数理解析研究所講究録 1290 巻 2002 年 121-126

121

DEFINABLE FIBER BUNDLES AND AFFINENESS OF DEFINABLE
MANIFOLDS$C^{r}$TOMOHIRO KAWAKAMI

1. INTRODUCTION

Semialgebraic sets and semialgebraic maps have been studied and results on themcan be seen in [1]. Let
denote an -minimal expansion of the standard structureis aof the fieldof real numbers. Every definable category on
generalization of the semialgebraic category and the definable category on 7? coincides
with the semialgebraic one [22].
Some of recent results concerning -minimal categories are [4], [5], [6], [7], [8], [10],[11], [12], [13], [14], [15], [16] $[18]$ , [21]. Semialgebraic sets and semialgebraic vector
bundles are studied in [2], [19], [20].
In this note, we are concerned with homotopy property of definable fiber bundles andaffineness of definable
manifolds. Throughout this article, the term “definable” means“definable with parameters in”and definable maps are assumed to be continuous.
The homotopy property for topological vector bundles is established in [9]. Its semialgebraic version, its equivariant semialgebraic version and its equivariant fiber bundle
version are known in 12.7.7 [1], [2] and 2.10 [17], respectively.
We have the following as adefinable fiber bundle version of this property.
$\mathcal{M}$$0$$\mathcal{R}=(\mathbb{R}, +, \cdot, <)$$\mathcal{M}$$\mathbb{R}$

$0$

$G$

$G$

$)$

$C^{r}$$\mathcal{M}$

Theorem 1.1 (1.1 [15]). Let $\eta=(E,p, X, F, K)$ be a definable fiber bundle over a definable set $X$ with fiber and srructure group K. If two definable maps , : $Yarrow X$
are definably
betw $een$ definable sets are homotopic and is compact, thenand$isomo\gamma phic$ .
bundle
fiber
$F$

$f$

$Y$

$h$

$h^{*}(\eta)$$f^{*}(\eta)$

Let $X$ and be definable sets. Two definable maps , $h:Xarrow Y$ are called de finably
homotopic if there exists adefinable map $H$ : $X\cross[0, 1]arrow Y$ such that $H(x, \mathrm{O})=f(x)$and $H(x, 1)=h(x)$ for all $x\in X$ . By 1.2 [11], if two definable maps between definable
sets are homotopic, then they are definably homotopic. Hence two definable maps in
Theorem 1.1 are definably homotopic.
definable in
We say thatis polynomially bounded if for every function :, there exist anatural number and areal number
such that $|f(x)|\leq x^{k}$ for any$x>x_{0}$ . Otherwise,
is called exponential. One of typical examples of polynomiallyis exponential, then thebounded structures is 72. By aresult of C. Miller [18], if
exponentially bounded
exponential function,
is definable. We call
if for every function :, there exist anatural number and adefinable in
$Y$

$f$

$f$

$\mathcal{M}$$k$$\mathcal{M}$$\mathbb{R}arrow \mathbb{R}$$x_{0}$$\mathcal{M}$$\mathcal{M}$$\mathbb{R}arrow \mathbb{R}$$h$$x\mapsto e^{x}$$\mathbb{R}arrow \mathbb{R}$$\mathcal{M}$$l$$\mathcal{M}$.
,2000 Mathematics Subject Classification
Keywords and Phrases. Definable sets, definable fiber bundles, definable vector bundles,
definable
manifolds.
$14\mathrm{P}10,14\mathrm{P}20,57\mathrm{R}22$$C^{r}$$58\mathrm{A}05,03\mathrm{C}64$$0$minimal,122
TOMOHIRO KAWAKAMI

such that $|h(x)|\leq exp_{l}(x)$ for any $x>x_{1}$ , where $exp_{l}(x)$ denotes the Ithreal number
iterate of the exponential function, e.g. $exp_{2}(x)=e^{e^{x}}$$x_{1}$

is exponentially bounded and $0\leq r<\infty$ , then every
Theorem 1.2 (1.1 [10]). If
definable manifold is affine.
$\mathcal{M}$$C^{r}$

