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Milnor bration and bred links at

De
6 pages
Milnor bration and bred links at innity Arnaud Bodin January 29, 1999 Introduction Let f : C 2 ! C be a polynomial function. By denition c 2 C is a regular value at innity if there exists a disc D centred at c and a compact set C of C 2 such that the map f : f 1 (D) n C ! D is a locally trivial bration. There are only a nite number of critical (or irregular) values at innity. For c 2 C and a suciently large real number R, the link at innity K c = f 1 (c) \ S 3 R is well-dened. In this paper we sketch the proof of the following theorem which gives a characterization of bred multilinks at innity. Theorem. A multilink K 0 = f 1 (0) \ S 3 R is bred if and only if all the values c 6= 0 are regular at innity. We rst obtain theorem 1, a version of this theorem was proved by A. Nemethi and A. Zaharia in [NZ] (with \semitame as a hypothesis). Here we give a new proof using resolution of singularities at innity.

  • everywhere dened

  • reduced

  • link then

  • crossing divisors

  • component

  • dened morphism

  • polynomial function

  • hopf link


Voir plus Voir moins

Milnor
ration
and
ration
singularities
j
red
and
links
ts
at
sp
init
factorial
y
ha
Arnaud
the
Bo
value
din
r
Jan
this
uary
w
29,
and
1999
S
In
can
tro
enables
duction
in
Let
1.
f
in
:
(0)
C
b
2
if
!
decomp
C
\
b
en
e
is
a
v
p
]
olynomial
follo
function
on
By
init
deition
.
c
e
2
of
C
binatorial
is
of
a
is
r
c
e
of
gular
R
value
ther
at
alue
inity
not
if
of
there
reduced
exists
en
a
is
disc
=
D
ee
cen
K
tred
m
at
f
c
alue
and
0
a
y
compact
v
set
f
C
not
of
w
C
y
2
0,
suc
critical
h
resolution
that
init
the
metho
map
to
f
re
:
drom
f
Milnor
1
of
(
v
D
a
)
.
n
ther
C
critic
!
inity
D
0
is
homotopy
a
j
lo
S
cally
f
trivial
S
ration
exists
There
The
are
ma
only
regular
a
ma
ite
what
n
it
um
is
b
olynomial
er
an
of
o
critic
otherwise
al
m
r
K
irr
1
e
3
gular
1).
)
ts
values
are
at
y
inity
of
.
osition
F
If
or
regular
c
init
2
6
C
a
and
at
a
W
suien
Rudolph
tly
in
large
the
real
(0)
n
R
um
In
b
theorem
er
do
R
e
,
yp
the
v
link
particular
at
e
inity
alue
K
.
c
of
=
at
f
y
1
This
(
d
c
us
)
describ
\
the
S
and
3
mono
R
y
is
the
w
ration
elleed
terms
In
com
this
in
pap
arian
er
of
w
resolution
e
f
sk
Theorem
etc
If
h
e
the
no
pro
al
of
at
of
outside
the
=
follo
then
wing
the
theorem
class
whic
f
h
f
giv
:
es
3
a
n
c
1
haracterization
!
of
1
red
e
m
a
ultilinks
ation
at
v
init
0
y
y
.
e
Theorem
or
A
One
multilink
y
K
ecify
0
kind
=
ration
f
is
1
f
(0)
a
\
p
S
then
3
is
R
op
is
b
r
ok
e
osition
d
it
if
a
and
ultilink
only
of
if
0
al
f
l
(0)
the
S
values
R
c
paragraph
6
The
=
eigh
0
of
ar
0
e
giv
r
b
e
the
gular
ultiplicities
at
the
inity
decomp
W
of
e
.
st
0
obtain
a
theorem
v
1,
at
a
y
v
c
ersion
=
of
is
this
critical
theorem
alue
w
init
as
,
pro
Neumann
v
L
ed
pro
b
ed
y
R
A
that
N
link

1
emethi
\
and
3
A
is
Zaharia
red
in
the
Z
wing
ith
2
emitame
e
as
not
a
v
h
an
yp
h
othesis
othesis
Here
the
w
alue
e
in
giv
0
e
b
a
a
new
v
pro
at
of
y
using
1Theorem
2.
Supp
r
is
disjoin
ose
C
that
in
c
re
6y
=
1
0
dicritic
is
:
a
order
critic
morphism
al
value
D
at
1.
inity
D
for
of
f
,
,
The
then
restricted
the
b
multilink
~
K

