デファイナブルt正則性定理

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Definable t-regularity theorem

Tomohiro Kawakami

Department of Mathematics, Faculty of Education, Wakayama University,

Sakaedani Wakayama 640-8510, Japan

[email protected]

Abstract

We consider locally deﬁnable C∞ _{manifolds, locally deﬁnable C}∞ _{maps and study}

t-regularity of locally deﬁnable C∞ maps.

2000 M athematics Subject Classif ication. 14P10, 14P20, 03C64.

Keywords and P hrases. O-minimal, locally deﬁnable sets, locally deﬁnable C∞ _manifolds,

locally deﬁnably homotopic.

1 . Introduction.

Let _{M = (R, +, ·, <, e}x_{, . . . ) be an}

ex-ponential o-minimal expansion of the stan-dard structure R = (R, +, ·, <) of the ﬁeld

R of real numbers. General references on o-minimal structures are [1], [2], see also [7]. For example, the Nash category is a spe-cial case of the deﬁnable C∞ category and it coincides with the deﬁnable C∞ _category

based onR = (R, +, ·, <) ([8]). Equivariant

definable category is studied in [3], [4], [5]. In this paper “definable” means “defin-able with parameters in _{M”, everything is} considered in M, “countable” means finite

or countably inﬁnite and each locally de-ﬁnable map is continuous unless otherwise stated.

A subset X of Rn _{is called locally def}

i-nable if for every x_{∈ X there exists a}

defin-able open neighborhood U of x in Rn such that X_{∩U is a definable subset of X. Clearly} every definable set is locally definable, every compact locally definable set is definable and any open subset ofRn _{is locally definable.}

Let U ⊂ Rn _{and V} _{⊂ R}m _{be locally}

deﬁnable sets. We say that a continuous map f : U → V is a locally definable map

if for any x _{∈ U there exists a deﬁnable} open neighborhood Wx of x in Rn such that

f_{|U ∩ W}x is deﬁnable.

Two locally deﬁnable maps f, h : X → Y between locally deﬁnable sets are locally def inably homotopic if there exists a locally

deﬁnable map H : X _{× [0, 1] → Y such that}

H(x, 0) = f (x) for all x _{∈ X and H(x, 1) =} h(x) for all x_{∈ X.}

Let Mn_{, N}p_{be locally deﬁnable C}∞

man-ifolds of dimension n, p, respectively, f : Mn

→ Np _{a locally deﬁnable C}∞ _{map, N}p−q

1 a (p− q)-dimensional locally deﬁnable C∞

submanifold of Np_. _{We say that f is}

t-regular on N₁p−q if for any x ∈ f−1_(Np−q

1 ), (df )x(TxMn) + Tf (x)N1p−q = Tf (x)Np.

Theorem 1.1.

Let Mn_{, N}p _{be locally}

de-ﬁnable C∞ _{manifolds of dimension n, p,}

re-spectively, f : Mn _{→ N}p _{a locally}

deﬁn-able C∞ map, N₁p−q a (p− q)-dimensional locally deﬁnable C∞ _{submanifold of N}p_{. Let}

−₁− −₁−

Received September 5, 2018

Definable t-regularity theorem

Tomohiro Kawakami

Department of Mathematics, Faculty of Education, Wakayama University,

Sakaedani Wakayama 640-8510, Japan

[email protected]

Abstract

We consider locally deﬁnable C∞ _{manifolds, locally deﬁnable C}∞ _{maps and study}

t-regularity of locally deﬁnable C∞ maps

2000 M athematics Subject Classif ication. 14P10, 14P20, 03C64.

Keywords and P hrases. O-minimal, locally deﬁnable sets, locally deﬁnable C∞ _manifolds,

locally deﬁnably homotopic.

1 . Introduction.

Let _{M = (R, +, ·, <, e}x_{, . . . ) be an}

ex-ponential o-minimal expansion of the stan-dard structure R = (R, +, ·, <) of the ﬁeld

R of real numbers. General references on o-minimal structures are [1], [2], see also [7]. For example, the Nash category is a spe-cial case of the deﬁnable C∞ category and it coincides with the deﬁnable C∞ _category

based onR = (R, +, ·, <) ([8]). Equivariant

definable category is studied in [3], [4], [5]. In this paper “definable” means “defin-able with parameters in _{M”, everything is} considered in M, “countable” means finite

or countably inﬁnite and each locally de-ﬁnable map is continuous unless otherwise stated.