2. DEFINABLE sets, DEFINABLE FIBER BUNDLES AND DEFINABLE$C^{r}$

MANIFOLDS,
be astructure expanding:is the additive (respectively the multiplicative) function$n(i)\in\mathbb{N}\cup\{0\}$,is a,is afunction, each:of , eachrelation, and each is aconstant. We say that (respectively ) is an $m$-place function
is afunction (respectively
symbol (respectively an $m$-place relation symbol) if :arelation).
isAterm is afinite string of symbols obtained by repeated applications of the following
two rules:
1. Variables are terms.
,
are terms, then the concateand ,2. If is an $m$-place function symbol ofis aterm.
nated string
Note that if $m=0$ , then the second rule says that constant symbols (0-place function
symbols) are terms.
, where eachis either avariable, aAformula is afinite string of symbols, one of the, $,$ ,,
symbolsofthelogicalsymbol,one
function symbol, arelation
brackets $(, )$ , or comma ,. Arbitrary formulas are generated inductively by the following
three rules:
andare formulas.
1. For any two terms and ,2. If is an $m$-place relation symbol and , , are terms, then $R(t_{1}, \ldots, t_{m})$ is aformula.
, and the, the disjunction
3. If and are formulas, then the negation
avariable,thenandare formulas. If is aformula and isconjunction
are formulas.
) if it is defined by aformula (with parameters).
is de finable (inAsubset $X$ of,suchand elements ,Namely, there exist aformula
$X=$
,
,
,
,
,
in
is
true
,
that
}.
{ }
be definable sets. We say that acontinuous map : $Karrow L$
andLet
is definable. Adefinable) if the graph ofis de finable (inmap : $Karrow L$ is called adefinable homeomorphism if there exists adefinable map: $Larrow K$ such that $f\circ h=id$ and $h\circ f=id$ .
. We,
An open interval means something of the form $(a, b)$ ,-minimal(order-minimal) if every definable subset of is afinite union of points
call
open
intervals. Remark that 7? is -minimal[22]. For example, $N$
andis an expansion of 7? but not -minimal because adefinable subset of in $N$ is not a
finite union of points and open intervals.
Notice that one can consider adefinable category in astructure which is not 0-minimal.
But this category does not have satisfactory properties

Let$\mathcal{M}=(\mathbb{R}, +, \cdot, <, (f_{i})_{i\in I}, (R_{j})_{j\in J}, (c_{k})_{k\in K})$$\mathcal{R}=(\mathbb{R}, +, \cdot, <)$$\mathbb{R}^{2}arrow \mathbb{R}$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}+(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}\cdot)$$\mathbb{R}$$f_{i}$$\mathbb{R}^{n(i)}arrow \mathbb{R}$$R_{j}\subset \mathbb{R}^{n(j)}$

$R$

$f$

$c_{k}$$n(j)\in \mathbb{N}$$\mathbb{R}^{m}arrow \mathbb{R}$$f$$R\subset \mathbb{R}^{m}$$\mathcal{M}$$f$$t_{1}$$\ldots$$t_{m}$$f(t_{1}, \ldots, t_{m})$$s_{1}\ldots$$s_{k}$$s_{i}$$=$$t_{2}$$t_{1}$$t_{1}$$t_{m}$$\ldots$$\phi\vee\psi$$\neg\phi$$\psi$$\phi\wedge\psi$$\exists,\forall$$t_{1}>t_{2}$$t_{1}=t_{2}$$R$$\phi$$\wedge$$\neg$$(\exists v)\phi$$v$$\phi$$(\forall v)\phi$$\mathcal{M}$$\mathbb{R}^{n}$$\phi(x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{m})$$\{0$$\ldots$$a_{n})\in \mathbb{R}^{n}|\phi(a_{1}$$\ldots$$a_{n}$$b_{1}$$\ldots$$b_{1}$$b_{m})$$\ldots$$\mathcal{M}$$f$$L\subset \mathbb{R}^{m}$$K\subset \mathbb{R}^{n}$$b_{m}\in \mathbb{R}$$f(\subset K\cross L\subset \mathbb{R}^{n}\cross \mathbb{R}^{m})$$\mathcal{M}$

$f$

$h$

$a\in \mathbb{R}\cup\{-\infty\}$$\mathcal{M}$$b\in \mathbb{R}\cup\{\infty\}$$\mathbb{R}$$\mathit{0}$$=(\mathbb{R}, +, \cdot, <, \mathbb{Z})$$\mathit{0}$$0$$\mathbb{Z}$$\mathbb{R}$123
DEFINABLE FIBER BUNDLES AND AFFINENESS OF DEFINABLE$C^{r}$

MANIFOLDS

map : $Uarrow V$
be definable open sets and $0<r\leq\omega$ . ALetandmap if it is definable.
is called ade finableis aLet
be adefinable set and $0<r\leq\omega$ . Adefinable map :mapadefinable
and
open
set
map
if
there
exist
adefinabledefinable
such that $A\subset U$ and $f–F|A$ .
The following theorem states some of useful properties of definable sets and definablemaps.
$U\subset \mathbb{R}^{n}$$C^{r}$$V\subset \mathbb{R}^{m}$$f$$C^{r}$$f$$A\subset \mathbb{R}^{n}$$Aarrow \mathbb{R}^{m}$$C^{r}$$U\subset \mathbb{R}^{n}$$C^{r}$$F:Uarrow \mathbb{R}^{m}$cell decomposition (e.g. 7.3.3.2 [4])]. Suppose that.
into
, tfiere exists a decomposition of,(a) For any definable set,,,.
definable cells partitioning, there exists a decomposition into,(b) For any definable function :iscells partitioning
such that each restr iction $f|C$ :definable

Theorem 2.1.

