. Edmundo, G.O. Jones and

N.P. Peatﬁeld, Invariance results for deﬁnable extension of groups, Arch. Math. Logic 50 (2011), 19–31.

[4] P. Heinzner, A.T. Huckleberry, F. Kutzschebauch, A real analytic version of Abels’ theorem and com-plexiﬁcations of proper Lie group actions. Complex analysis and geome-try, Lecture Notes in Pure and Appl. Math., 173, Dekker, New York, (1996), 229–273.

[5] E. Hrushovski and A. Pillay, Groups de-finable in local fields and pseudo-finite fields, Israel J. Math. 85 (1994), 203– 262.

[6] T. Kawakami, Deﬁnable Cr _{groups and}

proper deﬁnable actions, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 58 (2008), 9–18.

[7] T. Kawakami, Deﬁnable G CW com-plex structures of deﬁnable G sets and their applications, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 54 (2004), 1–15.

[8] T. Kawakami, Equivariant diﬀerential topology in an o-minimal expansion of the ﬁeld of real numbers, Topology Appl. 123 (2002), 323–349.

[9] T. Kawakami, Locally deﬁnable Cs_G

manifold structures of locally deﬁn-able Cr_{G manifolds, Bull. Fac. Ed.}

Wakayama Univ. Natur. Sci. 56 (2006), 1–12.

(4)

和歌山大学教育学部紀要　自然科学　第62集（2012）

−34−

[10] T. Kawakami, Nash G manifold struc-tures of compact or compactiﬁable C∞_G

manifolds, J. Math. Soc. Japan 48 (1996), 321–331.

[11] A.L. Onishchik (Ed.), Lie groups and algebraic groups. Springer-Verlag, (1990).

[12] A. Pillay, On groups and ﬁelds deﬁnable in o-minimal structures, J. Pure Appl. Algebra 53 (1988), 239-255.

[13] M. Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Mathematics 150, Birkh¨auser, Boston, (1997).