Pub
.
Ma
.
UAB
N° 25
Juny 1981
BLOWING
UP
FIXED
POINTS
F ancisco
Gómez
Ruiz
Facul adde
Ciencias,
Uni e sidad
de
San ande
and
Seccióde
Ma emá iques,Uni e si a Au ónoma
de
Ba celona
.
Spain
.
Rebu
el
15 de
Feb e
del 1981
This
no e
shows
how
o
use
he
echnique
o
blowing
up
submani olds
o
gi e
easie
p oo s
o
some
heo ems
conce ning
ixed poin
se s o
o al
ac ions
on smoo h
mani olds
.
None o
he
esul s
gi en
he e
is
new
.
The
main
heo em
( heo em
3)
as
well
as
he
applica ions
(8),
(9)
and
(10)
a e
well
known
.
Ne e heless
I
belie e
p oo s
p esen ed
in
his no e
a e
no
he
usual
ones
.
(1)
Le
us
begin
by
desc ibing
he
equi a ian
blowing
up
o
in a ian
subma-
ni o1ds
.
Le
G be
a
compac
Lie
g oup
ac ing
smoo hly
on
a
smoo h
mani old
M
and le
B
be
a
G-s able
closed smoo h
submani old
o
M
.
Conside
he
induced
ac ion o
G
on
he
angen
bundle
TM
o
M
.
I
is
gi en
by
a
.
=
(dT
a
)
x
( )

x
E
M, a EG, ET
x
(M)
whe e
TX(M)
deno es
he
angen
space
o
M
a
x
and T
a
is
he
di eomo phism
o
M
de ined
by
T
a
(x)
=
a
.x
.
The
abo e
ac ion
o
G
on
T
M
es ic s
o
ac ions
o
GonT
M
IB
and
TB,
since
B is
G-in a ian
.
Thus we ha e
an
induced
ac ion
o
G
on
he
no mal
bundle,
:E
n+ B,
o
B
in
M
.
Le
P( )
:P(E)
->
B
be
he
p ojec i e
bundle
associa ed
o
(i s
i-
b e
o e
x
E
B
is
he
p ojec i e
space
associa ed
o
he
quo ien
TX(M)/TX(B))
The
bundle
P( )'inhe i s
an
ob ioús
ac ion om
he
ac ion
o
G
on
.
We
alsoconside
he
canonical
line
bundle,
É

P(E),
on
P(E)
(i s
ib e
o e
z
E
P(E)
consis s
o
al]
ec o s
o z)
.
The
abo eac ions
o
G
on
P(E)
and
E
induce
an
ac ion
o
G
on
E
such
ha
he
map
a
:E
-
E,
gi en
by
a(z, )
= ,
is
G-equi a ian
.
Obse e
ha
a
is
su jec i e
and
es ic s
o
a
di eomo phism
É-P(E)
~
O
'
E-B
(we
iden i y
he
base
B
o
i s
image
unde
he
ze oc oss-sec-
ion)
.
U
o
B
in
M
oge he
wi h
a G-equi a ian
di eomo phism
!
E-P
U
such ha
82
I
is
no
di icul
o
cons uc
a
G-in a ian
open
neighbou hood
es ic s
o
he
iden i y
on
B
.
Se
a
=
sp
oa
:E
}
U
C
M
.
Thus
a
:E-P(E)
,
i
U-B
is
a
G-equi a ian
di eomo phism
.
Le
M
be
he
space
ob ained
by
a aching
É
o
M-B
ia hemap
E-P(E)
á
.
M-B
(i .e
.
Ñ
is
ob ained
om
he
disjoin
union
o
É
and
M-B
by
iden i ying
he
poin s
o
É-P(E)
o
hei
images
on
M-B
unde
o
) .
Endow
M
wi h
he
ob ious
smoo h
s uc u e
o
which
he
inclusions
É

M,
M-B
J+
M
a e
di eomo phisms
on o
hei
images
.
I
is
clea
ha
he
ac ions
o
G
on
É
and
M-B
inducean ac iono
G
on
M
.
De ine

a-
:M
-,
M
by

Q(i(x))
=
o(x)
i
x
E
E
and

á(j(x))
=x
i
x
E
M-B
.
Thé
map
Q
is
G-equi a ian
and
su jec i e
.
Fu he mo e
á
:M-P(E)
M-B
is
a
G-equi a ian
di eomo phism
.
Obse e
ha
M
is
ob ained om
M
by bowingup
B
on o
a
G-in a ian
hype su ace
P(E)
.
In
pa icula
a
poin
o
B
has
been
blown
up on o
a
eal
p ojec i e
space
.
In
case
B is
a
single
poin ,
M is
simply
he
connec ed
sum
o
M
and
a
eal
p ojec i e
space
.