(1) [Definable$C^{r}$$0\leq r<\infty$$A_{1}$$\ldots$$C^{t}$$A_{1}$$f$$\ldots$$Aarrow \mathbb{R}$$A_{k}$$A\subset \mathbb{R}^{n}$$Carrow \mathbb{R}$$A$$C^{r}$

definable$\mathbb{R}^{n}$$A_{k}\subset \mathbb{R}^{n}$$a$

map for eachof the decomposition.
be a definable set and(e.g.
(2) [Definable triangulation(8.2.9 [4])]. Let,,
definable subsets of S. Then there exist a finite simplicial complex $K$
definablyinsuch that maps and eachand a definable map :homeomorphically onto a union of open simplexes of K. If is compact, then we
can take $K=\phi(S)$ .
$9.1.2[4])$ ]. Let $X$ and $Y$ be definable sets and(3) [Piecewise definable trivialization$XarrowY$:a definable map. Then there exist a finite partition
of $Y$ into
definable sets and definable homeomorphisms : $f^{-1}(T_{i})arrow T_{i}\cross f^{-1}.(y_{i})$ such that$f|f^{-1}(T_{i})=p_{i}\circ\phi_{i},$ $(1\leq i\leq k)$ , where
denotes theand :projection.
$C\subset A$$C^{r}$$S\subset \mathbb{R}^{n}$$S_{1}$$\ldots$$S_{k}$$\mathbb{R}^{n}$$\phi$$Sarrow \mathbb{R}^{n}$$\phi$$S$$S_{i}$$S$$(\mathrm{e}.\mathrm{g}.$$\{T_{i}\}_{i=1}^{k}$$f$$\phi_{i}$$y_{i}\in T_{i}$$p_{i}$$T_{i}\cross f^{-1}(y_{i})arrow T_{i}$An equivariant version and an equivariant
version of Theorem 2.1 (3) are proved in[14].
Agroup is ade finable group if is adefinable set and the group operations $G\cross Garrow$
and $Garrow G$ are definable. Asubgroup of adefinable group is ade finable subgroup of
it if it is adefinable subset of it.
Let be adefinable group. Adefinable set with a de finable action is apair
consisting of adefinable set $X$ and agroup action 0: $G\cross Xarrow X$ such that 0is a
definable map. This action is not necessarily linear.
Adefinable space is an object obtained by pasting finitely many definable sets together
along open definable subsets, and definable maps between definable spaces are definedsimilarly (see Chap. 10 [4]). Definable spaces are generalizations of semialgebraic spacesin the sense of [3].
$C^{r}$

$G$

$G$

$G$

$G$

$G$

$(X, \theta)$( 1) $X$ , $F$, $K)$ is called adefinable
Definition 2.2. (1) Atopological fiber bundle$F$$X$
fiber bundle over with fiber and structure group $K$ if the following two conditions are satisfied:(a) The total space $E$ is adefinable space, the base space $X$ is adefinable set, thestructure group $K$ is adefinable group, the fiber $F$ is adefinable set with aneffective definable $K$ action, and the projection $p:Earrow X$ is adefinable map.
(b) There exists afinite family of local trivializations
is afinite openis adefinable open subset of $X$ ,of such that each$\eta=$$\{U_{i}, \phi_{i} :$\eta$$U_{i}$

p^{-1}(U_{i})arrow U_{i}\cross F\}_{i}$$\{U_{i}\}_{i}$124
TOMOHIRO KAWAKAMI, where: $p^{-1}(x)arrow F$,
covering of $X$ . For any $x\in U_{i}$ , let$U_{i}\crossFarrow
F$
, the
.
any
For
and
with
projection
stands for the: $U_{i}\cap U_{j}arrow K$ is adefinable map. We calltransition function
these trivializations de inable.
Definable fiber bundles with compatible definable local trivializations are iden$\phi_{i,x}(z)=\pi_{i}\circ\phi_{i}(z)$$\phi_{i,x}$$i$$U_{i}\cap U_{j}\neq\emptyset$$j$$\pi_{i}$$\theta_{ij}:=\phi_{j,x}\circ\phi_{i,x}^{-1}$$f$tified.

