NOTES ON TO WLOGICAL STABILITY
by
John Mather
Ha rvard Unlvcrrity
July 1970
Thara notea a r o part of rh. flrrt c h p t o r of r marlam of lacturaa
givon
-tor
t
b autbor la tb. r p r k d 1970.
Tha uLLmrta aim of thooo
will bo to prove rha theorom th.t Ch. a r t of topobglcJly a t a b h
mapplngo f o r m a dona. auboot of C ~ ( N , P I for .of f h l t a dlmonriond
mrnLloldr N md P whora N ir compact.
The flrat & & p a ri 4 a atldy
of tho Thorn-Wlaltmoy thaory of otrulflod oata rrrd atratlfiad m&pplry..
Tha cotuaoctbn of tho m a t a r i d h theoa mtoo with tbo thaorom on tha
dondty of t o p o b g k d l y .tabla mrpplwa appaare b 4 11, r b o r o wa glvo
Tbom'a .ocond lmotopy lamnu. Thim roa J t glvoa aufflclwlrt comdltbn~for
two a p p i r y r to ba topologkdly oquivdoat.
p
k t
I
8
,
porLHvo numbar p r w
rbicb rill bo h o d throu#hout
By lfemooth" wo r i l l mean dlflormtiablo of c l a r r
thlr chaptar.
Let
i
bo
M be
8
e)n-mmlfold rlthout boundrr7.
mmooth [i. e.,
c')
r mmooth (I. 0..
.
C'
M
rubnudfold of
I
.
r e rlll moan
.
muboot
By
X
of
H mueb t b t for ovary r E X tboro d r t m a coordinate chart ( e , U )
cP
of cham
r € U m d flX
much that
R~
.uItabls coordlnata plmo
r e do not amrurne t b t
1. clorod.
X
X
daRzlltlon of mubmanlfold that
S
ha. a aai#bborboed U
If
in R'
n Ul = R k rlflUt.
. In th.
detlalnon of mubmmlfold,
Howovar, It foilowm from tho
m
l locmllv clomod L.
in M
much rb.t
X fl U
l r m r-dlmenrioiml rubmmlfold of
X
&a tangent rpmes
TXs
bundle of r-planar In
of
TMr
X
.
rt
for a
r
M
0.
, oach point
1. clomod in
md
rEX
,
Ln
.
U
tbon
I D r p d n t in the Granrnumlan
In whrt follorr 'konvor#snco" mama
convergence In the rtandard topolon on thlr bundle.
Let
Set
X
and
r = dim X
.
Y
bs amaoth mubrrunlfoldm of
M
and lot
yEY
.
3.
Condltlon b.
( 2.
for rubrrunlfoldo of
W I
nn
.
wlU bogla by doflnhg Vhltaoylr cmblUon b
Than wo ertond thlr daflnltloa to rubrrunlfold8
af an arbitrary mmlfold. udng tba doflnltlon In
that condltion b Impliar condltion a
w m ~ l o1.2.
C
3
.ht
the . - d m
wltb
Y
(Wbltamy { 6 ])
bo tho m - u d r md lot
dalotod.
X
and
k the mat
doaota coordinate. for
{mx2
-
0)
(X,Y)
ratlrflom cundltion a at .U polntr of
condition a a t ovary polnt of
(X, Y )
Y
.
an
X
C3
.
.
+ y , then tho r n . ~ n t
x
whlcb L m pmrallol to tho llao jolnlng
through tho orlgln.
For any
I
I
d l 1 donot. tho U.0
and
€ Rn wo 1dr.h
Wo wlll 81.0 ah-
and palam0
f
TxR0
dth
I. In
tho rhndard way.
r
= dlm
ba (rrnooth) rubrmnlbldrn of
X.Y
k t
Y
u c o ~ tho
t arigtn, and that It doom mt matlwe condltion a tboro.
Wo wlU may t b t tba palr
la
and
4th
arm c a m p l n a u l y H c mubnunifoldr of
Y
1.1I
1f
x. y, m
(In Figure 1, wo &va mkotcbod tho LntorrocClon of
R ~ . m
) en X
It lo 0amL1y mom that
.ht
.
Rn
b
g
convarglnn to
y
-to y . sumorn.
. Lot
and
-
f
I
L
&
l
(I(,Y )
rathflom condltlon
rl bo a m.qtlmco of polnt8 In X ,
a mo~ueacoof polnta ln
Y
,
aImo.comarflng
for d l
Lot
y DE Y '
Y'o ray t b t tho p d r
If th. follo4afi hold..
y
rl Sl y,
yCY
L.t
.
X
DEFINITION 2.1.
ratlrllmr condltlba a If It mrtimflsr
ma.
( X * , Y ')
.
bo a rocond pmlr of mubrrudfoldr of
mn ,
and 1st
4,
Then t h r r r exlrtr a l l n s 1 E
tha t
-
1 g TY
(X, Y
but
Y
yl € Y
1& T
. Slnce
yl & x i .
ruch thmt
DEFINITION
- y6Y .
Let
Lot
-
2.2.
W e ray that
M
be a manlfold and
(X,Y)
X.Y
padrfler condltlon b
(pU) about
morns coordlnato chart
.
y
,
rubmaaupldl,
(X, Y )
-
htrlnce
yI x I
and
f k r , thir
0. E. D.
ratlrflsr condltlon b at
y
rubmanlfoldr and let
y€ Y
y
, then
X I . Ip(Un Y]
X and
be a manlfold and
Y
.
PROPOSITION 2.4.
It ratlrfler condltlan a
ratlrfler condltlon b a t
M
y
.
Let
r
- 88 =
ratlrflsr condltlon b ~t
y
conrtsnt.
For. by deflnltlon, the angle
then tbs tsngentr
1
,
'IXy
Lrt
Y
ruppore that
X
and
requence of polntr ln
TIR" = IRn
.
Y
X
a rnd b a r e purely local, we may
a r e rubmanlfoldr of
ruch that
V's murt rhoa that
x,
-
y
TY 5
Y
and
T
.
lFtn
. Let
TXxl * V
xi
.
be a
for moms
Suppore otherwire.
rlcb
(X,Y)
door not ratlrfy condltlon b
convergrta n Uno
(Yhltney ( 6
he the =-art..
.
Let
1).
Q
(X,Y)
T
rlth
.
=
2
9, and
T
n
kL converger
.
3
Let
x. y.
X be the rat
(Y
8
2
be coordlnater for E
+ x3 - m *
2 2
a
0) d t h
(In F l p r e 2 r e have oketched the lntermectlon of
R 3 ) It l r earlly rsen that the palr
and the palr
the condltlon
a betrean the llne TXx and the recant
r h l c h n u k e r an angle
Exampla 2.6.
than
denned
In polar coordlnater, thlr rplral lo glvsn by
Then the p l r
the 8-axlm deleted.
Prnof: Slnce both candltlonr
-
X be the rplral lo EL2
n u k e r a conrhnt angle 4 t h tho radlal vector. and
X
Y be the orlgln.
to a llns
(X,Y)
.
yE Y
&at tbe h n p n t of
.
For the r e r t of thir rsctlon, 1st
ratlrfler condltlon b if Lt ratlrfler cond(tlon b mt
Example 2.5.
r r b v e that the pair
(0.V) about y , r s have that (T(U0
m coordinate chart
(X. Y)
s v s r y polnt
let
Y
y
r o can choorr a r.sumce of polntr
y
P(Vn Yl) ratlaflau condltion b g f l y ) .
In d e w of Lemma 2.2. lf
for e
yi
,
Obvlour.
We ray
( d U 0 X)
p r r l n g through the o t I d n , much that
1 C TY
Y
.
contradlctr condltlon b
Proof:
nn ,
(X.Y)
ratlrfler condltlon a.
ratlrfler condltlon b a t all polntm of
orlgln and that l t doer not ratlrfy condltlon b there.
X
Y
except the
.
7,
6,
PROPOSITION 2.5.
fi
condltlon b
ProoZ:
. &T
dim Y
4
X ,
to
y
there e r l r t r a nequence
x,
In
8
hl
-Y
a
mm
.
depend. on the notlon of blowing up a mmlfold along
Slnce
7
r mm
(where r
yl
on
= dim XI.
.
TY
TqI
Let
converger to an
recant
1
ylxl
57"
Slnce
yl
-
(N U) U %
In
N
TYyl ; hence
Slnce
(X,Y ) ratlrfier condltlon b mt y
rhown
TY
dim X
=
Y
dim
+L G +
and
L
,
Ir orthogonal to
+ > dim T Y = dlm Y
Y
.
x
the
1'
Im orthogonal to TY
J
we have
TY
Y
1
.
.
where
Pt+,
Ry the natural projectlan
mintml=er the d i r a n c e to
I m orthogonal to
ba a nunlfold and
deflned In the followin# r a y .
Since condItlon b Impllem condltlon
.
.
N
T
.
Y
We have
U
a clored rubnunlfold.
By tha
manlfold BUN obtained by blowlng up h' along U, v a wLLI maan tbe rnanlfold
r
a rubmequence if necerrary, v e may muppome the mecantr
to a llne
rubmmnlfold, r h l c h r e
l
define In thlm rectlon.
S T
For i rufflclently large, there m
I a
Y
whlch miaimlrar the dlmtrnca to xi
By parrlng to
Y
Thlr formulation
vhlch converger
By the compactnerr of the Crarnmannlan, we may ruppore, by
(Propoeitlon 2.4).
polnt
Y
In the next rectlan, r e will glve mn I n t r l n d c
B l o w l n ~up.
formulrtlon of condltlon b r h l c h will b e u r d u l Imter on.
dim
parrlng to a rubmsquence If necerrary, that
plane
0 3.
(X, Y ) ratlrflar
It Ir enough to conrlder the c a r e when
y E
.
y
-Y
X.
7E X
Sumore
defined by letting
.
-
U ~ NU
. hence
r
1 mu
B UN
Am a r a t
denoter the projectlva normal bundle of
* : BUN
-
b e the projection of
be the Inclurlon of
N
-U
into
N
To define the differenttable 8 t ~ c h I r aon
the cmre when
N
fr open In
lFtn
and
U
mu
Flrmt,
BUN
mn
x
d-'.'
d x ) =(r.fl(x)] , where
homogeneouo coardlnater
I
- an
,
= IRn flN , where R'
n
-.
-r
Ir the
coordinater.
x=
dth
(7,
i r the polnt In
( x ~ + ~ . xn 1
A
and lettlng
r ~ d - ~ : l d e f l n aam
d follovr.
. 9econdly. If
B(x)
U
v e flrnt conrider
aIF++, I r the rtandard ldantiflcatlon of
U x mpn*'-l
then
.
a : BUN
on
.
coordlnata plane defined by the vmnlrhlng of the lamt
m e n r e have a m a p p h g
U
N , v a mean the mapplng
@
Q. E. D.
f r the dlrjolnt unloa
.
!x
.
I
pl~-'-l
€
nn - nr ,
vlth
It la earily verlfled t b t
rnx
R Pa-:-?# follolr. Let 4 ,
mn .
Let
For
r
,Xn
Xr+Im
+ 1 S 1c n , l
n
cW
Ie a
,x
n
rubrnrnlfold of
denote the coordlnatr of
)
pn-!-'
denote the hornogreour soordlnater h r
2,
a : BUN c.N Ir the aahrral proJectlon. the n r r t r h t m m a t I r
obvlour.
denote (he n b r o t of R ~ ~ . ~ - l d e f l nby
ed
40 ,
and let X
be the r u l -1ued function X,{ * XJ/XI on Z1.