.
Blowinp
up
he
ixed póin
se
o
a
p oup
ac ion
.
I
is
easily
leen ha
he
connec ed
componen eo
he
ixed poin
se ,
F
G
(M),
o
he
ac ion
o G
on
M
a e
closed
smoo h
submani olds
o
M
.
Blowing
up in
u n
each
o
hem
we
ob ain
a
smoo h
mani old
M,
ac ed
on
by
G,
oge he
wi h
a
G-equi a ina
su jec i esmoo h
map
&
:M
-
M
.
(2)
Lemma
.
0
Le
G
be
he
connec edcomponen
o
he
uni
in
G
and
suppose
ha
0
G/G
has
an
odd
numbe
o
elemen s
.
Then
he ac ion
o
G
on M has
no
ixed
poin
.
P oo
:
Clea ly
M
-9
- 1
(FG(M))
does
no
ha e
ixed
poin s
since
i is
G-equi-
a ian ly
di eomo phic
o
M-F
G
(M)
.
The e o e
i
is
enough
o
show
ha
Q-1
(F
G
(M))
does
no
ha e
ixed
poin s
.
Le
B be
one
o
he
connec ed
componen e
o F
G
(M)
and
le :E
-
u->
B
deno e
i s no mal
bundle
.
Endow
wi h
a
G-in a ian
Riemannian
me icand
assume ha
an
elemen
z
E
P(E)
exi s
such
ha
a
.z
= z
o
all
a
E
G
.
Fix
a
ec o
o
no m
1
belonging
o
z
.
We
mus
ha e
a
.
= E(a)
.
whe e
E
:G
~
{1,-1}
is
a
g oup
homomo phism,
cons an
on
each
connec ed
componen e
o
G
.
0
De ine
E
:G/G
-
{1,-1}
by
e(á)
=
E(a)
(a
deno ing
he
clase
o
a
in
G/G)
.
Se
c-1
(1)
=
{a
l
,
.
.
.,a
}and
E
-1 (-1)
=
{Bl,
.
.
.,bs}
.
We know ha
-1
(-1) ~ 0
because
i
no one had
a
.
=
o
all
a
E
C
and,
i
we iden i y
E
o
a
G-in a ian
ubula
neighbou hood
o B
in M,
his
would yield
a
con-
adic ion,
since
B
is
a
connec ed
componen
o
FG(M)
.
The e o e
E
-1 (-1)

=

{a
1B1
, .
..
,a
b 1 }

=

{B1'
. .
.,BS}
.

Hence
= s

and
0
G/G
has
an e en numbe o
elemen scon adic ing
he
hypo hesis
o
he
lemma
.
83
(3)
Theo em
.
Le
G
be
a
o us
ac ing
smoo hly
on a
compac
smoo h
mani old
M
.
Then
he
Eule -Poinca é
cha ac e is ic,
x(FG(M)),
o
he
ixed poin
se
o
he
ac iono
G on M
coincides
wi h
he
Eule
Poinca écha ac e is ic,
x(M)',
o
M
.
P oo
:
Le
B
1
,
. .
.
,B
be
he
connec ed
componen s
o
FG
(M)
wi h
no mal
bundles
i
:Ei

n
i
-

B
i
(i=
1,
.
.
., )

and

le
w
i
:E
i

+
U
i
(i=
1,
. .
., )

be G-
equi a ian di eomo phismswhe e
U
i
is
a
G-in a ian
open
neighbou hood
o
B
.
in M
.
Se
U =

u
U
.
.
i=1
yield
The
co esponding
Maye -Vie o is
sequences
o
he
open
se s
{M-F
G (M), U}
o
M
and he
open
se s
{M-á-1(FG(M))
=
M-FG(M)-
c
1
(u)
o M
(4)
x(M)
+
x(U-F
G (M))
=
x(U)
+
x(M-FG(M))
(5)
x(M)
+
x(U-F
G (M))
=
x(°
-1
(U))
+
x(M-FG(M))
whe e
x
deno es
he
Eule -Poinca écha ac e is ic
and
Q :M
->-
M is
ob ained,
as
explained
be o e,
by bowing up
in
u n
each
o
he
B
i
.