0
P
=
f
w
1
1
(0)
)
\
z
S
,
3
(
R
,
is
gr
not
tains
a
w
r
er
e
:
d
f
multilink
critical
W
eac
e
W
b
in
egin
!
with
!
deitions
h
the
to
second

part
is
is
2
dev
2
oted
w
toy
the/
pro
irreducible
of
w
of
is
theorem
ving
1.
e
W
D
e
D
conclude
).
with
C
the
al
pro

of
v
of
D
theorem
divisor
2.
onen
1
dic
Deitions
=
As
ite
in
comp
N
D
a
D
multilink
at
L
alues
(
c
m
the
)
;
(
c
m
bam
=
increase
(
wingps
m
a
1
p
;
and
:
p
:
suc
:
2
;
~
m
w
k
w
))
w
is
to
a
.
link
we
ha
.
ving/
eac
C
h
f
comp
w
oneno
t|
Ly
iy
w
C
eigh
or
ted
onen
b

y
L
the
L
in
line
teger
2
m
equation
i
0)),
.
three
The
w
m
=
ultilink
e
L
=
(
(
m

)
)
is
1
a
a
r
omp
e
restriction
d
to
multilink
ramid
if
the
there
e
exists
the
a
restriction
diren
h
tiable
these
ration
is

divisor
:
3.
S
D
3
2
R
is
n
um
L
suc
!
ts
S
crit
1
1
suc
:
h
g
that
v
m
y
i
the
is
1
the
:
degree
and
of
alues
the

restriction
D
of
v

divisor
on
is
a
union
meridian
o
of
no
L
n
i
of
.

A
minimal
re
,

obtain
1

(
P
x
p
)

is

a
P
Seifert
that
surfac
P
e
suc
for
that
the
f
m

ultilink
extends
The
a
link
elleed
K

0
from
=
w
f
C
1
1
(0)
This
\
the
S
ak
3
esolution
R
C
is/
a
f
m
ultilink
P
the
~
w
eigh

ts

bo
eing

naturally|
givy
eny
by
yy
the
C
m/
ultiplicities
P
of
F
the
an
factorial
comp
decomp
t
osition
of
of
1
f
(
.
1
A
(
r
1
e
the
d
of
link
P
is
ha
a
the
red
(
m
=
ultilink
w
ha
distinguish
ving
cases
all

its
(
comp
)
onen
1
ts
w
w
denote
eigh
1
ted

b
w
y
1
+1.
2.
Then
w

D
is
=
called
P
an
,
op
is
en
dicritic
b
c
o
onent
ok
the
de
of
c
w
omp
D
osition
a
.
co
Next
ering
w
de
e
e
giv
of
e
is
deitions
degree
and
this
results
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ab
whic
out
con
resolutions
all
see
comp
[L
ts
W
the
Let
al
n
D
b
.
e

the
(
degree
)
of
c
f
C
and
there
F
a
b
n
e
b
the
of
corresp
h
onding
onen
homogeneous
collected
p
D
olynomial
=
with
c
the
[
same
:
degree
[
The
c
map
.
~
irregular
f
alues
:
init
C
for
P
are
2
v
!
c
C
;
P
:
1
;
,
g
~
the
f
v
(
of
x
map
:
w
y
to
:
dic
z
moreo
)
er
=
h
(
D
F
i
(
a
x
t
y
of
;
b
z
os
)
e
:
w
z
the
n
um
)
er
is
blo
not
of
ev
w
erywhere
a
deed
w
nev
y
ertheless
in
there
to
exists

a
:
minimal
p
comp
C
osition
2
of

blo
=
wing
f
ups


:
w
p
:
C

1
w
h
!
the
C
2
1
p
(0)
1
regular
e
cuts
with
the
simple
divisor
=
D
ts
dic
a
transv
gr
ersally
ulus
and
can
is
t
a
th
normal
(
crossing
of
divisor
ab
This
dicritic
is
dic
the
1
p
is
artial
e
r
ose
esolution
With
for
d
the
similar
v
at
alue
smo
c
al
=
annulus
0.
onen
W
This
e
f
con
one
tin
\
ue

with
onen
blo
1
wingps
e
in
C
order
d
to
eac
obtain
(

)
t
giv
;
p

restriction
t
D
;
If

,
t
cal
suc
e
h
with
that
divisor
eac
in
h
of
re
with
of
D

ords
t
eac
cuts
y
the
t
divisor
of
D
C
dic
,
transv
With
ersally
e
and
onent
all
?
the
.
res
for
of
e