Ji
Then the InterrecHm of
dl N rr ZI Ir the rat deflned by
Xl
where
To prove the renuinla# two rhtementr, r e rot
obeervi thmt there e d r t functionr
r +Is1
Slnce the nupplag
BUN
on
by pulllag bnck the mmalfold rtructure on
Nor. l e t
end let
a Ir InJecHve, r e may deflae m nunlfold rtructure
1"
, Iet
N C be r recoad open rubeet of
-
q : (N,U ) (N'. U')
rp+ : DUN -* BUeN'
fl
be a
a&,N]
dlffeomorphIrrn.
.
U' =
,
c"-I
xi
,
cp and
for
a j n , euchthat
Thir 10 proved am follo+r.
vanlrhee on
of el.
q
U
n
N
mr ,
fl
Slnce for
r
+1
15 n
,
r e have tbmt
h
r e #at that
mrn N' ,
Let
be the Lduced mapplng, deflned by lottlng
be the nupplnl Induced by the dlfferenHm1. and
' P + I ~ :~Yu
lettlng p, IN - U : N - U -. N C - U' be the re.trlcHon of p . Then
%
-mud
TO .how thIr, we first obrerve that
,
Thetsfota, It mfflcen to n h w tlut
-pan.
j
d= 1
r t 1 5i
. Slnca
t!mt
n
.
*ye
+ boldr, where
.
l r r dlffcornorphlrm of c l r r r
I . ennugh to .how
ro that
140f chs.
rnd that XI,
cp,
F,
Is
blject10n and
1. of clrrm
d"
.
d-'.
L l i j n , that
I r of clanr
c''~
(?,1-'
TO ahor
a
(q3-l).
~ i r lt.
In view of
-1
(+I
, e, (7.() n Zk Is the rubnet of
Zk deflned by
( p ; ' ) ( ~ ~I.)
for r +lsf.n
md
m d hence 11 open. It followm that
7,-1 Zl
Ir open.
It r1.o follorr from
14. An Intrtndc formdatlon of condltlon b
m a t f o l d Let AN
donot. lh. diagonal In
-a -111 maan the mlalfold
T h e normal bundle
tangent h a d l a
TN
N2
F(N) obtalnod
of
W
hN
In
.
Let
N
CBB
be a rmooth
By tho fat ..urn
of
N2 along
blmlng up
N2
In a canontcat -.y,
.
N
,
&N '
bo tdentlff ed 4 t h tha
am follorr.
If
x E AN
,
then by
deflnltlon
N o r let
X
Suppnrr Y
The mapping of
Inducer an Imomnrphirm of
identify
x
Into
TNI Q TNx
vitb
TNI
7
x
TNr
with
rhlch r m d r
TNx
.
vQr
to
r
nf
PT(N)
I m clorod.
bo rmaath rabm.nIfoldr of N
PROPOSITION 4.1.
,
F(N) a r e nf two klndr: pairm
and tangsnt dlrectlnnr cm
N
.
(x,y)
4th
x,y € N
N
. Thua. pointm
and
x C y
.
la rlw of the p r d o u r p r a g r a p h . r o obbln the
'Iho pair
(X.Y)
and only if the follorlnl( condttlon holda.
denntem the prnjectlve tangent bundle of
yEY
and l e t
We ure &IDIromorphtrm to
manihld alnng a rubmmlfold, It follorr that
where
Y
-r
Frnm thir ldantlffcatlon and the daflnltlan of the procerr nf blaring up
8
and
polntm In
X
Suppore
x
s d
-
y
(7,)
,
y )
ratloflam condlHon b
y
,
(
y I
much that
Y
COwOrRe. to
and ( T X ~ ) c o n v o r g e ~(in the Gramrmamlan of
r h e r a r = d l m X ) t a . n r-&a
r s T NY .
kT
y
bo any r q u a n c e of
LA fx,)
any requonce of p t n in
~
-
g
r
l
x,
4 y,
lfne 1s my
p h n a r in
f S T -
TN
.
*
.
14.
45.
v%ltner pro-rtratlflcatianr.
mmibld dthout k n d a r y .
8
of
S
.
We ray
X
of
8
M
.
, p.t42].
prove by an argument due to Thorn [ 4
By a pre-mtratlflcaWon
We rkmtch Thom'r argummt for thm two otrata c a r e hmrm.
by p d d w dirjolnt rrnooth
non-trivial care i r when the h o rtrata matidy
.
We will ray &at 8
x
i r locally
and
a r e In
r'
.
X
X
In tblr care
X <Y
Thm only
and the two p o h t r
i r clemd and
.
?=Y UX
X
ham a nmlghborhsed whlch meetr at moot flnltsly
flnite if each p i n t nf
many otrat..
c')
h a rmoath (La.,
5
M , which llm In S
-
W
bm a rubrmt of
S
r e r i l l mom a corer of
rubmanifoldr nf
rtracum
Let
Let
8
rrtirfl.r
For rlmplIclty, r e will ruppore that
the condlHon of h e frontierif for mach
-
itr frontier (Jt
M
Ir compact, though It lw not
M
dlMcult to modlfy the argument to make It work in the carm
X ) n S 10 a union of otrat..
Ir
non-compsct.
We r i l l ray
10 a Fshlhay prm-rtratiflcatlan If it Ir locally flnltm.
8
ratirfler thm conrHHon of the frontler. amd
(X, Y 1 of rtratm of
fnr any p i t
8
.
be a r m w t h retraction. and lmt
~
Let
M
.
*
bm a Whitnwy pre-rtratlflcatlnn of a rubret
Suppore
the frontier of
dlrn Y < dim X
X and
X
. In
Y
a r e rtrat..
W s write
view of ProporlHon 2.5.
S of a rnanlfold
*X
Y
Y <X
If
. It f o l l a r earlly that the rmlation "<"
If
Y
10 in
8
Lat
M
he a manifold.
a Whltney pre-otratlflcatinn of S
x
In the mame con nected component of a rtraturn of
8
x'
M
. and
be two point.
. Then there exlrtr
a homeomnrphlrm
h
ouch t h t
. Thlo follnrr frorn Thom'r theory 1 4 1 and wa will
h(x) r x'
prove it below.
of
and
M
In the care 8
onto l t r d f r h l c h premerver
.
0X 3
(p = 0 ) ,
x
M
,
be a mrnooth functlon nn
and at a point
X
the normal plane to
P
r € X ,
let
T
M
ouch tbat
:N
X
Ir non-degmerate on
In thm renrm t h t the Hmrrlan matrix of
ham rank equal to the codlrnrnrlon of
Now let
definer a partlal
S a clomed rubaet of
. Let
1
p
In
p
at
.
.
X
then
mrder nn 8 ,
Remark.
N be a rmall tubular nmlghborhood of X
Let
watlrflem condition b
(X, Y )
S and
8
har only two strata, It IDgulte eawy t o
of
X
of
v
.
Fnr
x
Let
and
vX bm a omooth vector field on
rtartlng at
c >
xC be two polnts In the rame connectmd component
x
arrlvem at
r C at
compact, and
r :M F
X
11 a rubmerrlon.
r
oufflclently omall.
vector field
on
M
v
-X
and an
rl
.
Mc = (P = F 1 of
Furthermore,
lm cnmpact, and It follovm frorn condition b that
rubrnerrlon for
wuch t h t the trajectory
tlme t = 1
ouMclently omail, the rubret
(r
X
r :Y
F
-.
N
YF =
Ir
M FCI Y
X ir a
It follovo emoily that there l r a
* fI
ouch that
v
l r tangent
Let
M
be a manifold and
r mbmmifold.
X
A tubular neiubborbood
DEFINITION.
(E.c , p J , where
r :E
rmooth function on
X
-
X
T
non-degenerate on the normal plmna to
X
of
m
i m Inner product bundle,
, and p
1. r dlffeomorphlrm of
k vhich commute. witb the r e r o rectlon
m b r e t of
in
i r r triple
M
c
ir
pomitiva
A
B
onto An open
c
of E :
p
n u t r i r of
U
If
r h a m rank equal to tho co-dlmenslon of
at
Ir a rubret of
(E(u,
U
T
= (E, c o y )
X
in
M
.
X
We s e t
the nupping
T
,
IT (
= @Be)
r
rT
.
By the projection armoclated to
cp - 1 : IT1
-
X
T
.
)1%.
dBc-
pT ( 9 ~= ~pT,-
and
T - T'
write
we mean the non-negative r e a l valued function
.
T
X
cC on
with
TI
X
,
1. a.
,
the compoeltlon
-x
1. the identity.
vanirhem only on
Aimo,
X
.
Inclusion
-
.
lpBc
Y. 8 B ~ Y T
X
IT1
IT1
on
-
i r the 0-ret of
and (in tha c a r e
p
PT
than
f : M -. P
X
and
M
a mubrmnifold of
A tubular neighborhood
x
T'
a d
with
-
the differential of
pT
f
if
f
.
IE X
, pT
rT I f 1
to be compatibia with
6 :E
-
E* will be
f
(y~eC
T
to
J ~
.
T'
will be raid to be a mubmcrmion If
x E M
T
of
X
I.
A mapping
if
f -h = I .
in
h
.
f : M -. P
be a rmooth mapping.
M
will be r d d to be compatible
of
M
into ltreU will be sald
-
H :M x 1 M
A homotopy
into itrcif wiil be raid to ba compatlbie with
.
ir
.
ir
t E I ( = [O, 1))
U
a r e i ~ o m r p h i cU
@
2) a t a point
to
rT(pBeC rn mT,
Lf there edmtr a n 1romorpMsm f r o m
TPf(X, 1m onto for each
1
T
T
c' 5 rnln(c, c*] and
such that
Throughout the r e r t of h i s rectlon, let
retraction of
of
if there exirtm a poaitive
Note h a t if thir hold.,
A rmooth mapping
df : TMx
rT 1
..
IU
v a mean
. By the tubular function arroclated to
It L l l o v r from there deflnltion. that
T
m inner product bundle iromorphirm
continuour functlon
rn
the rertrictlon
.
T* = (E*.
c * . ~ ' ) a r e two tubular nelghborhoodr
sad
raid to be an imomorphlrm of
9'
.
X
~ I u ,( P ( ~ I .
dcfiiadar
of
in the r a n r e that the Hcmrian
X
By an irotopy of
f
if
f
Ht = f
of
for aii
M , we will mean a rmooth mapping
M
H:Mxl-M
m8chrh.t
t 6 I , If
dlffsomorphlrm l o t all
the .upport of
if
H :M
h f M :
If
xl
-
h
im
md
l
dll mean tb clamuro of
h
-
ia4
{x 6 M : h(x) 4 x )
Flrmt, under thr hypothamir b t
M
iato itreU,
.
Lik~ae,
H dll m o m tho clorure of
and
f
f
1%
im
Z
mubmurlfold of
MC , m d
To
and
rubmermloa. wr can choora
. Secondly, if
to iemva
4
l
To lU
M
clomed'mubmet of
Hx,tl+r).
M * 10 a mecond mrnibld and XC im
h : (M,X)
Ht:L(-L(
dlffeomatphimm of
i r m imotopl, the m u m r t of
M
t € I .
Ho=ld:M-Y
-
TI lU
T 1 a r e comptlblm with
h
~d
for mom8 open met
Z
much that
n X5 U ,
f
H to bm compatible d t h
U In
X
, and
thea we can choomm
h
Z
1.
l
H
md
point-vlra flxad.
The follorlnp proporition impllem theme mtrtarnslltm, m d ham mome other
(M*,XC) i r a dlffeomorpbimm, thea for any t u b d r r nelghkrhood
wrinklen mr well.