A subset X of Rn _{is called locally def}

i-nable if for every x_{∈ X there exists a}

defin-able open neighborhood U of x in Rn such that X_{∩U is a definable subset of X. Clearly} every definable set is locally definable, every compact locally definable set is definable and any open subset ofRn _{is locally definable.}

Let U ⊂ Rn _{and V} _{⊂ R}m _{be locally}

deﬁnable sets. We say that a continuous map f : U → V is a locally definable map

if for any x _{∈ U there exists a deﬁnable} open neighborhood Wx of x in Rn such that

f_{|U ∩ W}x is deﬁnable.

Two locally deﬁnable maps f, h : X → Y between locally deﬁnable sets are locally def inably homotopic if there exists a locally

deﬁnable map H : X _{× [0, 1] → Y such that}

H(x, 0) = f (x) for all x _{∈ X and H(x, 1) =} h(x) for all x_{∈ X.}

Let Mn_{, N}p_{be locally deﬁnable C}∞

man-ifolds of dimension n, p, respectively, f : Mn

→ Np _{a locally deﬁnable C}∞ _{map, N}p−q

1 a (p− q)-dimensional locally deﬁnable C∞

submanifold of Np_. _{We say that f is}

t-regular on N₁p−q if for any x ∈ f−1_(Np−q

1 ), (df )x(TxMn) + Tf (x)N1p−q = Tf (x)Np.

Theorem 1.1.

Let Mn_{, N}p _{be locally}

de-ﬁnable C∞ _{manifolds of dimension n, p,}

re-spectively, f : Mn _{→ N}p _{a locally}

deﬁn-able C∞ map, N₁p−q a (p− q)-dimensional locally deﬁnable C∞ _{submanifold of N}p_{. Let}

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和歌山大学教育学部紀要　自然科学　第69集（2019）

−₂−

A be a locally deﬁnable closed subset of Mn

such that there exists a locally deﬁnable open neighborhood U of A such that f_{|U is} t-regu-lar on N₁p−q. For every positive locally deﬁn-able continuous function δ : Mn _{→ R, there}

exists a locally deﬁnable C∞ _{map h : M}n _→

Np _{satisﬁes the following conditions.}

(1) h is locally deﬁnable homotopic to f . (2) h is a δ-approximation of f .

(3) h is t-regular on N₁p−q. (4) h|A = f|A.

Theorem 1.2.

Every n-dimensional lo-cally deﬁnable C∞ _{manifold X is locally}

de-ﬁnably C∞ _{imbeddable into} _R2n+1_. Theorem 1.2 is proved in [6] the case where r is a positive integer.

2 Proof of results

Remark that for any locally deﬁnable map

f between locally deﬁnable sets X and Y , if X is compact, then f (X) is a deﬁnable set

and f : X _{→ f(X) (⊂ Y ) is a deﬁnable} map.

Note that the maps f1, f2 : R → R de-fined by f1(x) = sin x, f2(x) = cos x, re-spectively, are analytic but not locally defin-able in _{R = (R, +, ·, <), and that the field} Q (⊂ R) of rational numbers is not a lo-cally definable subset of R. For example, if

M = Ran,exp, then f : (−1, 1) → R, f(x) =

sin 1

1−x2 is locally deﬁnable but not deﬁnable.

Let U ⊂ Rn _{and V} _{⊂ R}m _{be open sets.}

A Cr _{map f : U} _{→ V is called a locally}

def inable Cr_{map if f is locally deﬁnable. A}

locally deﬁnable Cr_{map f : U} _{→ V is called}

a locally def inable Cr _{dif f eomorphism if}

there exists a locally deﬁnable Cr _{map h :}

V → U such that f ◦ h = id and h ◦ f = id.

Definition 2.1 ([6]).

Let 1≤ r ≤ ω.

