Pub
.
Ma
.
UAB
Vol
.
28
2-3
Se
.
1984
In oduc ion
SEIFERT
ORBIFOLDS
AND
THEIR
UNITARY
TANGENT
SPACE
i .e
.
in
a
sec o
Ma ía
del
Ca menGazólazA e a
In
hispape we de ineSei e
o bi olds
as
geome ic
2~ R
i
s uc u eson
su aces,
allowingsingula i ies
o
angles
n
.
i
e
le

lao
.
small

2~

ni
p
.
1
n
.
we iden i y
wo
pa s

e
¡o

and

e

1

and
we
call his
co-
cien

p
/G
i

(Gi

is
he
g oup
gene a ed
by

i
,
gi en
as
21
R
i
(x)

= x
.

e

al

) .
We o o e ha heses uc u es
can
be
de ined,
in
a
na u al
way,
ompolygons
in
H
2
,
IR
2
o
S2
wi h
he
same
angles
and
ha
he
uni a y angen bundle
o
a
polygonp ojec s
o
a
Sei e
o bi old
.
This
pape dealswi h
a
geome ic
in e p e a-
ion
o
he
cons uc ion
o
olia ions
on
Sei e
bundles
as i
appea sin "Folia ions
on
Sei e
Bundles",
Ph
.D
.
Disse a ion,
Uni e si y
o
Chicago
1977 o
he
au ho
;
and
a
p ep in o
he
same
i le
in
1980,
wi hsome
co ec ions
on
he
compu a ions
and
mo e
gene al
esul s
.
The
same
p oblem
o
exis ence
o
ans
e se olia ions
is
app oached
wi h
di e en me hods
in
he
pa-
pe
"T ans e se
Folia ions
o Sei e Bundles
and
Sel
Homeomo -
phisms
o
he
Ci cle",
D
.
Eisenbud,
U
.
Hi sch,
W
.
Newmann,
Co
men
.
Ma h
.
Hel
56,
1981,
638,
660
.
I
.-
Sei e o bi olds
and
hei
uni a y
angen
spa
ce
1 .1
.
De ini ion
.-
A
Sei e
o bi old
is
a
closedsu ace
o
ge-
nus
g,B
9
,
wi h
he
ollowingaddi ionals uc u e
:
a)
Bg
is
co e ed
by
open
se s
{U
i
},
whe e
each
Ui
is
asso-
cia ed
o
a homeomo phism
o
i
:
Ui
--,
i/Gi
.
.
0

2II(3
.
whe e
P
.
=
{ e
l
,
0
6
9

1
,
>
0
being
a
small
numbe }
n
.
1
o
some
pai
o
in ege s
(R
i
,
n
i
)

wi h
0 <
Qi
n
i
,

and
G
i
2IIp
is
he
g oup
gene a ed
by
he a ion
ola
angle
o
b)
Whene e
U
i
U
j
,
he e
is
an
injec i ehomomo phism
ij
:
G
i
-
G
j
and
an
isome ic
embedding
p ese ing
he
canonical
o ien a ion
so
ha
he
ollowing
diag am
commu es
:
pi/Gi

`
p
j
/
ij
(G
i
)
Gi en
a
Sei e o bi old
s uc u e
on

Bg
,

we
can
choose
an open
co e

{U
a
}

o

B
g

and
a
subcollec ion

{U
i
}
¡=l,
. .
.,N'
such ha

na
=
1

o

a
q!

{l,
. . . .
N},

ni
*
1

o

i

=
1,
. . . .
N
.
We deno e
he
Sei e o bi oldby

B
(g,{(n
l
,R
i
)},N)'
wi h
he
con en ion
ha

g

will
be
posi i e
i

Bg

is
o ien a
bleand
g
will
be nega i e
i
B
g
is
no
o ien able
.
Le
us
suppose
ha
P
is
a
geodesicpolygon,
ha
can
be
ealized
in

H
2
,

IR2
o

S2
,

wi h
sides

{Si}i=l,
. . . .
2N+4g
posi i ely
o ien ed
and
angles
ai
,
be ween
S
i
and
Si+l'
wi h alues
2TW 1

2I1
nl
0
4g+N
.
2 W
2
2 P
N
2II
n
2

n
N
_

'