We dll umm it in ltm full genermlity.
(E.t,pl of X wm deflne a tubular neighborhood h+T of XC by
h
+
e
,
t
hml
, h
PROPOSITION 6.1 (Uaiquonemm of hbu1.r
t p) .
X $ M
rubnunifold
We d U hegln by mtrthg m d p r o h g a uaiqumemm theorem for tubulmr
U be an open rubret of
nalghborhoodr, and than we dll derlve M eximtence theorem from the
urriqueaeem thsorem.
Thlm procedure of deducing the erimtence theorem
lSae Figure 3 . ) LA
To
and
T,
then there exlrtm a dlffeomorphirm
polat-wlee fixed much that
that tbara 10 an lrotopy
point d m e flxad.
h,TO
H of
arm tubular nelghborhoodm of
h
- .
TI
of
in
M onto ltrelf which l e a v e
Moreover.
M with
X
h
can be choman mo
hl = H which ieavea
We crn generaliae thlm rarult in variour wayr.
M
X
,
,&
M
,
fix :X
&
U'
-
P
i r a rubmermlon.
be ciored rubrmtr of
V'
and muppome
TO md T,
U* C U
&
be tubulmr nehhborhoodr of
p0 : T0lu
-
vlth
-
l e a v i n ~ X polnt-wire fired. and d t h mupprt in
f
.
T,Iu
h,To JV' u U *
1.
. Then there i r on i r o t o q
T ~ ~ V
u U* *
,
where
(Ht(x), x) f N
for may
tf 1
there i r an imomorphirm
9IU'
bOIUo
h
Hi.
H :M x I - M
.
we can choome
& xC
M
. Airo.
M
p : ~ , T ~ I V ' U U'
.
Moreover. lf
xM
nei~hborhoodof the diamonml La
X
V' G V
X
.
X
.
M
mnd rupvorr there La m ImornorphLam
f
which a r e compatible with
Tha rlmplert unlqusnerm theoram for tubular aeighborhoodr atatem thmt
in clomed and
X
ba m open rubret of
V
from the unlquenerr theoram wmm ruggemted to urn by A. Ogur.
11 X
i r clomad. and
Suppome the
noirhborb6adm).
H
compmtible
V
,
N
much thmt
we can cbomo
T ~ ~UvU.'
much that
H
much thmt
ro thmt
Proof.
mk
let
Let
he mmbeddrd am
in the local csme when
M
9 of
l
,
m = dim M
R
k
c
x Om-*
of
ln
and
mm
p
= dlm P
.
For
k <m
,
WIU ray tb.t r r a r a
,
i r compact and tharr exlata a diffeomorphlrm
V'
onto an open rubret of
diffeomorphl.m
,
cod X
c
mm
.
#(XI
much that
P onto an open rubret of
r
follorrlng diagram cammuter, where
i r dven by
a
mrn-' n +(MI ,
mP
ruch that the
r(xi,
- - am)=
and
b l P m - * , x)
firdrmrmorm. r e m y 8uppoms arch
P
Then
{M
i r rmlatlvmly compact. and h t
Wx
- X ) U { w ~ ) i r a covmr of
finltr refinement of it. vhlcb wm nrvy taka
where ench
There a r e two rtepr in the proof:
Step I.
W
From the hypothemla that
f IX tr
we dlrcard all
W
i
Index the remaining
W
x
Etm
f(Vix)
of
x
Ln
much that
n
XI
= OWxl n mm-'
onto an open rubret of
EtP
for which
bm of thm form
Wl
x
Ux
of
much that the foiioving diagram commuter
sU U
W;U W; U
Wi
ia coumtablm.
n V*
W;'
... . S h c e
compact, it followr thnt
W;
C; Wi
foraii
nX
W; C; Wx,i,
i r compact.
i , by
M
ham
Nor
0 , and r e
r,
n r e can choora clored retr
V'
or
X ) U {W,)
Slncm
W EV
1
.
onto an open submet of
. and a diffeomorphlrm
n U*4
{wi)
.
-
Than we havm
V 0 ~ U U W i U Q , U - * - and
W
xi E X
for romm
{M
Wiam by the poriHve lntmgrrr.
x E X there exirtr an open neighborhaod
M , a diffeomorphirm ex of
@(Wx
(a
a countablm b a d r for it* topology, the coUmction
Reduction to the local care.
a rubmerrlon, It foiiovs that for each
W
411
l r contained in
i
M , ro that there mrirtr a locally
much thnt
, m d the latter 10 rmielvely
.
0
of
Ld
ml&borhoodr.
let
qO
We let
-
#O.#l,pt,.
lato i t d f and rtqumce
of lrotopiar
of iaomorpblmma of tubular
* idnuty ,
be d a f l r d by :
H
H0
.
z , *.
1
H ,H .H
Now r e coartruct by lnductloa a aquence
0 5 t 5 1 , and
be am glven in the atatemart of L e propomitton.
For Qe Lnductive rtep, we muppore t b t
0
H ,H
,.
1
..Hi-l
mucb that
and
I
Now if it i s true that the rtquence
i
H
'Lbea it follorr from the l o u l care of the prapomitlon that
be an op-
compact in
U * U W; U
W,
Wi
,
U* be an open neighborhood of
aad let
. .U w i
In
much that
5 WO
rubmet of
W;
ud
i
W:
much that
becaure
f
.
X
r h o r e clorure liar In
leaving
i I-1
begm TOIF;
I:i~Ol~;-l
n wi
in a neighborhood of any poiat
x € M
,
we can eat
1. relatirely
W:
1
0
U r m l U Wi
local care. lt f o ~ o w rh a t we can conmtnrct an imotopy
compatibla with
fii
ia evenbrally conrtrat
For. let
can be chorm mo that the coaditionr of tbe Induction a r e ratirfled.
WO
1
and
GJx)
-
X
nW
Wl
-
T~I$,
i
H'
of
.
From the
W~
.
(since the latter im erentuaUy conrtmnt In
palnt-vim8 flxed. and with #upport la
Tl
n W,
fl W i ' where
hi = Hii
.
(Thir I D
u;-,c xP J oreo over.
r e may
wa choore
N
ro that the projection
rt
l
neighborhood of any poht).
:N -M
14 proper (where
~f
'
*2
denote8 the pro)*ction on the recond factor). then It l r *arily meen chat
the mequence
xEM
i
l eventually conrtrnt In a nelghborbood of m y point
Gt(x) m
, and tbrt H and
$ have tha required ptopartle4.
Then there eriatr m unlque porltlve deflnita relf-rdjolnt iinerr mrpplng
Tblr C0mpht08 the reduction to the local care.
Proof in the locml care.
Let
To
= ( E O , a O , ~ Oand
)
We will f l r r t conrtruct m lromorpbirm
rhlch e~tOnd8 +0
.
Iu'
-5
4 :E
TI
.
H :W
= (EI,al,~I
H
to havo the
m y lnvertibla rrutrix
r
H-%
H
L : V -. V
prererver Inner productr.
It i r emally reen t b t tblr Ir equivrlent to the arrertlon Out
Remmrk I.
L
requlred propertlam.
much that
W
-
of Lnner product bundiem
mnd then conrtruct Qe irotopy
4
of r e r l numberr ham m unique decomporltLon
L
H ia a p r l t i v e deflnlte a y ~ m e t r l cmatrix m d
where
U
l r an
orthogonrl m t r l x .
The tubular n d ghborhood
a,
Ti
(i r 0,l)
Ei with the normrl bundle vX of
of
r E X , the rartrlctlon of
a1
to the flber
giver a naturrl identificmtion
X
ln
E
1, x
M
.
i r the compoaition
0
-1
% : Eo
B
,
lro(E E )
0' 1
-
El
. We may conrider
H~ : v
C'
E0,=
, only of clam c'-I
lnto
E
.
barir for
r
X
Then
P
by r rectlon
V
Ir tho
ln generrl.
will not he
I. r
; however, we m y approximate fl
arbitrarily clorely on any compact ruhret of
-
Dl
of clrar
a
e.
,
and let
Erirtence.
A a
-
e L1
product.
i
snd
V
j
.
a s
W
LA L :V
be vector rpacer, provided with inner
W
be a vector apace iromorphiam.
L"HL.
be rn orthonormrl
(a . ) be the m t r i r given by
.
aij (Lat. h j l j
{J
It followr from the rpectrrl
ij
brair
el,
-- ,
en
ro Out
(a.) 11 m dlrgonai matrlr:
iJ
Lr the Kronccher delta rymboi).
La an orthonormri brria of
H(fi) a f i / q
.
Then
V
.
Let
Let
f1
U
s
If there were t w ,
.
i r orthogonri.
-1
(H'L)
h k l n g adjoint..
-
W
.
Thm
be given by
H ham the required propertler.
Uniquenerr.
(HL)
a = ai6Lj
U
=Uei)/q
H :W
Ilnerr aigebrr.
LEMMA. LA
5
m d thmt
Let
much thrt
theorem for rymmetric p r l t i v e defhlte rnatricer Out we m y choore tho
fim' ' ' ,fn
# , we will need the followlng well known lemma ln
Hi : V -. V
La rymmetrlc m d poritive definite.
there 8
ij
To conrtruct
w preserver ~ m o product.,
r
Proof of the lemmr.
r r r rectlon of
&era the l t t e r 11 the bundle whore fiber over
.pace of iromorphlrma of
of c h r r
f3
Simllrrly, i t Lr earlly verified thrt there e d r t r a unlque
porltive definite reU-rdjoint linear nupplng
L
Let
2.
Remark
Explicitly, if
r e then obMn
H
Then
H'u-'
md
uH'L
H*
8
.
we would hrve thrt
HL
= H ro that H
.
'
ro
uH'
=
1
H
.
2
= H'U- UH* = H
FUrthermre,
nelghborhood of
dth
HI
?O
r
. Thum
U * U V*
To prove the propomltlon la gsrrrr8l. we take a locally finite f a d l y
Ln r muULdentiy r m l l
5 r (PO r f i 9
g la an lmomrphlmm of
M
of open retm In
H , + T ~ ~ uU* V*
(W )
i
having tbm follorlng propertlor:
TIJu'uvC.
It m
i clear from the conmtructlon thmt
lervem
p o l a t - d r e fired.
X
thr eonmtruction of
G
we may arrange for
Finally. by cboomlng the function
o
ured in
to have rupport in a vary r n u l l aelghborhood of
H
t
topology) a r r e like.
H l r compatible vlth f and
(o
V*
,
be am clore to the identill Iln the compact-open
0. E. D.
N m we atate and prwm the axlrtence theorem for tubular ndgbborhoodm.
PROWSlllON 6.2.
be .a open rubret of
X
. &&
U'
X
Suppore
and let
be a mubset of
a tubular nelghborhood
f i x :X
To
Im a mubmerrion. L A
P
be a tubular nei~hborhoodof
which m
l clomed ln
U
M
X
T
-
X
.
U
(b) each
&
U
Tben there axlrtr
(c)
WI
(w, n X )
im compact, and
1. a cover of
X
.
.
T~u'-T~IU*
much that
Furthermore. we can choore clored retm
Proof.
-
It L m enough to conrider the came when
For, In general. there la an open mubmet
clomed rubmet of
Mo
.
tubular nelghborhood of
mince
X
in
X
Mo
ln
X
M
Im locally cloned ln
Mo
m
I clomed ln
much that
M
M
.
Is a
X
. Clearly a
m
i a tubular neighborhood of
X
i r a cover of
family
(Wi)
X
.
and
M
i 4 countable.
poaitlve integerm.
