B - 1. ,4cA B - 1. 24cC-. IDA cC - 1. A A A =C -. '11. =C -. '11â¢1 c C -. 'C-. 'DÅC- 1. DcB - 1. A , we have B- 1. A =C -. 'D and then. (C -. ID) 2 c C - 1. C-. ID DcC - 1.
J . M ath. K yoto Univ. ( JM KYAZ) 14-2 (1974) 287-318
Dependence of local homeomorphisms and local Cr-structures By
Shuzo Izumi (Received July 30, 1973)
Introduction L e t E (X ) b e th e s e t o f a ll C r-structures on a topological manifold X . T h e s tu d y o f t h e diffeomorphism c la s s e s o f E ( X ) h a s b e e n an im portant subject i n differential topology. W e , however, consider E(X) itself paying atte n tio n to its d e p e n d e n c e relation ( c ) defined below. W e give som e results w hich a r e chiefly reduced to a lo c a l theory of homeomorphisms o f R " . W e b e g in b y th e following problems. P ro b lem I G . F o r g i v e n C r-structures 9 , g ' e E ( X ) , c a n w e fin d a th ird g " e l ( X ) su ch th a t 9 c g " , 9 ' c .9 " ? Problem I I G . F o r g i v e n Cr-structures g,
e E (X ), ca n w e fin d
a th ird g " e l ( X ) su ch th a t 9 " c 9 , 9 " c g '? These problem s a r e q u ite ra w a n d m o re suitable presentations will b e fo u n d ac co rd in g to th e sta g e s o f o u r stu d y . F irst, w e lo c a liz e the problems. B y a local C r-structures o n R " w e m ea n th e g e rm a t 0 o f a C r structure o f a neighbourhood o f 0 e R" (w e shall give a m o r e detailed definition in S ection 1). B y a local homeomorphismw of R n w e m ean t h e g e r m a t 0 o f t h a t homeomorphism between neighbourhoods o f 0 (1 ) W e use this term following Sternberg, who investigated local homeomorphisms in connection with the theory of flow an d found normal form s o f conjugate classes of local diffeomorphisms.
288
Shuzo I zuini
w h ic h le a v e s 0 fix e d . A ll the local hom eom orphism s of R " fo rm a group , c w it h respect t o th e operation (o) induced by the composition o f m a p s . If tw o local Cr-structures 3 a n d g ' a re given, take admissib le ch a rts f ( t ) a n d g ( t) o f their representative C r-structures in neighs
bourhoods of O.
The germ f og ' of f og q t) is a local homeomorphism -
-
of R . W e s a y t h a t g is dependent o n 2 ' a n d w rite 2 c 9 ' when fog ' i s o f c la s s C r . T h is defines a n o rd e r in t h e space o f lo c a l Cr structures. T h e n w e o b t a i n t h e lo c a l fo rm s , (I L ) a n d (II L ) , of th e problems (I G ) a n d (II G ) in a n obvious w a y . G lo b a l dependence is defined by the pointwise dependence of admissible charts (see Section -
4). ( I L ) a n d (II L ) a r e e a sily re d u c e d to p ro b le m s a b o u t th e subsemigroup „E OE„C that consists o f th e germ s o f c la ss Cr. Church's sm oothing lem m a (2.1) gives a sufficient c o n d itio n fo r ( II L ) to b e (2 )
r
answ ered in th e a ffirm a tiv e . N e x t, w e sh o w b y t h e examples (2.4), (2.5) a n d (2 .6 ) neither (I L ) nor (II L ) is unconditionally answ ered in th e affirm a tiv e . These exam ples reveal som ew hat com plicated aspects o f „ C relative t o E . I f w e re stric t o u rse lv e s to the 1-dimensional c ase w e o b ta in a sh a rp positive result ( a s a consequence o f Theorem 3.1): Theorem 0 . 1 . I f n = 1 a n d if ( IL ) is an s w e re d i n t h e affirmative f o r a p a i r ( 2 , 9 ') then ( II L ) is also.
In substance th is assertion gives th e sim p le e x p re ssio n E,7 o f th e subgroup o f , C generated by E ( C o r o lla r y 3 .7 ) . A difficulty in th e proof lie s in th e removal o f th e singularities o f a function which is sm o o th a lm o st e v e ry w h e re . W e re m a rk th a t th e c o n v e rse o f th e 1
1
1
r
above theorem is false fo r every R" (Example 2.5). A ll o f these local results yield corresponding answers to (I G ) and (II G ). T h e p o sitiv e results for (H G ) a re best explained by a reduction th eo ry o f s tru c tu re s (4 .1 a n d 4 .4 ). W e can also treat subsem igroups of the hom eom orphism g ro u p o f a C r m anifold (see (4.6), (4.7) and (2 ) A su b sem igro u p o f a g ro u p is a subset closed with respect t o multiplication.