(1) A locally deﬁnable subset X of Rn _is

called a d-dimensional locally def inable Cr

submanif old of Rn _{if for any x} _{∈ X there}

exists a deﬁnable Cr _{diﬀeomorphism φ from}

some deﬁnable open neighborhood U of the origin inRn_{onto some deﬁnable open}

neigh-borhood V of x in Rn _{such that φ(0) =}

x, φ(Rd _{∩ U) = X ∩ V . Here R}d ₌ _{{x ∈}

Rn_{| last (n − d) components of x are zero.}}

(2) A locally def inable Cr _{manif old X of}

dimension d is a Cr _{manifold with a}

count-able system of charts {φi : Ui → Rd} such

that for each i and j φi(Ui ∩ Uj) is a

deﬁn-able open subset of Rd _{and the map φ} j ◦

φ−1_i _|φi(Ui ∩ Uj) : φi(Ui ∩ Uj) → φj(Ui ∩

Uj) is a deﬁnable Cr diﬀeomorphism. We

call these atlas locally def inable Cr_.

Lo-cally deﬁnable Cr _{manifolds with}

compati-ble atlases are identified. Clearly every de-finable Cr _{manifold is a locally definable C}r

manifold. A subset Y of a locally deﬁnable

Cr _{manifold X is called a k-dimensional}

locally def inable Cr _{submanif old of X if}

each point x ∈ Y there exists a locally

de-ﬁnable Cr _{chart φ}

i : Ui → Rd of X such

that x ∈ Ui and Ui ∩ Y = φ−1i (Rk), where

Rk _{⊂ R}d _{is the vectors whose last (d} _{− k)}

components are zero.

(3) A locally deﬁnable Cr _{manifold is}

af f ine if it can be imbedded into some Rn

in a locally deﬁnable Cr _way.

Since a locally deﬁnable set X is para-compact, for any countable deﬁnable open cover {Uα} of X, there exists a partition of

unity _{fα} subordinate to {Uα} such that

each fα is locally deﬁnable. Thus we have

the following theorem.

Theorem 2.2.

Let X be a locally deﬁn-able C∞ _{manifold. Every locally deﬁnable}

open cover of X has a subordinate locally de-ﬁnable C∞ _{partition of unity.}

Definition 2.3.

Let X _{⊂ R}n_{, Y} _{⊂ R}m

be locally deﬁnable sets, f, h : X → Y

lo-cally deﬁnable maps and δ : X _{→ R a} posi-tive locally deﬁnable function. We say that g is a δ-approximation of f if dm(f (x), g(x)) <

δ(x) for any x ∈ X, where dm means the

standard metric of Rm_.

Proposition 2.4.

Let X be a locally de-ﬁnable C∞ manifold. Then every C∞ map f : X _{→ R}n _{is approximated in the C}∞

Whitney topology by a locally deﬁnable C∞ map h : X _{→ R}n_.

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Definable t-regularity theorem

−₃−

P roof . By Theorem 2.2, we have a

lo-cally deﬁnable C∞_{partition of unity}_{φ j}∞j=1

subordinates to some locally ﬁnite open de-ﬁnable cover _{Xj}∞j=1 of X such that X =

∪∞

j=1supp φj and Xj is compact. For any

j, take an open neighborhood Uj of supp φj

in X such that Uj is compact. Applying the

polynomial approximation theorem, we have a locally deﬁnable C∞ _{map h}

j : Uj → Rn

which approximates f|Uj. If our

approxima-tion is suﬃciently close, then∑∞_j=1φjhj is a

locally deﬁnable Cr _{approximation of f .}

P roof of T heorem 1.2. By Whitney’s

imbedding Theorem, there exists a Cr

imbed-ding f : X _{→ R}2n+1_{. Since imbeddings} from X to R2n+1 _{are open in C}r_(X,_R2n+1_), we have the required locally deﬁnable Cr

imbedding h : X → R2n+1_.

For a positive number k, Cn_{(k) means}

the open ball ofRn _{with center 0 and radius}

k and Cn_{(k) denotes the closure of C}n_(k).