In
Since
w; E Wi
ha. a countable bad. for itm topology, thm
We 411 muppome that It l r indexed by the
UL= U U WI
For each porLtLve integer we let
U* = U * U w ; U - - . U V ~ ; .
i
(w;]
much that
We let
UO= U
md
u;=
u
...U P
U'.
M.
Now we conmtruct by Induetlon on
The local case of this propomition m
i trivial.
U;
in
X and a tubular naighborhood
i
m open neighborhood
Ti
of
U;.'
We take
U;'
To
of
am
Cmtrol data. Throughout &lo roctlon. lot
47.
liven. P o t tho inductive rtep, r e mupporo Ul:',
ctmotructed.
u,"
Wo let
that
Wl
U;'
= AU
8
4
W b l t a q pro-mtrmlflcatLon of a muboot
Supporo t h t for oach rtratum
thoro d o t open met.
B s uC1
and
W
.
S of
be a mmlfold a d
.
M
ln X rhlcb
U;
neighborhood
- uLl ,
B ,
c - uim1
A
beon
-
ULl
S Wl
U;
Ti,l
bo any open neighborhood of
ir rektlvoly compact in
Sinco
and
h4
A and B
ln U;
ouch
Sinco tho eximtence thoorom
.
M
TX or X In
arrociatod to
IT~J
.
TX
-
r o a r e given
% : ITX I
k t
pX :
projection mmmociated to TX rad
8
X of
-
R
tho
X
&
tubular
donoto tho
mbul.r h c H o m
for tubular ndghborboodm i r true in the Local care, we may choome a tubular
ndgbborhood
T;
of
Wl tl X la
IVl
.
M
,
ndghborboodr of U" n W, t l X La
1-1
rad
Tlml
. Sinco
An B
fhod ouch that
II,T,,~A
!r compatible r l t h
f
ndgbborhood of
n o i g h b o r h d of
U;,
IA
T;
.
I
IA
md
Fbrthermom,
eamily that there m
i
T
-
Ti
b
T;
h
of
IA
k
tl B
of
4
-
if
,
rl V i rl X
. hrthermaro. we may muppore
b
h , T i _ l / ~tl B
AU B
U
:
.
Ti 16 .- h,Ti,lIB
Ti
- u;,~)
X rad Y
X <Y
4re otreta and
,
ht=
h
I.
in
Clenrly
i r the identity in
T; JA n B
-
4
for 4 U
there 10 a
M
much that
Ti
im compntibit 4 t h
m~
f
ouch that both ridem of the oqumtlon 4re defined. i.
) r x l n ~T,I
If
.
m
f
n4po
compmtiblt 4 t h
Ti,1
in
4
nolgbborbood of
tubular ndghborhood
in a neighborhood of
if the follodng( comrnuhtion r t l t l o n a arm m4tirfl.d:
8
i r the ldontity outrido 4n nrbitrmrlly mmall
. Since
Ti
(u:
control d a h for
Ti
{ T ~ ) of tubular nelrbborhoodr r l l l bo u l l d
o n b ltmoU lemvln~ X pointwime
B ; ln partlcukr. that
A
tubular neighborhood
Ti
-
nB
.ad
tl
namely tho rertrlctionr of
i r rektively compact In
we n u y find a dlffeomorphirm
DEFINITION. The famllt
Tb.n r e have two tubular
for alI
U;
neighborhood m
i compstlble with
f
.
T
i
of
U;-l
X in
.
It foUorr
M
frX(m)r f(m)
M
f
into
rylmIE
P
if for all
X
familx
{TX) wtll be 44ld to be
E S and d l m E ITX I
.
we h4ve
.
PROPOSIT~ON 7.1.
4
, a11
IT%(.
, then the fmmlly
ouch that
, and that thir tubular
Q. E. D.
ouchthat
0.
f :M
{TX) of control &tn
-
P
for 8
Ir
4
rubmeraion. then there e d m t ~
which 10 compatible 4 t h
f
.
F o r (he proof of tba proporitlon, we will naad Lomnu 7. 3 bdow. 'Ibe
proof of Lemma7.3Qpmadr on L r ~ 7 2 , r b i c bmale (roupbly rperking) that
rpbare bundla in
T
:E
-
X
Rm-c
noc 5 mm,
t r l d r l bundle over
m.c
wanurntbetrl~le ( E . c . ~ )?
.hr
Rrn"
lunar productl, t = 1
T
.
ldantlflcrtlon mapping
mC
d t b flber
o : BC
md
x
Rm-'
mC
-
-
mm
Rrn
nelghborbood of
In
U
la opan in
U
Rm
m
i the ramtriction map of the
.
,
X
X
neiahborhood of
q:U-R
m
,
.&
r
&
4 mubrnmlfold of
x
E
X
.
lrovenln
U
md
0
(TIt.
t *(.(v))}
T
.
rhare
m
l a mrnootb manifold
Wa r i l l
m.f
l a thlr casa tha tubular ratrsctlon
c
Ir
-
wT : ITI,,
X
that the palr
(Y,XI
2 X
&
X
Y
ba dlrjoint mubmanl foldm of
matlmflar condltlon b
X
M
.
.
&
T
M
ba a tubular
Than thara axirtm a porltlva smooth function
much that tha m r p d n g
.
IU
M
.
then thera axt.tm
M
I
Claarly
1
the r t m b r d tubular
m, c
LEMMA 7.2.
.
LA
LEMMA 7.3.
neighborhood of
T
vlll m a a
cr
('
[V
Irtha
(prodded d t b Itm rtandard
Rm"
{v E E :
Im a propar mapping
E
(*
More g e n a r r l l l if
-
Sc.
a ( ~ 1 ~ and
. Intarlor
admirrlbla Lf
By tha mtandard tubular aalgbborhood
D-INTTLON.
, 1. m.,
denotar the projaction.
with boundary
avary tubular naipbborbood i m locrllv like r m t m b r d axampla.
E
TX
11
8
i r a rubmermlon.
tubular
r coordlnrte chart
xEU,
Proof.
-
c
Lat
be h a rat of
yE
/TI
ruch that the rank of tha
rucbthrt
mapping
d X flUl
= d U )n
Rm"
c = cod X ) and rucb that
(vbara
at
Proof.
If
lmmadlate from the daflnltlon..
T = [ E . c . ~ ) I8
8
I?\*. 0
= @B,
n
B ~ . ) and
X
,
we lat
m
I
for any
tubular aalgbborhood of
m y rmooth p o ~ l t i v afuactloa on
y
X
in
( T I C r= dB(
M
n
BC.)
a 1 ~ 1 =~ p. ( n S(.I
~ ~whcra st.
c*
md
t.
x 6X
=6.
(R x X )
. The lamma l r aquivalent to the ammartLon chat
h a r e axlmtr a neighborhood
Since thir m
i
l
N
of
r
In
C'
X =
mrn"
x 0c
,
and
T
H. ruch that
purely local rtmtamant. It follorm from
Lemma 7 . 2 that it Im enougb to prova the propomltion when
,
m
i the
N fl &
c dim
M = Etm
m
l tha mbndard tubular neighborhood
.
Tm,
mm
. h &I.
~ In- R~
~
R
of
on
mmmC,
pT
and
wT
care
Y <X
1. the orttmgonal projection of
Ir tho function whlch la glven by
TX
y
y
E
IT(
- EtmmC.
near enough to
m
n
R
~
I , LO..
Y C
.
L:.
ln place of
T' T
8
and
y
my
y
of
'I"Y
m
.
-
Rm-c
near
+I
c
plmer
Hence for y
Y
1. tranrrerral to the k.m.1
, ro thrt ( r T , o T J 1 ~ la a rubmerrion
b t
S
1
I
We
we may do lt one rtratum at a dm.,
Let
Ik denote the famllt of rtrata of
and
Is k
dim Y 21
,
.
.
TX
In two otepr, a r follorr.
U1
denote the umlon of all
We let
XI
Uf
of
XI
TI
1
Y <X
for
. In the firat atop. r e conrtruct r
nX
by decremlng Induction on
I
.
1
. la tha
but W r la pmrmlttad, rlnce
lnductlve rtep, we will ehrlnk variou8
ITy
we do lt only a finlte number of tlmer.
Then in tha oscond atop. r m antrnd
to a tubular nelghborbood
F i r r t rtep.
conetruct.
For
I
=k ,
TX
of
we &re
X
.
2
0
,
r o theto l r nothing to
For the lnductlve rtep, we auppore L b t
TI
bar been
gk
.
conrtructsd and t h t the following rpsclrl caaer of the commutntlon
rmltione r r a ratirfled:
and
~ ~ + ~E ( I m
Ty)1
.
if
Y <X
where
.
dim Y 21 + I
+l = T T+l
~
X of
TX of
k
be a atraturn of dlmenrlon
v e 1st
tubular oelghborbood
8
k that the proporltlon Ir true for Ik and
< k , we a r e given a tubular neighborhood
X
construct the tubular nelgbborbwda
For macb
To
For the lnductlve rtep. we ruppore that for each rtratum
dlmsnrion
.
k
To conrtruct tho
I at
m e
2 k , and let Sk denote the unlon of all rtrata ln
Y e vlll rhov by lnduction on
Sk
at
p
Q. E. D.
Proof of Propodtlon 7.1.
of dlmenrlon
whlch lie8 ln
we
- h ~ n that
~
(wT,oT1
of thedlffmrsntial of
at
1. clorr In ~ b a(irarr-mlan
- c + 1 plane
(T
x 0 ~o
1 ywT(yl Ln
hypbtharlr t b t condition b 11 r a t i f i e d impUsr t b t for
in m rpnce to a
.
ITX 1 f l ( T 1 m~ O
, than
on the rtrata of dlmonalon
of the game dlmenrlon.
The kernel of the differeatid of
1. thm o r t h o ~ o l u complemmnt
l
of
P.
MI^^^ x oc1 0 y=
(11
T
hold.)
rlnce there a r e no commutation relatione to bs raHofled among the r t r r t s
~ ( y=l d l r t . ( y , ~ m - c ) L .
Let
X <Y
nor
X
.
.
.
m
E
tl ( T( ~
then
and
thir family of tubular nelghborhoodr matlrfler the commutation reltlonm.
By mhrinklng the
Y
TX
a r e rtrata of dimenrion
If necemrary, we may ruppore that If
<k
X
.
and
vhlch a r e not comparable lime, neither
By replrcing
may ruppore thrt for
with a rmrller tubular neighborhood if nOCdr88ry. r e
m
E IT1+I 1 there im
Z
X
with
dlm Z > I
such
To conrtruct
u c h atralum
T it Im enough to conrtruct
1
* X of dimenrion I reperately, mince if
Y
a r e two rtrata of dlmenrion
Y'
and
ITY 1 fl X
T on
1
1 , we have
ITY (
n
ITy.
1
0
Y
O
.
for
mnd
Y'
rlnce
Y
IT Y (
I ~ , , ~ l n1 ~ ~ 1 . 1
( V e may have to mbrink
au
m~
a r e not comprable,
to guarantee that there bqualltiem hold for
Furtherrnora, by rhrinking
Ihum. we wirb to conrtruct a tubular nelghborhoad
TX,
Ty
further lf necermary, wr mmy
of
auppore h t
ITy 1 M who" reatrictlon to
TI
amtlrfled:
m C ITX,
%,Y
-
lTYI ') XI+l
, much that the following comrnutatlon relation Ir
rertrlctlon of
if
Ir lmomorphlc to the
1 fl
ITy
1
and
r
(ml C ITy
x.y
1,
where
Tx. Y '
Ir a rubmeralon.