L ocal hom eom orphism s an d lo c al Cr-structures
289
(4.8)). I n t h e la s t section w e rem ark on the diffeom orphism class of a C - s tr u c tu r e in some dependence relation w ith a n o th e r. B y the way, transform ing S choenflies theorem , w e sh o w th a t M iln o r 's g r o u p „A o f exotic spheres is a quotient group o f a subgroup o f „C (Theorem 5.3.) T h e essential p a r t, S e c tio n 2 a n d 3 , o f this paper depends upon real analysis. E specially w e u s e a b a s ic knowledge o f measure theory a n d t h e sim plest c ase o f S a rd 's th e o re m . W e r e f e r to r e s u lts in ordinary differential topology only in t h e additional section 5. I w is h to express m y thanks to P rofessor S . M izohata for various (3 )
advices an d to P rofessor R .C . K irby fo r kind encouragement. Preliminary consideration
1.
F ir s t w e c la r if y t h e relationship betw een local homeomorphisms and local C r-structures. W e sh a ll o fte n e x p ress a property o f a germ b y one of its representative. L e t u s f i x a n a f f in e co o rd in ate sy stem t =(t' , t ,..., t") o f th e Euclidian n-space t assins naturally a C r-structure on R ' . T h e n it d e te rm in es t h e following subsemigroups o f t h e g ro u p „ C of local 2
homeomorphisms at 0 e R : ,,D, = {the elements o f „C o f class
e
except at O},
,,Er = {the elements of „D o f class Cr} r
„J = {the elements o f „E r-fla t at 0} . r
r
H e re a (lo c a l) h o m e o m o rp h ism o f c la s s C r d o e s n o t m e a n t h e Crdifferentiability o f its inverse. r-flatness m ea n s v a n ish in g o f a l l the partial derivatives o f o rd e r u p to r. L e t „D,! (resp. „E:, „J!) d e n o te s th e s u b s e m ig r o u p o f ,,D r (resp. „E , „J ) consisting o f all t h e e le m e n ts w h o se Ja c o b ia n d o not v a n ish e x c e p t a t O . L e t „E* * b e the subsem igroup of „C .K consisting o f a l l t h e elem ents w hose Jacobian d o n o t v a n is h a t O . W h e n w e d o n o t n e e d to re stric t th e d im e n sio n o f th e Euclidean space, th e left r
r
r
(3 ) A c co rd in g to S a rd [8 ], this case is d u e t o M . M orse.
290
S huz o /z um i
su b sc rip ts n o f „ C , ,,D e t c . a r e o m itte d , r is alw ays assum ed to be a natural num ber o r co. D : a n d E,.** a re subgroups o f C a n d satisfy r
D:
-
I
=D,*,
E:*-1 =E' k*. r
It is also easy to see that
E . E; = J .J ; ', r 1
r
r
E ; .E , = J;'Jo , E: ' 0E: = -
r
N ow take a n element f e C a n d its representative f ( t ) defined o n a neighbourhood U o f 0 . Regarding (U , (tof )(0) a s a c h a rt w e c a n d e fin e a n o th e r C r-stru c tu re o n U . T h e s e t e r ( f ) = ( f ) o f th e germ s o f a ll th e C r fu n c tio n s a t 0 in t h e new sense is determ ined by th e germ f a n d is independent o f th e choice o f t h e representative ( U, a n d only f(t)). I t i s c l e a r t h a t g r (f ) r ( g ) (resp. g r (f ) =g ,.(g )) i f if f e E . g (resp. fe E **.g). T h u s a local C r-structure defined in th e r
r
introduction c a n b e n a tu ra lly id e n tifie d w ith a s u b r in g g r ( f ) o f th e r in g o f germ s o f continuous functions at 0 e R . Dependence of local Cr-structures ju s t corresponds t o in c lu sio n o f su b rin g s. T h e left coset n
space E* \ C can be also identified w ith th e space of local Cr-structures. W e rem ark that E * * is n o t a n o rm a l subgroup o f C , which we see in the following: r
Example 1.1.