and ( $=(E’,p’, X’, F, K)$ be definable fiber bundles whose, respectively. Adefinableanddefinable local trivializations are$EarrowE’$map :is said to be ade finable fiber bundle morphism if the following
two conditions are satisfied:.
(a) There exists adefinable map : $Xarrow X’$ such that$x\in
U_{i}\capf^{-1}(V_{j})$, the mapand for any(b) For any , such that$f\{j\{x$ ): $U_{i}\cap f^{-1}(V_{j})arrow K$ is a
: $Farrow F$ lies in $K$ , anddefinable map.
Adefinable fiber bundle morphism : $Earrow E’$ is called ade finable fiber bundle
isomorphism if $X=X’$ , $f=id_{X}$ and there exists adefinable fiber bundle morphism. We say that
, and
: $E’arrow E$ such that $f’=idx$ ,is definably trivial if is definably fiber bundle isomorphic to the trivial bundle
pro;, $X$ , , $K$ ), where proj : $X\cross Farrow X$ denotes the projection onto the(first factor.
is a(3) Acontinuous section : $Xarrow E$ of adefinable fiber bundle
map.
map
:
is
,
any
adefinable
the
if
for
definable section
is aprincipal definable(4) We say that adefinable fiber bundle$K$
fiber bundle if $F=K$ and the action on is defined by the multiplication of $K$ .

(2) Let$\eta=(\mathrm{F},\mathrm{p}, X, F, K)$$\{V_{j}, \psi j\}j$$\{U_{i}, \phi_{i}\}_{i}$$\overline{f}$$f\circ p=p’\circ\overline{f}$

$f$

$i$

$U_{i}\cap f^{-1}(V_{j})\neq\emptyset$$j$$:=\psi_{j,f(x)}\circ\overline{f}\circ\phi_{i,x}^{-1}$$f_{ij}$$\overline{f}$$\overline{f’}\circ\overline{f}=id$$\overline{f}\circ\overline{f’}=id$$\overline{f’}$$\eta$$\eta$$F$$X\cross$$\eta=(\mathrm{F},\mathrm{p}, X, F, K)$$s$$U_{i}arrow U_{i}\cross F$$\phi_{i}\circ s|U_{i}$$i$$\eta=(\mathrm{F},\mathrm{p}, X, F, K)$$F$.
Definition 2.3. Suppose thatsubmani fold
(1) Adefinable subset $X$ ofis called a -dimensional definable$x\inX$(a definablediffeomorphism
existsadefinable
there
any
if
for
of
of thefrom some open definable neighborhood
homeomorphism if $r=0$ )such that $\phi_{x}(0)=$of inonto some open definable neighborhood
origin in. Herethose which the lastdenotes the subset of,$(n-d)$ components are zero.
manifold with afinite system(2) Ade finablemani fold $X$ of dimension is a,
is an open definable
and
that
for
each
such
of charts:
is aand the map
subset of
).
$r=0$
call
this
homeomorphism(aifWe
diffeomorphism
definable
definable
manifolds with compatible atlases are identified.
atlas de finable . Definable$X$
charts { :manifold with definable(respectively ) be adefinable(3) Letmap : $Xarrow Y$ is said to be a). A:
(respectively
andis open and definable inmapany
if foranddefinablemap.
:
is adefinable
the map$X$$X$
manifolds. We pay that is definably $C^{r}dif$ feomor(4) Let
and be definablemaps
phic to (definably homeomorphic to if $r=0$ ) if one can find definable$XarrowY$:
and $h:Yarrow X$ such that $f\circ h=id$ and $h\circ f=id$ .
diffeomorphic
manifold is said to be $af$ fine if it is definably(5) Adefinablesubmanifold of some .
(definably homeomorphic if $r=0$ ) to adefinable$0\leq r\leq\omega$$C^{r}$$d$$\mathbb{R}^{n}$$C^{r}$$\mathbb{R}^{n}$$U_{x}$$\phi_{x}$$V_{x}$$\mathbb{R}^{n}$$x$$\phi(\mathbb{R}^{d}\cap U_{x})=X\cap V_{x}$$\mathbb{R}^{n}$$d$