The commutation relation that wr mumt verify 11
preclmely tbe condltloa that
(py. r y I : ITy
1 n XI+,
-
T
X. Y be comptible 4 t h the mapping
R x Y
Therefore from the genermllred tubular
.
neighborhood theorem, we get thst If
wbome clomure liar In
By rhrlnking
IT Y 1
m E ITlt1 1 fl ITy
I
If nacemmary. we may arrange that If
mnd
wf +l(m) E ITy
m
i already ratldied (with
Sinca
and
mE
IT^+^(,
In place of
thereexirta
.
rI+L(ml E ( T ~ ( Slnca
rpace 10 not empty; hence
rertrlctionr
Y c Z
.
1,
Y
ZcX
then thir comrnutatlon relation
r
x. Y ) for the following rearon.
with
1
rI +i(rn) E ITy rl
and
Therefore
d i m Z > I , mC ITZ[
IT^ I .
the lamt named
0
XI+1
ia an open rubmet of
XI+1, than thare e ~ I r t r T
x. y
to
the rertriction of
TI+,
. Nor r e replaca
tubular ndgbborhoodn
Ti
defined analogourly to
XI+1 , but 4 t h
T
x, y
much thrt
TZ
X'I +l E XI0
T;
which matimflsr the
IT^ 1 n %I0+l
commutation relatlonm and whore reatriction to
for
.
11 1momorphic
Z cX
where
Ln place of
X
by smaller
X;+l
TZ.
Then
ham the required propertisa.
Z a r e comparabie. and by dlmenrlon
Thir completer the firmt atep: r e conclude thmt there exlrtr a tubular
neighborhood
To of
Xo
matidying
IbO) for any Y c X
.
Second atep.
eomprtibls with
From
f
.
la0)
it follorr that r e n u y armma thrt
Formby repLcln5
wahavs
me J
ir
To with a rnullar tubular
nel#hborhood if nacearary. r e m y rrrume t h t if
Y cX.
To
T and
~ ~r O ( m ) € i T y l .
m E ( T o ) , than for roma
Than
48.
M
Abrtrrct pro-rtrrtiflsd ratr.
To
la compatibla with
f
,
f
To
to a tubular neiahborhood
prodour rsclon.
Thlr proddam
Ty
b
V
with conolderrbla r t a c t u r r .
?br
r i o n u t l r e &a wort of o t a c b r a rhlch occurr.
We dapirt only rlightly from Thom'r notion of rbrtrrct rtratlfled rot
r e m y extoad a ruitable rertriction
T
of
X
I
md I 4
4 t h poaribly rrmrller tubulmr nalghborhoadm (am in Step 11, r e
DEFINITION 1.
(Al) V
TX , and therefore alro completer
An abatrrct pro-rtrrtlflad met La r triple iv.8. 3)
rrtlrfying the follor*laa rrlomr, Al
net thrt the compatibility condltlonr r r e aatirfled.
Thir completar the conrtaction of
11.
vhlch lo compatible with
, by the @enerrllrsdtubular nalahborhood theorem. Than, by replacin#
the
Ir r clorrd rubrat of a nunifold
Sectlun 5) then r e e m find control &ta for thir prr-rtrrtlflcatlon by tho
([3
of
V
whlcb rdmltr r W M t a o y pre-etrrtiflcalon (In the renrm deflned Ln
purpora of Lbio ractlon i r
Since
If
- A9.
11 r Hrurdorff. locrLly compact bpologicrl rpacr with
8
countable brrlr for itr topolow.
the proof of the proporition.
Tlrla rxiom lmplhr t h t
l a matrlublo.
V
compact, l t ia r a p b r . ro the metrlsrbillty of
metriratlon theorem (Kelly [ 1
1).
X
of
X
can be rsprroted by open rat.).
V
Sbce
V
For. rlaca
V
V
i r locrLly
f o l l w r from Uwroba
i r metrlrable, avow mubrat
i r n o r m 1 (in the ranas that m y two dirjolnt elowad rubratr of
We will oftan urns thio fmct without
a q l l c i t vention.
IALI 8
i r r frmlly of locally clorsd rubrstr of
m
t the dimjoint union of the membarr of 8
.
V
, much that
V
8
'Ibe memberr of
will be called the rtraLloT
V
,
Of courre,
(A31 Each rtratum of
X
Into
=x.y
ordrfleo the a d o r n of the frontier: If
X md
Y
the mapping
If
Y
SF
tr~mftive: Z c Y
(A6J J
Y4X
and
I, r rmbotb rubmatrion.
Ir
l
trlple
Imply
TX
onto
X
,
X
and
(A91 F o r any r t r r t .
& :X
[o,-)
i r a contlnuour
the tubular nei~hborhoodof
the givm mtructure of a p r e - r t r a t l f k d met on
retraction of
TX onto
X rnd
X
, Y , and
dlm X < dlm Y
Z
,
when
TX,
4
rs bava
whmever both rider of tblr equation a r e defined, I. e . , whenever
v
TX
Thio lmpiier
1, {&I) whero for each X E a ,
In V , \ ir a contiauous
function.
We wlU call
X x ( 0 . eJ)
~
.
{{T~I.
TX m
i an open neighborhood of
retraction d
. Tblm relation ml obvloumiy
Z <X .
, wo wrlte Y < X
Y <X
and
,
X, Y E 8
Y ~ % C O ,then Y s z .
and
.
l r locrlly flnlta.
---?r
8
(0.m)
cC).
( T ~ , ~ @ 4 :1T, ~ ~
l
(AS) Tbe famlly
Into
rruy be empty, in whlch caae thema a r e the empty
TX,
(A01 F o r any r t r r t .
8
TX,
Ir r t o p o l o ~ c r murlfold
l
(in tbe induced
V
topology). prodded with r rrnoothnemm rtruchrro (of clrmr
(A41 The famlly
bSyIa a mapping of
md
pX
V)
.
E Ty,
the tubular functlon of
X
.
(a).
NO
V
Z(V) E
.
TX,
We ray that two stratified r e t r
(V.8. J ) @
a r e equivalent if the foilowina conditionm hold.
( ~ ' ~ 83')
'.
the @
i
ry,
DEFINITION 2.
X (with reapect to
wX
and
=
V*
, 8 = 8' , m d for each stratum X of 8 = 8' , tha
rrnoothnean rtructures on
X
given by the two rtratificmtionm a r e
the rame.
If
wX,
X
and
= wX I TX,
Y
a r e any rtrata, we let
. and
PX,Y =
T X , Y = Tx"
fxITx,~
Then
x. Y
Y'
m
i a mapping
(b). If
3
=
for each mtratum
{{Tx?,
X
{oxll
and Z ' =
{{GI,
($1,
, there exirtr r neighborhood
7';
of
bill
X
in
.
then
.
Henca it follwm from Propomitlorn 7.1 thrt m y Whtlnmy pro-mtrmttflad
mot admib the mtructuro of ur mbmtract pre-8tratlfl.d
mot.
From tha aornullty of arbitraw mubmetm of a atratiflad met, i t follmm
t h t ray (abmtract) pra-atratiflad oat 18 o q d n l m t b one whlch matlmtiom
the following condieloam
'
+ 0 , than
(A101 If
X. Y ara mtrmb rad
T
(All) If
X,Y mra mtrmh and
TX 0 TI
comprrabla, I. a.
, on.
X .nd
Y
X cY
if and only if
order on 8
.
XcY
It m
i enough to vorifi
mIlmrlt.neoumly.
X
T
x, y
X cY
But (A81 Impliam
X
<Y
and
and jp : V'
and
.
Y cX
mta comprmblo if and only if
Note t h t from (A81 It followm tbmt rho relation
.
,
.
XcY
0 . tbaa
of the f o l l o r i n ~boldm: X c Y
From (AlOl. it follow. that
from (All) that
x,
II (V,1,3 1 m
i a pra-mtratifld mat,
Y
X= Y
or
+0 ,
.
if
m y topolofical mpca,
i a homaomorpbimm. them tba mtructura of a mtratified
V m
of a rtrattfiad mat on
( ~ * . 8 * ~ 3 'm
) d
a bomaomorpbirm
(V,I, 3 )
cp : V*
-
V
v'
.
mre abmtract pre-mtrrttflsd rstm. than
m
l maid to ba an ioomorphimm of mtrmtifiad
daflaem a partial
Tha uniquwemm ramult that wa rill prova below lmpllam tho followlag:
Y c X do not hold
* dim X * dlm Y
18
V "pulls back" ln m obdoum way to fivm a mtruchrra
(v',fl,&]
and
n Ty + 0 .
T,
mat on
are
-
V'
.
if
(v,gm3)iml Whitnay pro-mtratiflad met, m d 3
and
3'
mymtamm of control data, tbon tbo abmtract pro-mtrat1fl.d matm
A
m an eumplo of m (abmtrnct}pro-mtratiaad mat, 1st V ba a mubmet
of a nunifold
I
,
mnd let
wX: T i - x
M
mnd muppomo
V mdmltm a Y bitnay pro-mtratlficmtion
{ T i ) be m family of control &t. for 8
and
pX:~;-(O,mJ.
m
i m abmtrmct pro-mtrmtifiad mat.
of control & h for
8
Sot f = { T X ) Then
V
(V.8.Z)
In thim way. we ammoclmte vltb any my8hm
H'hlmey pra-mtrmHfied mot
mbmtrmct pre-etratlfied met on
. and let
.
V
,
a mtructure of an
md
(V.l.3)
ara lmomorphic.
a r e two
(V.1.3 )
49. Cantrolled vector fleldr.
Throughout thlr rection. we let
be an (abrtract) pro-mtratlfled rot.
We mppore p 2 2
DEFINITION. By a atratifled voctor Aeld
collection
%
(qX : X E 8 )
q
.
V
, we mean a
{{TX), b X ) , {pX))
Tx,~
T
X,Y ' and
&,Y
,
the .pulvrl.nce
cp-'
and for hvo mtrab
X and
.
Y
tbero lo a nelghborbood
vC T
;
f(v) r fmX(vJ lor all
Y
Ti
ol X
.
cham of tbe pro-rtralfied aet
(V.I,3)
(V,I. J )
,
1. o., If
(v.8, J * )
, thmn a controlled
.
pro-rtratifled retm i r the rame a r a controlled vector fleld (or controlled
mubmermion) with rempct to the other.
let
be dafmed am Ln the prevlour mection.
&!
PROPOSlTION 9.1.
there d r t m a nd&borhood
for any mecond mtratum X > Y and any
q
V vill be raid to be
Ti
for any mmootb vector fleld
& Ty much thnt
Y
vE T
; il X
,
f : V -. P m
i a controlled rubmarmion. then
we have
7
V
much that
f&v)
c
,
P
there l r a controlled vector field
c(f(v)) for all
Proof. By Laduction on the dlmenrlon of
-
vE V
.
V (.rhera the dimendon of
VJ.
V 11 defined to be the r u p r e m m of the dfmenrlonr of the rtrata of
By the
-
V of
hy'y,X)r+Wa ~ b y m X W )
(9.1-b)
DEFINITION. J!
P
Lo
l
rmooth nunifold and
coatinuour mapping, we will may t h t
f
f :V
-
-
P
k
dimenoLon
k
.
of
V
.
Clearly
Vk
we will mom the union of d l rtmta of
Vk
h m the rtructure of a rtratifiad met,
a r e the rtratm of
V which lie in
tubular neighborboodm mre the intermectlonr with
i r a controlled submerdon Lf
L m a rmooth rubmerrlon. for each mtratum X of V
Vk
mkeleton
where the mtrmtm of
P &
neighborboodm (in
the followlny conditionm a r e matirfled.