Hence
Gis
G-equi-
a ian
and he
ac iono
G
on
M
does
no
ha e
ixed
poin s
.
Choose nex an
elemen
h
o
he Lie
algeb a
o
G
such
ha
exp h
is
dense
in G
and le
Z
h
be
i s
co esponding
undamen al
ec o
ield
in
M
.
Explici ly
Z
h
is
gi en
by Zh (x) =
(dA
x
)
e
(h)
whe e
A
x :G
1M
is
gi en
by
A
x
(a)
= a
.x
and
e
is
he
uni
elemen
o
G
.
.The
éc o
ield
Z
h
has
no
ze os since
he
ac ion
o Gon M
has
no
ixed
poin
.
The e o e
x(M)
= 0
because
o
Hop
heo em
(see
co olla y
3,
page
399
o
[11)
.
On
he
o he
hand
x(á
-1
(U))
= E
x(P(E
j ))
and
since
he
ac iono
i=1
G
on
P(E
j )
has
no
ixed poin
he
same
a gumen
as be o e
yield
x(P(E
j
))
= 0
(i
=
1,
.
, )
.
84
The e o e
(5)
can
be
w i en
as
ollows
(6)

x(U-F
G
(M))
=
X(M-FG(M))
.
Finally
we deduce
om
(4)
and
(6),
using
also
he
ob ious
ac
ha X(U)
=
X(FG(M)),
(7)

X(M)
=
X(FG(M))
.
We
show
now
some
applica ions
o
heo em
(3)
.
(8)
P oposi ion
.
Le
Tbe
a
maximal o us
o
a
compac
connec ed
Lie
g oup
G
.
Then
X(G/T)
is
he
numbe
o
elemen s
o
N(T)/T
whe e
N(T) is
he
no malize
o
T in G
.
P oo
:
The
ixed poin
se
o
he
ob ious
le
ac ion
o
T in
G/T,
is
gi en
by
F
T (G/T)
=
{xT
i
x
E
N(T)}
.
The e o e
FT (G/T) has
he
same
numbe
o
ele-
men s
as
N(T)/T
.
The
p oo
in
now
inished
by
applying
(3)
.
(9)
Co ollá y
.
Any wo
maximal
o i
o
a
compac
connec ed
Lie
g oup
a e
conjuga e
.
P oo
:
Le
T,T'
be
maximal
o i
o
a
compac
connec ed
Lie
g oup
G
and
con-
side
he
le ac ion
o
T'
on
G/T
gi en
by
'
.xT
=
( ' .x)
.T( ' E
T'
xE
G)
.
We
know om
p oposi ion
(8)
and
heo em
(3)
ha
X(FT,(G/T))
=
numbe o elemen so N(T)/T
.
In
pa icula
FT' (G/T)

0
.
The e o e
he e
exis s
xEG
such ha 'xT
= xT
o all
'
E
T'
.
Thus
x-1
T'x
=
T
.
(10)
Theo em
.
Le
Kbe
a
closed
connec ed
subg oup
o a
compac
connec ed
Lie
g oup
G
.
Then
X(G/K)
=0 i
ank
K <
ank
G
and
)«G/K)=
n
G/n
K
i
ank
K=
ank
G (nG
is
he
numbe .o
elemen so
N(T)/T
o
T
being
a
maximal

o us
o
G
and
n K
is
he
co esponding
numbe o
K)
.
8
5

P oo
:
a)
Suppose
i s
ha ank
K
c
ank
G
and le
T be a
maximal

o us
o
G
.

The
le ac ion
o T
on
G/K
has
no
ixed
poin s
because
i
xK
=
xK
o
all
E
T
hen
x-1
Tx
C
K
which
is
imposible
since
ank
Kc
ank
G
.
We
use
hen
heo em
(3) o
conclude
ha
x(G/K)=
0
.
b)
Assume
now
ha ank'K
=
ank
G
and le
T be
a
maximal
o us
o
K
and
hence
o G
.
We
ha e
F
T (G/K)

=
{xK1
x-1Tx
C
K}
.

Bu
x-1Tx is
a
maximal

o us
in K, i x-1Tx
c
K
.
Thus by
co olla y
(9)

he e
exis s
k
E
K
such
ha
x-1
Tx
=
k-1
Tk
.
The e o e
xk-1
E
N
G(T)
(no maliza
o
T in G)
.
Hence
xK
=
xk
-1
.kK
=
xk-1 K
wi h xk
-1
E
NG
(T)
.
The e o e
FT(G/K)
=
{yKly
E
NG
(T)}
=
.
NG(T)AN
C
(T)
n
K) = N
G
(T)/N
K(T) =
NG
(T)/T
NK T
/T"
B
i
b
l i
og- ~p~Y_
The e o e
x(F
T
(G/K))
=nG
/n
Kand
we
inish
now
he
p oo _by
using
(3)
.
1
.-
W
.
G eub,
S
.
Halpe in,
R
.
Vans one
;
Connec ions,
Cu a u eand
Cohomolo-
gy
.
Vol
.I
.
Academic
P ess
.