U_{i}arrow \mathbb{R}^{d}\}$$C^{r}$$i$$\phi_{i}(U_{i}\cap U_{j})$$j$$\phi_{i}(U_{i}\cap U_{j})arrow\phi_{j}(U_{i}\cap U_{j})$$\phi_{j}\circ\phi_{i}^{-1}|\phi_{i}(U_{i}\cap U_{j})$$\mathbb{R}^{d}$$x$$\mathbb{R}^{d}$$C^{r}$$\{\phi_{i} :$\mathbb{R}^{n}$$C^{r}$$C^{r}$$C^{r}$$Y$$U_{i}arrow \mathbb{R}^{n}\}_{i}$$C^{r}$$\psi_{j}\circ f\circ\phi_{i}^{-1}$$Y$$C^{r}$$C^{r}$$\{\psi_{j}$$i$$V_{j}arrow \mathbb{R}^{m}\}_{j}$$C^{r}$$\phi_{i}$$f$$j\phi_{i}(f^{-1}(V_{j})\cap U_{i})$$\mathbb{R}^{n}$$C^{r}$$\phi_{i}(f^{-1}(V_{j})\cap U_{i})arrow \mathbb{R}^{m}$$C^{r}$$Y$$Y$$C^{r}$$f$$C^{r}$$C^{r}$$C^{r}$$\mathbb{R}^{l}$

125
DEFINABLE FIBER BUNDLES AND AFFINENESS OF DEFINABLE C’ MANIFOLDS

3. SKETCHES OF proofs

Theorem 1.1 is obtained from the following three results.
Lemma 3.1 ([15]). Let$A$

be a definable set, $X_{1}=\{(x_{1}, x_{2})\in A\cross[0, 1]|f_{1}(x_{1})<x_{2}\leq$, $X_{2}=\{(x_{1}, x_{2})\in A\cross[0, 1]|f_{2}(x_{1})\leq x_{2}<f_{3}(x_{1})\}$ and y7 $=(E,p, X, F, K)a$definable fiber bundle over $X=X_{1}\cup X_{2}$ , where : $Aarrow[0, 1]$ , $(1 \leq i\leq 3)$ , are definable
are definably trivial, then is definably
andfunctions with $f_{1}<f_{2}<f_{3}$ . Iftrivial.
$f_{2}(x_{1})\}$$f_{i}$$\eta|X_{2}$$\eta|X_{1}$$\eta$

Lemma 3.2 ([15]). Let $X$ be a compact definable set and $\eta=(E,p, X\cross[0, 1], F, K)a$definable fiber bundle over $X\cross[0, 1]$ . Then there exists a finite definable open covering
is definable trivial.
of $X$ such that each$\eta|(U_{i}\cross[0, 1])$$\{U_{i}\}_{i}$Theorem 3.3 ([15]). Let $X$ be a compact definable set, $r:X\cross[0,1]arrow X\cross[0, 1]$ , $r(x, t)=$$(x, 1)$ and $\eta=(E,p, X\cross[0, 1], F, K)$ a
definable fiber bundle over $X\cross[0, 1]$ . Then there.
exists a definable fiber bundle morphism $\phi:Earrow E$ with
$p\mathrm{o}\phi=r\mathrm{o}p$Let:$\mathbb{R}arrow \mathbb{R}$,$n\in \mathrm{N}$$e^{-exp_{n-1}(1/x^{2})}$be the function defined by $e_{n}(x)=\{$,$x\neq 0$$x–0$0,
where $exp_{0}(x)=x$ . Then elementary computations show the following proposition.
$e_{n}$

Proposition 3.4 ([10]). (1) For any polynomial function $P(x_{1},$\ldots ,$\lim_{xarrow 0}P(\frac{1}{x}, exp_{1}(\frac{1}{x^{2}})$(2) Every

Since$\mathcal{M}$$e_{n}$is a$C^{\infty}$,$\ldots$$x_{n})$’

in n variables,, $exp_{n-1}( \frac{1}{x^{2}}))e_{n}(x)=0$ .

function.

is exponentially bounded, asimilar proof of C.14 [7] proves the following

proposition.

andProposition 3.5 ([7], [10]). Let
be a non-empty compact definable subset of$f^{-1}(0)\subset
g^{-1}(0)$exponentiallyis.
, two definable functions onsuch thatIfbounded, then there exist a $nat$ rural number and a positive constant such that $e_{k}(g)\leq$$c|f|$ on
.
$A$

$f$

$\mathbb{R}^{n}$

$A$

$g$

$\mathcal{M}$

$k$

$c$

$A$

Theorem 1.2 is proved by using the above two propositions.

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DEpARTMENT OF MATHEMATICS, FACULTY OF EDUCATION, WAKAYAMA UNIVERSITY, SAKAEDANIWAKAYAMA 640-8510, JApAN$E$-mail address: kawa@center.wakayama-u.ac.jp