(1) f[X : X
in
vector flsld (or coatrolled rubmeraion) with rerpect to one of them0
controlled (by 3 ) lf the followln~control condiHonr a r e matiafied: for m y
mtratum
,
i r a pro-rtratlfled net which Lo oqulvrlmnt to
DEFINITION. A mtratified vector fleld
-
X
Note that both the nottonr t h t wo hmve furt introduced dapsad only on
X.
By rmooth vector fleld we mean a voctor f l d d of clam.
$
For my rtratum
(2).
TX much thmt
, where for each rtratum X , we hsve that
Lm a rmooth vector field on
Let
IV. 8 , 7 )
V ) of rtrat. in
tubulmr function on
.
tubulmr function. on
.
the
of t
b tubular
and the local retrmctlonm m d
a r e the rastrlclonr of the local retractions and
Vk
V
Vk
Vk
Vk
.
In tha car.
dim V
v.9
= 0 , tha rtatunart of the propodtion Ir trivlal.
-
(whera
Haaca. by lnductlon, It ia aaough to r b m tbat If &a proporltlon i r t m a
wbaravet
dim V
than i t l r trua when
k
(and do1 aaeume that
dim V
dlm V
k
+l .
hnh.-ora
Thur. wa may
= k + 1 m d that theta i r a controlled vactor
BY
thooaa h. T:
a. that
denoter tha frontier of
Y )
,
T:
rlnce
18 C I O . ~ ln
~
V
-
bY
rtratum
X
of
, It i r enough to conrtruct
much that
V
dim X
2
2
TY ~ T ~ = o .
mapratsly for each
neighborhoodr
ruth that If
Y cZ
eatlafled (with
that
f
of
2
Y ha
Ty
Tha control condilon (9.1) i r ratirflad b r ma).
Y. X
much that
Y cX
.
vk
i r -controllad. we can choore
(one for arch rtratum
s Vk
Y
)
and (9.2-a,)
for all r t r a h
X
for
I
v € Ty
n
Z , By the arrumptlon
i r controllad. wa m y chooma tbe neighborhoodr
f(v) = fwy(vI for all
1
v E Ty
T;
much that
.
Conalder a p l n t
I
Ty
(one for each mtrrtum
Y cZ
a r e rtrata In
Vk
Y
2
Ty
of
Y
C Vk) much that the following holdr: if
than
vE X
.
X
aatlofylng 19.2-b)
To prova thir clalm WIN c l e a r h ba
.
The rat
BV of atrat.
Y cX
ruth that
1. totrlly ordared by inclumion, rinca if Y and Z arm not
2
Z
If SV Ir not a m p y , &m thara h a
comprabia than Ty 17 T Z = O
v € T:
hrgemt mambar
lt l r aarily raan that we may choore neighborhoodr
Y <X
on
enough to prove tha proporltion.
a r e rtratr. than the control condltlonm (9.1) a r e
In p k c a of
:
2
vE T y n X .
V a c h l m L h t there ia a vactor f l d d
Ty1
nX on X
= k + I , becaura the condition that
a vector flald ba controllad Involver only rtratr
Slnce by the Induction aamumption
Ir m a t r l n b l e
- .
(9. &ay ).
TJ
-
and tharalora n o m l . and Y la clored ln V dY
Finally, we c m
2
l a Z , than
chooaa &a Ty mo that if Y ia not c o ~ ~ p r a bto
Now conaldar the f o l l d n g coadilonr on a vector flald
To construct
v
.
Y = Yv
.
in
.
Z<Y .
Suppome for the mombnt thlr Ir the caaa and (9. 2-ay) holdr a t
In the latter came
.
Z = Y or
2
)
Then
y ( v ) € T Z (by the choice of the T
Then (9. 2 - r Z ) hold. for all
Z € Sv
For either
-
v
410.
the parameter lroupr.
v
Let
bm
V
one-parameter l r o u e of homeomorphlrmr of
and
a :R x
mapping
vE V
and d l
V
4
much t h ~ t at+,{v)
V
. Now .uppore
p r e s s rvea each rtratum.
ray that
any
Thum (9. 2 - a Z ) holdr a t
v
for a l l
Z E8
v
(
r)
V
.
If
V
r)
im
m
topologicd rpace.
,
ve mean
atar(v)
rtrmcifled met
A
for d l
the mapping
4
4
atratlfied vector n a l d on
t -at(v)
of
D lnto
V
(V, B.3)
La
C'
l
t, l E R
a
and
V
a if the followlag condltlon la satlafled.
generatea
By
coaHnuour
la
mapping into the atratum which contain*
. firtherrnore
v
4
04.
.
=a
For
a
v ) mnd
Note that thin impller
Thur (9.2-b) holds a t
v
.
It 10 well known that any
c1
vector f l d d om
4
compact mralfold
without boundary generatam a unique one-parameter group ( r e e , e. I . .
Thie r b o r r that to c o n r t m c t
for a l l
v
Y <X
for all
vx
matidying 19.2-b) and (9.2-8
. i t i r enough to conrtruct
vE X
for wbich
nX
srtiafiing
Y
)
(9. 2-ayv) a t
[ 2 , p. 66
manifold.,
1) .
v e mumt genermlize the notlon of one p a r a m e t e r group.
im non-ernpv, and ratirfying (9.2-b)
8
v
at
v
for ail
a vector n a l d
vE X
\
for which
8
i s ernpv.
1 o r 19.2-bJ.
"
y
ratirfying the approprlrte condition in TX
nX
Cieariy, we can construct
v
in r nalghborhood of each point
the appropriate condition (9.2-a
It im 4 1 ~ 0k n o w that to extend thlr reault to non-compact
v
in
X
artimfying
Since the met of vector.
i r convex. we may construct
globally by means of a partition of unity.
Q. E.
D.
DEFINITION.
Lst
p a r a m e t e r group (on
of P1 x V
hold.
-
rnd e : J
V
& a locally compmct rp.ce.
V ) I r a pmir
-
V
(J.a)
, where
A local one-
J
I r an open muboet
i r a continuour mrpplng such that the fotlowing
(b). If
vE V
,
Jv * J
then the met
n (R x
v) C R
m
t 4n open interval
(av, by) *
Thlm gmerdisom the ordinarl, notion of v h r t It melnm for a voctor f i d d
to generate
Id).
e>O
For l a y
ruchtbat
vEV
and mny compact 4et
dv,tlf K
If
KG
V
,
4
local m e - p a n m e t o r group.
there ozrlmtm
Slnco
t E ( 4 V . 4 v + c ) ~ ( b V#-, b y ) .
(V,S,3)
18 8 pro-mtrmdfiod sot. l t m k o a #an00 to L.Ik of
controlled vector field on
From n o r on in thim rocdon, we muppome (V,B.3)
pre-mtr4tlfid set, and
q
I8
4
we may
q
&
rtratiflod w c b r flold on
peratem
(J, a ) Ir 4 10-1
a
V
.
one-parameter group (on
if the followinn conditlonm
4
-
k c h atraturn
X
of
V)
,
a [ J n (R a X ) ] c _ X .
For any
udqus local one-mramotor nroue
For each atrrtum
hence
qX
garteratom
dlffeomorphlmmm of
v
E
V
.
, tho romtrtctlon
.
gX
of
Q
to
m
l
X
(by tbs definition of mtrrtifiod vector fleld);
X
mmooth vector Held on
X
1J.a)
%
i.e.,
12, 1V, 4 2
(c).
4
Proof.
-
c a r e matisfled.
v I r invariant under a ,
q i s a controlled v e c b r flald on V
PROPOSlTlON 10.1.
4
(4).
(Soctlon 5).
V
Im an (abstract)
pineratam
DEFXMTION.
4
]
X
. Let
J =
4
mmooth local one-parameter group
.
by
mtmhrd romult ia differential gsornstv
be defined by
4J.a)
UJX
XEB
4
of
(Jxo%)
a s
UaX
XES
r e have
We amaert that
(J.a)
is
4
local one-parameter g r m p generated by
q
.
q
8
,b
group hold, and that lf
b:
I t I r c b r that
.
.
and c In the d e h l t l o n of local one-paramatar
I8 8 local one-paramatar group, then it I 8 .ganrrata
Unlquanar r l a obvlour.
by
v
J
I 8 open,
compact met
K
clore to
or
8
v
V
In
holdm.
&,v) E K
ruch that
exlrtr a requenca
v
{tl)
'.
.
v 1
which coataiar
V
vE V
v
.
bv
Let
(ramp.
O(t.v) E K
t
arbltrarlly
for valuer of
t
But thlr contradictr the hypc?herla thrt
contradlctlon prover
Then there
X
(rarp.
y)
.
a
mad
Y 1 denote the
group.
X
=
Y
,
Y <X
.
For l a r g e
1
. %,
at-t,
much that
I
and
py
mi
r
Y,X
= av( ti )
(a (t. 1)
v
.
1
[O,C]GJyl
Y
,
,
where
uy
yl
= uymX(mi), and If
1. the local retraction of
18 the tubular function of
Y
, then
ayi([O,i])
and the control conditlonr for the pair
m E toy,
= pY, Xlmi)l n,:-
,lt++[O,
i
1) n x .
T
Ty
ir
TX
TX
J
Am bafora, lat
X
-
y
-
j
88
m
.
Thla
be the rtratum vhlch containr
U
of
v
in
X
.ad an
c >0
danota the tubular aelghborhood of
on
X
ax([-c, t t c]
v
1
,
pX
and
[-a, t t
ouch that
X
.
the tubular functlon of
U)
l r compact, we may choora an
E=
{y E TX : %(y) 5 el
C]
x U
rX the local rattaction
X
.
Mnca
el > 0
ruch that the
folloving hold:
onto
~ ~ , ~ ( cym on
~ ) c
Y ,X
Let
(t )
(a). Let
Then
(b).
y
,
and
r X ( y ) E ax((-c, t t c] x u ) )
.
G la compact.
a r e ratisfled for
Slnca
.
ax Lr 8 local one-parameter group, there La a compact
nelghborhood
of
aufficiantly large. wa may ruppome that there axlrtr
tha tubular nelghborhood of
Y
m
I a one-parrmate
X(uv(tl)) and
a r e defined. and the control conditlona a r e matidled for
Thur, by taking
QX
.
v
( t , ~ E) J
la treated rlmllarly.
we get 8 eontradlctlon to the fact that
Othedae
d
b:
. W a d l 1 rhow th.t J l r a aelghborhood of 1t.v)
i r contlnuour a t ( t , v ) . Y a dll ruppore t 2 0 ; the other c a r e
from below. ruch that
Since
If
i r compact (bacaura
and a
Now l e t
= llm a ( t 1 axlrtr and llam In K
rtratum of
for valuer of
the other came i r treated mlmllarly.
convarglng to
x ( a y ~ o c]J
.
i t follovr from the control condltloaa that
If not, there exirtr
b V . We m y ruppora that
arbitrarily clora to b
y
d
%, X(ml)) n,:r
oY,X(ml) c t y On q i l [ O . el)) , and uv 8 1 . ~ 0in X (by dafinition).
A11 that renuinm to be verlfled l r that
a Im contlnuoua, and d holdm.
We begln by ahoviag that
=
by,
If
where
yE
Y
E
.
then the control conditlonr for the p r l r
i r the rtratum whlch containr
y
.
X. Y
hold a t
J
.
Cloarly,
rXtyI € U
a0 s e t r0
Ir
of
y
nolghborhood of
e Tx
in
v
ouch thnt
V
.
&(Y)S
If y E EO
.