4g+N'
. . .
o
pai s
o in ege s

(ni
,(3
i
)

wi h

0
<
Qi
<
ni
.
Le
us
deno e
he
polygon
by

P(g,
(nl'Rl)'
. .
.'
(nN'RN)
andle

{
i
}
i
,
=1,
. . . .
N

be an isome y
sending

S
2i-1
o

S
2i
while
we shall
deno e
by
g
i
.
he
isome ysending
-1
S2
N
+4j
+i+2
.

j=l,
.. . .
g,

i

=
1,2
.
Simila ly
we
can
conside
a
polygon
S
2N+4j
+i
o
P(g
.
(n1
,
R 1
) p .
.
..
(N
.R
N
))
9
1
wi h
angles

{a i
}
_
{
2
~
1
,

2g+N
,

2~
2
,
2g
n
N
, . .
.,

2~
N
,

2
g+Ñ
. .
.}
i=1,
. .
.N+2g
.

The

{
i
}

a e
de ined
as
abo e
and

g
i
sends
S2N+2j+l
o
S
2N+2j+2
j-0,
. .
.,g-1
.
Le
G
be
he
g oup
gene a ed
by
{
i
,g
i
1
o
a
gi en
polygon
P(g
;(ni
.0l)~
.
.
.,(nN#RN))The
isome ies
i
a e
con~
juga ewi h
o a ions
and he
isome ies
g
i
a e
hype bolic
unc ions
;
hey
can
be
conside ed
also
as
di eomo phisms
o
he
ci cle
es ic ing
o
he
bounda y
o
H
2
(in
he
caseo
S2
o
IR2

hey
a e
o a ions)
.

The e o e
hey
ac
on
he
bounda y
o
P
and
on
he
ib e
o
hesepoin s
i.e
.
G
ac s
on
he
uni a y
angen
space
o
he
polygon
T
1
(P (g
;
(n1,pl)
, . . . .
(nNYON)
).
1
.2
.
Theo em
.-
llowing
conmu a i e
diag am
:
9
2
Gi en
a polygon

P(g,(n1,Q1),
. . . .
(n
N
'0
N
)

we
ha e
he
o
T

T
1
(P)
1
íP)
--~

G
1)
p
is
a Sei e
bundlewi h
Sei e
in a ian s,
[3],
(0,

o,

g,

N-2+2g,

n
i
,

Ql,
. . . .
n
N
,

O
N
)
o
m
lo i + i
ni =
1,

0
<
Si
<
n
i
i

g
>
0
.
1')
p
is
a Sei e
bundle
wi h
Sei e
in a ian s
:
(0, n
.,
-1
g1
,
N-2+1
g1
,
n
1
1
11
. . .
l
o
N
)

o

m¡oi
+
7ini
=
1,
0
<
,
i
<
ni
i

g
<
0
.
2)
II

gi es
o

P/
G

an
s uc u e
o Sei e
o bi old
B
(g,
{
(ni,
Ri)
}
,

N)
T
(P)
G
~
D
3)3)

G

~>

admi s
a
connec ion
on
he
Sei e bundle
.
1
.3
.
De ini on
.-
A
connec ion
in
a
Sei e
bundle
:M
-->
B,
g
>
0,

is
a
one o
0
E
A'(M)
-sa is ying
:
2)
Rg-
e
=
6
whe e
R
g
is
gi en
by
he
ac ion
o
g
on M,
and
a
X
:
IR
-->
TXM
is
he
induced
map
on
he
angen
spaces
o
he
map
S1
---,
M,
gi en
by
he
ibe
in
a
poin
.
I
B
is
a
non-o ien ableclosed
su ace,
TI
:
B'
-->
B
is
he
o ie i ed
doubleco e
wi h
g oup
G
o
co e ing
ans-
o ma ions,
hen
a
connec ion
o
a
Sei e bundle

(M

P
""
B)
is
a
one
o m
9
wi h wis ed
coe icien s
A'(M
;IR{G]) such
ha
11
*
(j)
is
a
connec ion
o he
double
Sei e
bundle
o
o e
B'
.
We
call
he
image
o
aX
e ical
ec o s
o
he
connec
ion
and
he
ke nel
o
e
X
ho izon al
ec o so
he
connec ion
.
p oo
o
1
.2
.-
a)

Suppose
ha

E
R
1
<N
-
2

+
2g
(E~
i
<N
- 2
+
I
g1
)

he e
n
.

n
.

-
1

1
9
3

o e
P
can
be
ealized
in
hype bolic
o
euclideangeome y
.
G
is
a
subg oupo Moebius
ans o ma ions
o
H
2
and
he e o e
hei s
es ic ion
o
he
bounda y
is
a
subg oup
o
TOPS
1
(homeomo phisms
o

S1
) .
i)

We
claim
ha

[a2g
,9
2g-1
1

...