P roof of T heorem 1.1. Since N₁p−q is a locally deﬁnable C∞ _{submanifold, it is}

cov-ered by a system of chart of Nq _{such that:}

(1) N₁p−q _{⊂ ∪}∞ i=1Yi (2) (Yi, ki) is a chart of Np. (3) ki : Yi∩ N1p−q : Yi∩ N1p−q → Rp−q. Let Y0 = Np _{− N}p−q 1 . Then {Yi|i ∈ N ∪

{0}} is a locally deﬁnable open cover and {f−1_(Y

i)|i ∈ N ∪ {0}} is a locally deﬁnable

open cover of Mn_{. On the other hand, M}n₌

U ∪ (Mn _{− A) is a locally deﬁnable open}

cover. Thus there exists a locally deﬁnable

C∞ atlas {(Vj, hj)|j ∈ Z} such that:

(1) _{Vj} is a locally ﬁnite reﬁnement of

{f−1_(Y

i)} and {U, M − A}.

(2) hj(Vj) = Cn(3).

(3) Let Wj = h−1j (Cn(1)). Then {Wj} is

a locally deﬁnable open cover of Mn_.

Renumbering Vj, if necessary, j 0 if

Vj ⊂ U.

We can take a locally deﬁnable C∞

func-tion φ :Rn _{→ R such that:}

(1) φ(Cn_{(1)) = 1.} (2) 0 < φ(Cn₍₂₎_{− C}n_{(1)) < 1.} (3) φ(Rn_{− C}n_{(2)) = 0.} We deﬁne φi : Mn → R to be φi(x) = { φ_{◦ h}i(x), x ∈ Vi 0, x�∈ Vi . Then φi is a

locally deﬁnable C∞ _{function and for each}

f (Vj), there exists an i(j) such that f (Vj)⊂

Yi(j).

By induction, we construct the required map g. Let f0 = f . Then f0|U is t-regular

on N₁p−q. Assume that a locally deﬁnable

C∞ map fk−1 : Mn → Np is constructed

such that:

(1) fk−1| ∪j<k Wj is t-regular on N1p−q. (2) fk−1(Uj)⊂ Yi(j).

We now construct a locally deﬁnable C∞ map fk : Mn → Np such that:

(1) fk| ∪jk Wj is t-regular on N1p−q. (2) fk(Uj)⊂ Yi(j).

(3) fk is a ₂δk approximation of fk−1.

Put i = i(k) and λk := p2 ◦ ki ◦ fk−1 ◦

(hk)−1 : Cn(2) → Rq, where p2 : Rp−q ×

Rq _{denotes the projection onto the second}

factor. Then λk is a locally deﬁnable C∞

map. For any > 0, there exist (q, n) matrix

A and (q, 1) matrix B such that:

(1) The absolute value of any element of

A and B is less that .

(2) Put L(x) := Ax + B. Then 0 is a regular value of λk+ L. Deﬁne fk(x) = { k_i−1(ki◦ fk−1(x) + L(hk(x))φk(x)), x∈ Vk fk−1(x), x∈ M − Uk .

Then fk is a locally deﬁnable C∞ map.

Since we take suﬃciently small A, B, fk

is a ₂δk approximation of fk−1 and fk(Uj)⊂

Yi(j).

Thus fk| ∪jk Wj is t-regular on N1p−q. Let g(x) = limkfk(x). Then g is a locally

deﬁnable C∞ _{map with required properties.}

References

[1] L. van den Dries, Tame topology and

o-minimal structure, 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 structure,

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和歌山大学教育学部紀要　自然科学　第69集（2019）

−₄−

[3] T. Kawakami, Deﬁnable G CW

com-plex structures of deﬁnable G sets and their applications, Bull. Fac. Edu.

Wakayama Univ. 54 (2004), 1-15. [4] T. Kawakami, Equivariant diﬀerential

topology in an o-minimal expansion of the ﬁeld of real numbers, Topology

Appl. 123 (2002), 323-349.

[5] T. Kawakami, Imbedding of manifolds

deﬁned on an o-minimal structures on

(R, +, ·, <), Bull. Korean Math. Soc. 36 (1999), 183–201.

[6] 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.

[7] M. Shiota, Geometry of subanalytic and

semialgebraic sets, Progress in

Mathe-matics 150, Birkh¨auser, Boston, 1997. [8] A. Tarski, A decision method for

ele-mentary algebra and geometry, 2nd

edi-tion. revised, Berkeley and Los Angeles, University of California Press(1951).