U
t
of
are
1
normal
1
crossing
U
divisors
U
This
f
is
v
the
2
total
U
r
t
esolution
lo
.
suc
F

or
W
the
S
total
(
resolution
"
the
n
v
can
alues
the
c
)
1
whose
;
a
:

:
D
:
equations
;
the
c
w
g
ha
0
lo
coming
happ
from
comp
the
the
comp
1.
onen
p
ts
1
D
1
of
ach
the
omp
new
interse
D
crit
crit
is
with
other

in
t
0
(
dicritical
D
is
)
reduced
=
p
c
o
i
a
are
fact
the
v
critical
1
v
;
alues
is
at
v
init
argumen
y
pro
.
2.
2
c
Milnor
with
ration
crit
at
of
init
e
y
Fibration
Un
D
til

the
D
end
dicritical
of
and
this
e
section
oin
w
in
e
(
supp
[
ose
By
that
2
the
w
only
an
irregular

v
U
alue
1
at
;
init
regular
y
of
for
Let
f

can
parametrisation
b
F
e
p
the
U
v
ho
alue
co
0.
v
Let
that

e
=
u

u
t
c
coming
so
from
lo
the
b
total
v
resolution
)
In
a

e
t
b
the
facts
sphere
that


1
comp
t
(
(
ciated
S
a
3
consist
R
uli
)
ration
is
D
diomorphic
D
to
[
the
0
b
the
oundary
are
S
to
of
lo
a
case
neigh
e
b
us
ourho
v
o
to
d
ok
of
what

ens
1
the
t
onen
(
of
L
dicritical
1
Lemma
)
The
ee
oth
[D]).
oints
Instead

of
t
studying
L
f
)
=
e
j
dicritic
f
c
j
onent
restricted
nonmpty
to
ction
S
D
3
=
R
0
n
an
f
In
1
w
(0)
the
w
tersection
e
D
study
with
=
h
j
comp

t
j
empt
restricted
or
to
to
S
single
n
oin

Pr
1
of
(0).
is
Let
consequence

the
b
that
e
o
the
e
restriction
P
of
n
=
0
j
1g


j
a
to
co
S
ering
n
similar

ts
1
can
(0).
v
By
Lemma
c
Each
hanging
al
the
omp
sphere
D

D
1
D
t
=
(
is
S
de
3
e
R
1
)
2.1
in
on
to
(
S
)
w
D
e
D
only
Let
kno
b
w
a
that
comp

t
is
let
in
b
the
the
homotop
p
y
ts
class
D
of

f
t
=
L
j
)
f

j
(0).
.
lemmas
As
and
in
w
MW
kno
there
that
is
is
a
ann
corresp
and
ondence
j
b
:
et
!
w
P
een
n
the
0
irreducible
1g
comp
a
onen
co
ts
ering
of
order

.
1
u
t
C
(
b
L
a
1
of
)
.
and
or
a
h
W
oin
aldhausen
of
decomp
w
osition
c
of
ose
S
cal
n
ordinates

u
1
)
(0)
h
in

to
b
Seifert
written
three
(
manifolds
v
W
=
e
.
will
e
pro
ho
v
S
e
that
that
is
the
cally
restriction
en
of
y

j
to
j
the
"
Seifert
where
manifold
is

small
(
ositiv
D
real
)
um
asso
er
ciated
these
to
one
an
calculate
y
the
irreducible
of
comp
to
onen
Seifert
t
onen
D

of
D

asso
1
to
t
is
(
ration
L
res
1
of
)
ann
is
32.2
Fibration
in
a
v
previous
alues
neigh
the
b
=
ourho
j
o
ultilink
d
o
of
er
a
the
nonimple
surface
p
at
oin
v
t
0
In
a
a
D
neigh
of
b
(0)
ourho
ultilink
o
ose
d
j
V
Moreo
of
do
a
for
nonimple
(0):
p
Moreo
oin
2
t
at
i
0
a
1
p
still
oin
no
t
ration
b
all
elonging
j
to
urthermore
a
b
dicritical
reduced
comp
without
onen
g
t
surface
D
)
and
f
another
ers
comp
1
onen
ho
t
resolution
D
exists
0
p
2
onen

e
1
ration
t
on
(
1
L
trace
1
is
)
2
[
.