[
g2'
g1"
"'012"
l
is
a
ansla ion
by

N-2
+2g
i
O<-
i
(x)
-x
<
l,

0'!!!
1
(X)
-x
<l
.
ü)

1G

is_a
Sei e
bundlewi h
Sei e
in a ian s
iii)

b

=
[
g2g'g2g-l1

o
.
. .
o
20
1
To
p o e
ii)
obse e
ha ,
i
B
E
is
a
smallball
in
he
e ex

2
~
R
1
,

hen

BE n
P

is
isome ic
o
a
smallball
in
he
2

ie
21W
i
cen e o

H

wi h
poin s

e

,
0
<,O-<
n
..

Le

-
he
equi-
alence ela ion
(0,

o,

b,

n
11
(3
1,
. . .
,Q
N
)
.
2IW
,

2IT(i
( e
e
,ele)

^,
( ei
(e
+

ni
l)
,

e
l
(0
+

nli)
)
.
.
We
cons uc
a
co e ing
map
om
a
solid o us o
1 (B
E
ñ
P)

as
ollows
:
(B
FnP)
x
S1
T
-~
( ele,e( e
ie,Ri'

ei(~
-
wi h
e=el
+
2M
ose
,
<2
1
n
i
n
i
The
g oupo co e ing ans o ma ions
is
gene a ed
by
:
2
n
"i
2II
i
P(ni,m
i
)( e
le
,

e
le)

_

( e

(e

+

n
i
)
"

e
+
n
l
) )
whe e0imi
+
Tini
=
1
o
in ege s
1
~<
m
i
<
ni
.
b)

I

~
>
N
- 2
+
2g
(
ñ
i
>
N
- 2 +
I
gl
)
i

i
hen
he
polygon
can
be
ealized
in
S2
as in
case
a)
.
The
p oo
o
i)
and
iii)
is
jus
a
i ial
geome ic
a -
gumen
.
Thenwe ha ep o ed
1)
o
1')
.
2)
is
i ial
and
3)
is
gi enby
he
p oyec ion
o
he
s anda d
connec ion
o
H2,
I2
1
.4
.
Dé ini ion
.-
In
he
condi ions
o
heo em
1, 2,
we
call
T
1
(P)/G

he
uni angen bundle
o
he
Sei e
o bi old
B(g,{(ni,pi)},N)-
and
he
p oo
wo ks
I
g=0, N=1,
o
g
=0,
N=2,
n
1
*
n2
we de ine
he
uni a-
y
angen
bundle
o

B
(0,{(n
i
,
pi
)},N)

as
he
Sei e
bundle
o e
S2
wi hSei e in a ian s
(0,
o,
o,
-
1,
(n1
,p
1
))
and
(0,o,o,o,(n1,p1),(n2,p2))
1
.5
.
De ini ion
.-
A
Sei e
o bi old
B
(g,{(n
i
,
p
i
)},N)
associa ed
numbe
has
an
ha we
call
Eule
ch ac e is ic
o
he
Sei e
o bi old
.
Obse e
ha
1 .5
is
wellde ined
by
heo em
5
in
Sei e 's
pa-
pe
131
.
Lis
o
e e ences
.-
[1]
Magnus,
W
.
:
"Non
euclidean essala ions
and
hei g oups"
.
AcadenucO ess
(1974)
.
[2]
Kobayashi
and
Nomizu
:
"Funda ionso
di e en ial
geome y"
.
In e science
.
[3]
Sei e ,
H
.
:
"Topologie
d eidimensionale
ge ase e
Ráume"
.
Ac
.
Ma h
.,
60
(1933)
.
[4]
Siegel,
C .L
. :
"Topic
in
complex
unc ion
heo y,
2"
.
Wiley
(1969)
.
[5]
Thu s on,
W
.P
. :
"Non
cobo dan
olia ions"
.
Bull
.
Ame
.
Ma h
.
Soc
.
78
(1982),
511-514
.
Rebu
el 26
d'abn
U
de¡
1984
SPAIN
R
.
-(N-2+1g1
-

ñ
l
) .
g
<
0
.
i
Uni e sidad
Au ónoma
de
Mad id
Di isión
de
Ma emá icas
CiudadUni e si a ia
de
Can o
Blanco
Mad id