8,
and
it follo=e from
Diagram 10.1
tho control conditionr that
whore
forall
md
d
r f J
,
ruchthrt a ( r * ) € C b r 0 s r C c m . Fromthosefactr
Y
Y
I t folio-• thmt I-c, t + c) E Co J ; thum J contains a
ndghborhood of
(t, V )
.
f i ' . r ? € ( ~ - c , t + c ] r E 0 . thaa
.
we may choomo
continuour at
(t. v)
r>0
.
y * = o ( t ' , y ) E TX
a
El. Hence a la
m d a ndghbrbood
P , be a manifold, and
p r o ~ e r ,contrp.l~ed.~ubOmer~1ofi1
meq f
thl. came that there 1, a homeomorphlmm h : V
V
over
0
f : V -. P
be.
I 8 a locally trivial fibrathn7
Proof. It 18 enough to ca~~mider
the care when
-
denote. the fiber of
V
1 5 i _c Ir
,
. By
thoro 11 a carrtrolld rmctor field
ruch that
By P r o p o o i t i ~10.1, mach
[Jima,)
. Clearly
bi
t. v))
non-nnimblng entry i r la thm
Q. E. b.
*
L
on
.
Ir
on R
+(~*)58,, md
f
COROLWRY 10.2.
1
Proposltton 10.1, for u c h
a,, -.., bk
d+ur obowm thmt if
wX(y*)= ~ ( f * , - ~ ( y J lHence for an arbitrarily small neighborhood of
dt.d
dmnotam the projection on tho rmond factor.
Conoldor tho coor&nato vector floldo
b;
I h e r r p t a r a d r h - a r m &kJ a n *
r
2
-
P =uk and rhow In
Vo
x
nk ,
rkre
, ruch that the following dlrgram commuter:
m
generates a local o n e - p r a m s t a r group
--
=her. tho
flv) t (0.
0, t, 0.
Ith
?baa from the rrmumptlm that
phce.
0)
l r proper and condtlon d IP the doflnltion of one parametor group, It
followr that
Ji
-
R x
V
.
Lot
h
bo glvm by
From the fact that the
h%.
-
hh
ldentlty.
al 'a
Hence
Let
a r e one-parameter groupr, it followr that
h
10 a horneomorphlrm.
AD
required.
Q. E. D.
x E Xn W
ham a natural rtructure of a pre-rtrrtified s e t
Vo
. vhere
(VO,IO.JO)
and JO a r e defined
SO
(xn v0: x r 8 ) . a x c 8
collection
.xo
vx
a
Into
l txo
Xo
triple
mO
and
becaure
f
T
.
~
NO*. ~that
.
.
(mc0)l
VO x Etk
the
.
, we mry
n x)
TX
.
.
be the
.
px < c
on
where
(pX,uX) : TX
Z
of
TX
it 10 enough to conrider
By replacing
M
with a rufflclsntly
ruppore that
X
i m connected and
-x
~ h e r e mX : 1
;
TX
of
X
In
M
much
i. the'projectlon
From Lemma 7.3, It followr t h t by cboorlnp
rufficlently rmrll, we may ruppore that there e r i r t r
T ~ o
Z0
.
w n rx .$w
that
aaaoclated to
fxv0
,
.x0 "UP.
We let
r
x
-.
rE YnW
clored, and & e r a exlrta a tubular neighborhod
SO lo the
TXo =
l r a controlled rubmerrlon.
m x 0 1 {=x0)
Furthemnore
pX J
follovr.
x O = x n v Oi r
and
IO
, than we let
correrpondlnp member of
AD
TO rhov
vhat bppanm in a neighborhood of
r r m l l open nei~hborhoodof
Note that
.
(
>0
Tx
much that
, where pX ir tbe tubular hmctlon arroclated to TX ,
(0, C ) x X
i r proper, and where for each rtratum
8 , tbemrpping
ha. a o t n c t u r - of a pro-rtratlfied met
(defined in ul obvloum way).
l r a rubmerrlon.
COROLLARY 10.3.
Corollary 10.2,
Proof.
ah
h
hi. conrtructed a0 In the proof of
~ e ts* =
i m m Iaomorphlrm of pro-rtrrtined metr.
Imrnedtate from the conrtructlon of
h
. (See the end of
Section 8 for the definitlon of lromorphlrm. )
& M
COROLLARY 10.4.
h
rubnet of
X
rnd
Y
and let
8
be r t r a t r 4 t h
X
be a clomed
be a Whltnev pre-rtratlfication of
X <Y
.a
W
pre-rtratlficatlon of
of control data
IS n ITX
be a manifold. let
S
. _Let
be r rubmanlfold of
M
Cz n (T,
8
S)
S
for
- x),~*.J')
- XI : z E . her 8' tm a W h i t n e y
n (TX - X) . By Ptoporitlon 10.1, there i r a family
8 which l r compatible with (pX,.wX) . Then
i r an abrttact pre-stratlftad met and
controlled mubmerrlon.
Hence by CoroUaty 10.2,
locally trivial bundle over
(0, t ] x X
S fl ITX
, and by Corollary
trivlalirationm respect the atratiflcatlon.
(%."X)
- X)
Ir a
ir a
10. 3. the local
TX
(E, 9
(IT
(
-
a)
X. (px,
XI
{ t, n r R n X: 0 < t < r (n)). Thea tbo bundla
qx),x, I in locdly trivial and tho ~ oJc trLvl~isatlonrc m
X
be chooan to rerpoct tha otratUIcatioa
Tho noxt corollary rayr that a pro-rtracifiution which oatirfiao all tba
eondItionr of a Whitney pro-rtratlflcatlon ucmpt the condition of the frontier
alro wtkaflar rho eoadltion d the frontiar. proddad that Itr rtrata arm
conneetad.
& M
COROLLARY 10.5.
pro-rtratlficrtion d a clooad eubaat
ouch t l u t my p8lr of mtr-
8
be a mmlfold and
g M
V
be a IocaUv finite
whore rtrata a r e connactad
msn
eatirfv condition b.
8
io a M'hitney
pro-etratiflcation.
Proof.
Suppoee
10.4 rho-.
Y
,
and
It euillcae to rhow that the conditioa of the frontlar holdr.
Y
.ad
X
tht
Y
arm rtrata mrd
Y n 'JT 10 opan in
Y
ie connactad. thlr prove.
Y I1
.
+a.
Shco
Y
X .
Y
fi
The proof of C o r o U q
1. clearly clomed in
The proof of Corollary 10.1 a h 0 ehowr:
COROLLARY 10.6.
pra-atratlflcrtlon of
nal~hborhoodof
X
M
a
.
M
X
M
be a mmilold,
a rtrahrrn of
M
.
I
a W'hltney
T
such that for any rtraturn Z
X
a tubul~r
& 8 , tha
411.
' h a irotopy Iernmar of Thom.
In thlr rectlon. vs 4 1 1 state
'hom'm firat and recond irotopy lamnum,
We will prove the flrmt and
rkatch m proof of the recond.
Throughout tble section. we let
f :M * P
a rmooth mapping, mnd
M
and
P
be rmooth manifoldr,
S a clorad mubrat of
M
We may that
which
a nupplng
admitr a Whitnay pro-rtratification.
-
m
i trivial over
f
fo : Xo
Yo
Z
if there edmtr space.
nnd horneomorpblamm
Xo
and
X 2 Xo x 2
Yo
.
Y -' Y 0 x 2 much &st the following dlagrarn of #pacer and mappiagr Ir
Proporition 11.l.lhoaf. flrrtirotopy lammr. Suppors f IS : S
proper urd
f
Then the bundle
l : ~X -. P
(S
,f ,
im a mubmeraion for each rtratum
P 1.m
X
&
.
commutative:
PI m
i locally trivial.
-
Proof: By Proposition 7.1, we can flnd a rystem of control d a b for
S whlch la compatible with
f
. ?Wr provider
an abrtract mtratiaed met La much r way that
f
with a mtructure of
S
im a controUsd rubmersion.
Than the qonclurion of the theoram 16 ur immediate conrequence of
Corollary 10.2.
Remark:
Q. E. D.
Thorn conriderad the care P
We may
= R
.
If
a, b E R
, then
f
m
i locally trivial over
neighborhood
U
we have that
f
of
s
in
lf for any
s€ Z
Z ruch that ln the diagram
tbs proof of Proporition 10. I conrtructr an irotopy from the fiber
to the fiber
Z
Sb , whence the nmme #'irotopy lemma".
The second irotopy lemma m
i an analogour remult for mappings instead
of apacem. Conrider a diagram of rpacer and mappings:
m
t trivial over
U
.
.
there m
l a
.
L o u 1 W*lrlity of a mapping
f
over a rpacr
r h l c h uiU ba very importrat in w h t foUowm.
kmily
{fa : a € Z) of mmppingm, where
obtrinsd by rertrictlng
i r coarrectsd urd
b
in
2
,
f
f to the n b s r
l r locally trivial over
the nmppingr
fa
&era .dmthomeomorphirmr
tht
h'f8
Xa
md
\
2
Wa think of
-
f. : X.
of
X
To rtmte Thomlr macand imotopy lemma, we mumt introducm m o m l r
bar a conragusnee
Y8
over
f
condltlon
am a
l r the m p p i n g
a.
Z
If
Z , them for ~ n y
and
be a point in
h' : Ya
-
Yb
much
$,h.
.
h t
Y
. Suppore
We may the p d r
(X, Y)
5
L.t
familier of mapping. a r e locally trivi.1
g
bundle.
IX
s
r
h t
-
and lot
y
g J Y a r e of eoamtnat rank.
at
-
y if the follorlng
X
converglmg to
y
.
k e r ( d ( g ( ~ * lS~ )T < ~
men k a r ( d ( g ~ ~ '€1r~
condltlon
a6
(XI Y)
at evmry pobt
y
matlm8em condition
Y
of
a 0 if i t matlrflsm
.
in the mame deflnrd above. by
N w , we return to the mlmtlon of Magram 11.1.
M * 11 a mmooth manifoid and 5'
We uiU u y t h t
P)if the followlag condltlonr a r e ratlr8ed.
i r a eloasd
M * , whlch admit. a Yfhibmy pro-mtratiflcation 8
g : M'
M'
in the appropdate G r a m r m d m
TM&
i a m o m mapping (over
g m
rubmet of
and
ratlmfhm eondltlon
We may t h t ths pmir
an appllcatlon of Tbom'r rreond imotopy lemmm.
Now ruppors
be rubmaJfoldm of
Suppore that the raquence d p t n e r
topologicdly atable nmpping, mnd a rtap in the proof that the topologlully
rtabls nvpplngr form an opun denra rat d l 1 be to mhow t h t eertaln
Y
be any rsqwncm ofpointm ln
convargem to a plme
Thir i r the rahtlon of equlmlenes t h t l r ured in the definition of
and
X
hold.:
a r e oauivalant ln tha ranme t h t
h : X8 -* Xb rad
a6
.
M b r a rmooth m p p l n g and ruppome g(S') g S
(a) g ( ~ *and
.
f ( S a r e proper.
(b) For each rtrmtum
X of S
( c ) For each rtratum
X* of
, fIX
i r a mubmsrdan.
Thornqr recond lrotopy lemma glvem rufflclent conditionr for tha following
d l g r a r n to bs localiy trivial:
of
8 , and
g : X'
-
X
8
, g(X')
liar in a mtratum X
i r a mubmerrion(whance
glx'
l r of
conrtant rank).
diagram 11.1
(d) Any pair (X',Y')
of .tram of 8'
(which maker aenme in d e w of
(c)).
matimflar condltlon a B
In the came
P l r a point, we vlll drop
wer
P
.