Put a(x) =2 + sin [(42)— rt log Ix!], 13(x)=5 ot( ) d . o
T he germs f , g a t 0 o f th e maps f (t)=66(t ), t , t ,..., 1
2
3
,
g(t)=(eti, t , 1 ,..., 2
belong to
3
E :* respectively. Suppose th a t E ,” is a norm al subgroup
L ocal hom eom orphism s and local Cr-structures
291
o f D ` . h =f - 'o g o f b e lo n g s to E ,* * . T he first component 111 ( t) o f h(t) depends only upon t a n d then w e have 0 )> 0
Putting k (x )=h'(x , 0, 0, ..., 0) we have fiok(x) e fl(x)
(0 x 0)
0
(t =0)
—2exp(1/t)
(t< 0),
— exp (2/t)
—1110g t
(t > 0)
exp ( —1 /t)
0
(t =0)
1/log( — t)
(t < 0),
exp ( — l/t) 0( 1
#(0=
=
0
/i(t) =
(t > 0) 0
) < 0) , (t > 0) (t = 0)
—exp (1 /t).( t < 0 ).
Then w e have f = 0 ° 4 9 e 1 E „o tp ,
f= 0
g = cto tp e ,E ,,o tp ,
g= fl - iotke
- 1
. 0 e t E V o t lf ,
a n d hence
{16%0(f) n 1 (01 c { s (f) u 1
0
1
0
T h u s t h e problem s (I L ) a n d ( II L ) a r e answ ered in the affirm ative for the germs f an d g. N ow w e m ake som e notational agreem ents. W e put ItM =
t tt i= f o r t = (t 1 , t 2 t n ) e R . L e t f ( t ) b e a C r - f u n c t i o n o n a domain 52cR n , A a subset o f 0 , v =(v , v ,..., A/) a n elem ent of {0, 1, 2,..., r}" ± v 2 + ••• v . W e put ivi n
i
2
n
f(t) II =
sup sup
01 vlf (t )
(Ot )ri(8t
2
)v
2. • • • (
a tn )v-
N ext, le t I b e a n interval o f R . W e define M r(I) to be th e s e t o f all properly m onotone increasing function o f C r ( I ) . W h e n w e m ention of t h e m e a su re e [A ] o f a subset A c R " , it i s t h e ordinary Lebesgue measure. 2.
Dependence of local homeomorphisms
I n th is section w e sh o w a sufficient condition for the assertions of (II L') in S e c tio n 1 a n d a few exam ples o f f a n d g f o r which (I L')
294
Shuzo /zumi
o r (II L ') a r e answered i n t h e negative. We illustrate these situation in advance.
Here discs mean subgroups of C.
Lemma 2 . 1 (C h u rc h 's s m o o th in g le m m a ).
Let M
and N
be
connected paracom pact C r m a n ifo ld w ith o u t b o u n d a ry , A = {x ,} b e a discrete subset o f N a n d {(U i ,
hoods centered at x i . and C r o n M - f of N
-
t?..... t7))} coordinate neighbour-
I f f(t): M -■ 1 ‘1 is co n tin u o u s a n d p ro p e r o n M
'( A ) , th e n th e re e x is ts a C r homeomorphism g(t)
s a tis fy in g the fo llo w in g :
(i)
g .f( t) is Cr on M .
(ii)
g (t) can be expressed in the form (tj
i(Ilt ill), tt • a
ill),..., t7
s u c h th a t a i ( x ) i s f l a t a t 0 a n d c i ( x ) = 1 f o r x k
i
( > 0 ) w h e re {t:
I till -5 k } i
(iii) g ( t ) is a C r diffeom orphism o n N - A a n d th e ide ntity m a p on N - u U i .
I f N is a C s (s _ r) m anifold, g can be chosen to be Cs.
R em rak 2.2. The condition (ii) has a special meaning. The homeom orphism g ( t) induces a new 0-structure (p u ll back by g ( t) ) o n N.
295
L ocal hom eom orphism s and local Cr-structures
I t i s e a s y t o s e e t h a t (ii) m e a n s th is structure i s diffeom orphic to o rig in a l o n e . See (5.3) a n d (5.5). Corollary 2 . 3 . If r Since (E;!) - ' °E'r' f , g E C satisfying fog -
E',!` =( C.') 0E1% s it hold s that .14 = E,7 . E „ (II L ') is answered in the af f irm ativ e -1
1
Example 2 . 4 . H ere w e give f, g E C su c h th a t (II L ') is answered Er. in th e negative i.e. f o g
T ake a function a ( x ) defined o n [ —I, 1 ] w ith th e following properties.
-
(a)
a(x)e W
(b)
a(x) is not differentiable on a dense subset A o f [ — I, 1].
[
1].
(c) a ( 0 ) = O.
W e can construct such a function sum m ing m onotone increasing functions w hose graphs a re o p e n p o ly g o n s. T h e germ f of f(t)— (c((t'), t 2 , t 3 ..... t ) a t 0 belongs to C . I f E » .E = C we may r
assume th a t f = h ' o k fo r some h, k e E , . . F o r sufficiently small positive (5 w e have h o f(t) = k(t) (t e In = [ (5, s r ) a n d hi(t), k (t)e C V " ) . W e put t
l'(x)= 11 (x, t6, 1
)
.