PROPOSInON 11.2 4 Thorn'r mecond lmotopy lemma).
P , t h s g i s locally trivial over
Tbom w p v i n a over
The proof of thA4 requlrem new
machine^.
of control data for the atratiilcatlon 8
a myrtem
of
s*
{T')
3
of control data o
s
of
S
Let
{T)
g
P
&
.
n
v ET
of
8
,
.
~ c iTx.l
4 ~
~ c a t b t~
T
Furthermore, Lf
be a myrtem
. We need the notion of
for which both aldem a r e deflnsd, 1. s., a11
v
holdr f o r all
g ( X * ) and
g(Y')
1
l i e in the r a m s rtraturn
then the cornrnutatlon relation
for the rtratlflcatlon 8 '
{T)
.
for which both ride& of thlm equation a r e defined.
hold. for a11 v
CAU'RON: A ryrtsrn of control data
control data a s previoumly d e f i e d .
of control data over
over
(T)
lo not a mystern of
a l r o be a system of control data tout court
(T)
then the fundamental existence theorem for control data over
contalnr
lo a mtraturnof 8
X'
(b) If
If we were to requira that a rymtern
and
X
l a a rtratumof
S
which
g ( x * ) , then
IT)
(Propomition 11.3, below) would not be true.
for all
DEFINITION: Suppore
g
i r a m o r n r n a ~ p i n ~A
. mvatem
(T*)
ail
of control data for
indexed by
?
8
'
{T)
-0
, where
Ti
v
v E
for which both alder of thlm equation a r e defined, 1. e . , f o r
IT~,J n
g
-1 IT,^.
Ir a family of tubular neighborhoods,
lo a tubular nei~hborhoodof
X
with the f o l l o w l n ~propertiem:
-
a
Note that
M'
data in the case
1s weaker than the commutation relation for control
g(X')
and
g(Y*)
a r e not in the mame stratum of
P .
(a1 If
X'
and
Y'
a r e strata of
3
and
X* < Y
,
then the
PROPOSITION 11. 3.
comrnutatlon relation
(T)
of control data for
data for
8'
over
IT)
g
3
.
im a Thorn mappinu then for any myrtem
t h e r e e x i r t s a symtcrn
IT')
of control
The proof of thlm m
l rimilar to the proof of the admtsnce thaorem
for
for control data (Proporltloa 7.1).
Y ' < X*
md
dlmY'2f.
Ti'
mtap, r e canmtruct a tubular ndghbrhood
Proof (Outllna): Let I
s
dlmunmlon
,
k
and let
; be the Lmily of aU rtrata of
%
S;
8 ' and
ln p k c a of
k *
that L e propomition i r true for 8
.
T;
hesflrrt
X i by dacraaming
r h a r a nacerrary.
ThIr mtep m
i c a r d e d out fn a r r s l r t l d ~the ram. m y a 8 the firrt
;
5'
, mhrlnklng mriour
iaduction on f
of
x'.
'
dmota L a union of dl rtrata in 8
We rill ahow by h r c t i o n on k
and
8 ' of
= Ui n
X;
Welet
Vfe r i l l only outllne it.
mtep in tha proof of Proporition 7.1.
Thir r i U ruUlce to prova the
d a r e &era i r nothing to prove.
Vfe mbrt the induction at 1 = k
,
For tha fnductlve mtap, we muppors
proporition.
We obrerva that to conrtruct
h m been conrtructad.
The care
k
0
im trivial.
&at for s a d mtratum X'
a tubular ndghborhood
of
TX,
8'
For tbe tductiva rtep. we muppoma
ofdlmenmlon
< k , w a a r a givun
enough
conmtruct
of dimenmion f
X ' m d thmt thir famlly of tubular
of
a
Came 1.
-
nalghborhoodr utirflam con&tlonr (a) m d (b) abova.
T*
f on
maparataly.
If
B ( Y *)
md
IT^. 1
n
X'
T i it i r
1 < X'
for u c h atraturn
T h a thare a r a two camam.
g[X*) a r a in the rams rtratum of
,
than the construction i r carried out h the rams m y am the corramponding
By rhrinkbg the
X'
than
and
Y*
1
~
if necermary, we mey ruppome h t if
TXc
a r e rtrata of dlmanaion < k whieh a r e not comparabie.
ft
IT,.(
~ ~
= 01 . Toconrtructthe Tx.
onthertrata
X'
be
8
mtratum of
8'
of dimanmion k
lTycl
foilarm.
For each
f
sk,
we (at
U;
.
of I
TXO in two mtepm am
denote the union of all
Than
ITy.
Ti
mo h t the commutation rebtionm ( a ) hold.
X'
2. In tha cara
1
g ( y O ) and
g(X')
, the proof murt be modiflad. Lat X
containr
We conmtruct the tubular neighborhood
n
Came
-
relationr (a) mnd (b) impoma no con&tionm on pairm of mtrata of the
b t
on
In W r Way r a daiina
(Note that condition (b) followr from (a) h thir came. )
of dimenmion k , we m y do it one mtratum a t a time, mince the
mame dimamion.
conmtruction in the proof of Propomition 7.1.
g(X')
Y <X
G(IT,'I)
.
and lat
Y
By rhrinking
E ITy[
. Let
ara,not in the name mtratum
be the mtratum which
be the rtratum whlch contains
ITy.(
g(Y 'I
.
if nacerrary, we m y muppoma that
&.re
the flber product i r taken wiEh ramped b tba mrpphgr
By daf!nltlon.
'Ibea th. nupplng
x'
F C if .ad only i f
t b i w w i n g ir not onto.
Since
i r defined becaura tha fo11ovIng diagram commuter:
i r onto.
From condltfan
clowura
NWWO
it foUowr that Y' doer not mast the
g'
of the rat of polntr where tNw rrvppLng 11 mot onto. Q. E. D.
axtend
T;
a
ovar
IT^.^
holdr (the wamk (a)!) aad (b) holds.
by the lnductiva bypotharlr that (b) i r rntiwfled for thora tubular
n
X'
~ n r u e h a r r y t h . t(a)
W o may do thir by tha gsnarnlirad
adrtence theorem for tubuiar naighborhoodr m d L a m U. 4.
ndghborhoodr which a r e already defied.
LEMbU 11.4.
There adoto
8
GIN n x'
-
Thia completar the t n d u c t i ~rtsp.
neighborhood N
: N
n x*
Y*
&
(3'1
ruch &at
Now the rocand rtap (uxtenmlon of
v
T i from
U;
over aU of
iw carried out in s u c t l y tha mama way ns In tho proof of Proporltfon 7.1.
i e a rubmarsion.
Prool:
h t
dlffaranthl of
G be the mat of point. in
C
ITy
XC)
fl X*
1. not onto. It wufficew to wbm that
The rawt of the p r o d of Propowition 11. Z w i l l be cnrrlad out in
vhera tha
Y* fl
= Q
.
thraa stapw. Flrrt, we debna tha notion of a controllad vector flald
Q.
and if
-motfrat
controlled vector flald.
(WARNING: this i s not
8
t(X8) and
#(Y '1
a r e Ln the 4amr atrmtum of
8
tbsn r o have
mpecld
u r b of the nation of a controlled vector field. ) Thm r e p r w a a lifting
theorem for controlled vector fialda.
Flndly. we
&OW
that evsry
(Note that condition b i r relkmr thn the condltdaa r h t wm Impommd
coatrollad vector flald over another controlled vector fleld gsnar8ter
on
l
l
g (Y '1
cmtrolled vector fleld Ln Sactloa 9 l a the care
8nd
six')
local one parameter group.
a r e not in the name rtratum of 8
N w we muppora
g
11 a Thorn mapping. We nrppore thmt we a r e
PROPOSITION ll. 5.
flvm a ayrtem
{T)
of ecatrol data for
of control &ta f o r S m d a ryrtem
S* over
controlled voctor fleld an
T
.
&t
I)
m
{qx}xE8
IT*)
.I
There d m t r a controlled vector a d d on
S*
ovsr 7 .
be a
The proof i r completely malogour to the proof of ProporitLon 9. I,
5.
and r e omlt it.
PROPOSITION 11.6.
over t ) , t&
-
p'
11 7'
la r controlled vmctor flald on S*
ganeratem a local one parametergrarp. which
commuter with the one-varameter nroup on
S
generated bv
t)
.
The proof of thim in errsdtimlly the rrma am tho proof of Proponltion 10.1.
'Iha only rddltionrl rermrk to be nude i r that if
(b) For m y
N~
'
of
Y*
in
X'
,Y'
E 8'
4th
Y' < X* , there i a a neighborhood
( T ~ . J muchthmtfor y e € J T ~ . { ~ x *r e,h a v e
rtrrta of 8
X
ller in
,
with
im
r
,
than, in the care
atartlng a t a poht of
y
Y 8 < X'
X'
trajectory of fl
We omlt the proof.
and
g(Y')
Y < X
,
X* and
Ilea in
a trajectory
ccnnot approach
Y
Y
y*
Y
mrm
mnd
g(X')
of
t)'
beemure the Imagm of
and therefore cannot approach a point of
Y
.
Proof of Proporition fl. 2.
P , i t ruUlcer to coarider the care
P in thim care.
trivial over
IT)
of control &.ta
for
&ere d r t r a ryrtem
Let
b,..
.'*
8
IT*}
over
mP
~d prove tbrt
CY
Si
compaHble d t b
f
,
on
.
a r e proper m d
bi
over
nP.
S1
pi
and
.
IT}
By
"
to a controlled vector field
on
to a controlled vector field
By Proporitionr 10.1 and U. 6 the vector flaldr
g
and by Proporition 11.3
of control d.6 for &
lift
local o m parameter group.
ir
g
By Proporitton 7. l we can flnd a ryrtem
and by Proporltion 11.5 wa c m lift
5'
r
a ~be the coortflnrte vector nal&
P r ~ p o r l t l o n9.1. we
on
P
I# locally trivial over
6
To prove that
1
(pi
gi
nnd
5
. Since the nupplngr
S
.
1
a1
gemorate
f
generater a (global) one parameter group
and
,
It i. earlly verified that the above diagram conrmtem and Uut
I t follow. that
Let
So
i
and
(re8p.
a r e (global) one parameter group..
h C a r e homeomorphirmr.
s;)
denote the fiber of
S (reap.
5') over
0 .
To complete the proof, it m
i enough to conrtruct local homeomorphlsmr
h
and
h*
ouch Haat the following diagram commuter.
Q. E. D.
h and
REFERENCES
[I]
J. Kelley, G e n e r a l Topology, Van N o r t r a n d Co.
N. J . , 1955.
Pl
S. L a w .
,
Inc
., P r i n c e t o n .
Introduction t o differentiable manifolds, I n t e r r c i e n c e ,
New York (1962).
[3)
R . Thom,
Local topological p r o p e r t i e r of d i f f e r e n t i a b l e mappings,
'rColloquium on D i f f e r e n t i a l Analymis", pp. 191-202. T a t a
Inst. Bombay, 1964, Oxford Univ. P r e r s . London and New
York, 1964.
Ensemblem e t morphimmer mtratifi&.
Bull. A m e r . Math.
SOC., 75 (1969) pp. 240-204.
[5)
[61
H. Whitney, Tangents t o a n a n a l y t i c v a r i e t y , Ann. of Math..
pp. 496-549.
, Local properties
81 (1964)
of a n a l y t i c v a r i e t i e s . Differentiable and
combinational topology, P r i n c e t o n Univ. P r e s s , Princeton,
1965.