I f (li)' oc((a)00 (a E A n 0, there exists th e differentiable inverse function (19 - 1 (x ) defined i n a neighbourhood o f li oc((a) such that c((x )=(li) - 1 . k i(x , t
,
0 ,— , t) .
T h is c o n tra d ic ts to t h e a ssu m p tio n th a t a ( x ) is not differentiable at a. T hus w e have proved (P )'. a (x ) = 0 o n A nI. Since A is dense and a(x) is continuous, oa(x) = (0W /00)f (t)=0 a n d h o f i s c o n sta n t o n I x t6 x tax • • • x t 3 , a c o n tra d ic tio n . Thus f E'
296
S h az o I zum i
Example 2 .5 . I f r 1 w e h a v e E » o E E , o E a n d hence there is a pair (f , g) c C such that (II L ') is answered in the affirmative and (I L ') in the negative. I f r 2 , th is e x a m p le is w e a k e r th a n th e n e x t simple example 1
r
(2.6).
T h u s the reader can neglect this. Let A be a subset o f I =( 1,1) such that : -
(a)
A is a closed subset o f I.
(b) A has no interior point. (c)
A has positive measure.
(d)
0 is a density point of A , i.e.
[ Ati
11m
n(a, b)] b —a
a i 0 ,6 1 0
= 1.
( e ) A is symmetric with respect to O.
W e c a n r e a liz e s u c h a s e t u sin g a generalized Cantor set. Let (i =l, 2, 3, ...) be the connected com ponents o f I —A and p i , 2r i b e the center and the w id th of I . T a k e a C°-function a(x) such that a(x) =-- — 1 (x — 1), o ( x ) = l ( x and o (x)I E_ , , E M '[— 1, 1]. p(x) is defined by p(x )=o - a. (x). If we put -
1101x) IIR1
= max { P(x)II
i
and r ri
a(x ) -= E 2 i 1=1
ci
fi(x )= t' 2 i
ci
-
-
i= I
[P(
x
[a(
x—
;
ri
pi
p
,
) + sign p l , i
) + sign Al
,
M 0 (/ ). I t i s a consequence of the condition (b) that th e n a(x), 13(x) e Œ(x) a n d ig(x ) are properly monotone in c re a sin g . T o se e the smoothness, confer the proof of (3.2) given later. Obviously we have
0
If x e a(! i ) =fi (1) w e have
(i' i ± ri)=f l(Pi± ri).
L o ca l hom eom orphism s a n d local C r-structures r( - 1 oa- '( x ) = 2 - " c o- ' i x
13' ofl - ' (x ) =
op
-
{
[
—11 ,
—i} .
ri)] 2i
cr'
c
11
i)]
x — ( p r i )]2i
297
Thus we obtain t (x), /3 0 fi-t( x ) —a op 1 1
2
,
—1}
---, 0 (x1rx(p 1 —r 1 ) = 13(p i — ri )) . Similarly cx' oc i(x ) -
fl'
(x)
0
( x t c t ( pi+ ri)= f i(Pi+ rt)) •
W e put f '( t ) = ( a ( t 1 ), cx(t 2 ),. . a ( t " ) ) g - 1 (t) =O (t' ), fl(t 2 ),..., fl(tn))
a n d then fo g
- 1
E n
E,T i o n Er
Assume th a t th e r e e x is t h, k e „E , such
that f og = h o k
-
then f
k =h
Comparing each component on the 0-axis we obtain '.f l.k l(t', 0 , 0 , .., 0 )=1 1 '(t', 0 , 0 , .., 0)
fo r i =1, 2, 3,..., n a n d It' I .45. W e m a y assume that h'(6, 0, 0,
0)> 0
298
Shuzo /zum i
without loss o f generality. Putting
/(x)=k'(x, 0, 0, .., 0) w e have cx'occ'.(,60/)(x) -1
0(flol)(x)
_l ' ( x ) (Oh' 1 at )(x , 0, 0,..., 0) • 1
T h e numerater a n d t h e denom inator o n t h e rig h t a r e continuous on
J =[0, .5]. T h e le ft expression tends t o 0 w hen 1(x ) approaches pi + r from th e inside of I . T h u s
1'(x)=0 provided
1(x)= pi ± ri e K, w here K i s a c o m p a c t in te rv a l su c h th a t K 1 (J) a n d p[A n 1(]>0. Then B — { pi+ ri}
nK
is in c lu d e d i n t h e s e t o f t h e critical v a lu e s o f 1. latter includes the set
B eing closed, the
B =A n K a n d th u s h a s positive m e a su re . T h is contradicts t h e theorem o f Sard o n critical value and proves f
o
g- 1 (
1E1
n
Ei
q.e. d.
I
Example 2.6. W e g iv e a n elem ent f e n o t b e lo n g to E ,.0 E ' for Put
=
th a t d o e s
299
L ocal hom eom orphism s and local Cr-structures (i =l, 2, 3,..., n).
fi(t)= / pOlt11 2 ) • ti
Then the germ f of the map f(t) =(.f (t), f
f
l
n
(t))
a t 0 belongs to Dt , because o
OW I f • •
f
2
o(i', t2
9 . . . ,
n
tn)
d e t(titi + jilla
)
2
) + (Iltd 0
[N/ POIM n 2
2
011 +O(110 2n
2 n+ 2
2 n "
)
n
)
[Vp(11t11 ]" 2
But f
E . E,7' if r
2.
T o s e e th is let u s assu m e that
f e E, E,7 = J 1
e
r
-
r
cp e J, fo r so m e go e j . T hen VP[119(t)11 ] 9 ( t) and p[ii4 (011 ] • 119(011 a r e r-flat a t 0 . Putting
or
f o
2
r
-
l
9
2
2
(7(X) = 119(X, 0, 0 , — . O ) ,
t(x)= p[a(x)] • o (x) , -
we have T" = cr•(01 2 ' p (a)+ 201 2 ' p (a) "
'
+ • a" •p' (o ) +
•p (a) .
-
S ince a ( x ) a n d t (x ) a r e r-flat (r 2), t "(x ), o ( x ) , oi(x), o " ( x ) and p[cr(x)] approach 0 and p ia( x ) ] is bounded w hen x tends t o 0 . Then a•(cr') • p"(a), and consequently .
2
'N /
COS
0.
13
approaches 0 . On the other hand, putting 1
4;•(x )=
1
/
cos 13 d s s
-
300
S huz o Iz utn i
we have 1
li m
cos
[a (x )] =- lim
xto
—
s
3 d s —
co .
The previous observation means • d l i r n - - [ a ( x )] = 0 . dx
T hese tw o lim it equations a r e n o t com patible. T h u s w e h a v e proved f eD
,
E . E,. ( r
f
-1
r
2 ) .
3 . The case of R ' I n t h e previous section w e h a v e s e e n th a t th e asse rtio n s o f th e problems (I L') and (II L') suppose some condition on f an d g . Corollary 2.3 gives such a c o n d itio n . If w e restrict ourselves to o n e dimensional c a se , w e o b ta in a sharper result w hich yields Theorem 0.1 in Introduction. Theorem 3.1. Fo r r, s__1 w e hav e the following:
,E,.. ,E,7 E I
(i)
(ii)
r
.
If of
n
g
4)
then
,E,. o f fl 1 E 5 ° (iii) I f f , g for so m e e C .
i
0 4)-
E . go f or some go e ,C then f r
g
E
I
If
(iv) 1
,
,. ( f ) ,
i ' , . ( 9
g 0 ( 0 ) c i d' s ( g ) f o r some
)
f or som e go e C then 1
eC,.
T h e a sse rtio n s (i)— (iv ) a r e m u tu a lly e q u iv a le n t. W e arrange a
L ocal homeomorphisins a n d local Cr-structures
301
few lemmata before proof. Lemma 3 . 2 . Let Q b e a n o p e n su b se t o f R ", f(x ) e C ° (Q ) and let f(x) 0. T hen there ex ists a f unction cp(x) e Cc°(2) such that
0< q (x )< f (x )
i f f (x) > 0 ,
(p (x ) is c o - f l a t a t x,
i f f(x 0 ) =0 .
P ro o f . W e m ay assume that f(x) M < oo without loss o f generality. Put P = {x : x e , f(x)> 0}
a n d ta k e a countable covering V o f P , where V i i s a n o p e n b a ll of ce n ter p a n d r a d i o u s ri < I a n d t h e c lo s u r e Vi is c o n t a in e d i n P. ;
i
Let p (x) b e a C °-function on /in such that p(x)>O (M4 < 1),
1)
P(x)=0
Putting in f f(x ) ,
ci= Ilp(x)1114,
xcv,
we show that (p(x).=
p( x
ci
1=1
\
—
Pt
ri
has th e required properties. Since
Poo(
x p j
x — P i— < rt / 1
)1 (v )
w e have 2 ,n 1
d
p
ci \
( x p r i/