function
1
an
(0),
Similarly

ration
is
?
deed
y
in

appropriate
b
lo
1
cal
whic
co
1
ordinates
S
b
is
y
ab
(
is
u
osition
v
on
)
Nonbred
7!
of
u
y
d
with
.
con
Let
init
T
f
b
c
e
R
the
ultilink
tubular
S
neigh
complex
b
6
ourho
j
o
\
d
.
of
a
D
F
\
p
V
v
giv
onen
en
3
b
)
y
e
(
u
j
j
v
o
j
er
6
a
"
b
).
torus


j
n
T
,
dees
the
a
=
ration
radii
whose
ulus
res
n
consist

of
f
d
p
ann
p
uli


en
1
decomp
(

e
cally
i
F
)
6
\
W
T
conclude
=
and
n
=
(
is
u
a
v
o
)
\
2
and
T
\
;
co
j
n
v
so
j
j
=

";
S
u
ration
6
the
=
v
0

and
op
u
ok
d
a
=
dep
j
b
u
not
j
ultilinks
d
yp
=
2
e
of
i
e
o
f
:
<
F
es
or
critical
T
f
0
.
a
=
tubular
j
neigh
1
b
c
ourho
S
o
a
d
the
of
0
D
(0)
0
R
\
er
V
um
giv
with
en
!
b
j
y
links
(
!
j
3
u
cut
j
e
6
!
"
v
),
y
the
the
re
=

init
1
and
(
0,
e
dicritical
i
with
)
at
\

T
L
0

is
D
also
dicritical
a
where
union
p
of
q
d
=
ann
i
uli
:
These
v
diren
this
t
is
pieces
m
t
ration
nicely
ecause
on
a
the
D
torus

@
S
T
"
\
f
@
g
T
the
0
of
.
re
So
v
with
cst
a
p
plum
of
bing
ann
of
D
T

and
f
T
g
0
v
,
If

is
is
reduced
a
olynomial
ration
then
on
=
V
and
.
is
2.3
op
Fibration
b
in
ok
a
osition
neigh
,
b
is
ourho
lo
o
a
d
if
of
\
the
dic
strict
=
transform
.
Let
e
F
w
b
b
e
collecting
an
gluing
irreducible
results
comp
j
onen
j
t
a
of
in

neigh
1
ourho
(0)
d
n
S
D

0
(0)
hic
on
h
V
corresp
S
onds
h
to
v
the
S
ae

set
(0),
f
=
1

(0)).
:
F
n
can
1
in
!
tersect
1
D
a
0
F
or
with
D
discussion
dic
o
.
e
If
j
F
j
\
an
D
en
0
o
6
decomp
=
or
?
m
then
ration
lo
ending
cally
f
in
eing
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conclusion
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regular

v
e
alue
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init
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y
118
or
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ach
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m
a
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ory
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24
S
e
3
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R
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Math
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red
mono
when
fr
c
t
6
A
=
Indag
0
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their
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t
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and
v
,
alue
Unfoldings
at
282
init
aul
y
oratoire
.
onne
Ac
b
kno
Norm
wledgmen
141-169.
ts
L
I
C
thank
A
Professor
appr
F
the
ran
onje
coise
Ko
Mic
17
hel
F
for
C
long
On
discussions
omies
and
olynomial
for
C
her
,
man
Z
y
emethi
ideas
Milnor
References
inity
[A]
N
E
323-335.
Artalartolo
Complex
Une
curves
d
at

v
emonstr
98
ation
W
g
Rudolph

the
eom
Ann

and
etrique
knot
du
Math
th
349-351.

e
eor
T

I
eme
Emile
d
de
bhyankaroh
T
,
F
J
dinicardpslser
reine
Ec
angew
Sup
Math
(1991),
464
[L
(1995),
D
97-108.
^
[D]
and
A
W
Durfee
er
Neighb
ge
orho
al
o
o
ds
to
of
Jac
algebr
c
aic
ctur
sets
,
,
dai
T
J
rans
(1994),
Amer
W
Math
Mic
So
and
c
W
276
er
(1983),
the
517-530.
dr
N
of
D
p
Eisen
map
bud
om
and
2
W
C
Neumann
preprin
Thr
(1998).
e
A
eimensional

link
and
the
Zaharia
ory
r
and
at
invariants
,
of
Math
plane
S
curve
(1992),
singularities
[N]
,
Neumann
Ann
algebr
of
plane
Math
via
Stud
links
110
inity
,
In
Princeton
en
Univ
Math
Press
(1989),
(1985).
R
MW
Neumann
D
L
L
Unfoldings
^
knot
e
ory
F
Math
Mic
278
hel
409-439
and
Corrigendum
C
in
W
the
eb
,
er
Ann
Courb
(1988),
es
Univ
p

olair
P
es
Sabatier
et
oulouse
top
I
olo
lab
gie

des
Picard
c
route
ourb
Narb
es
31062
planes
oulouse
,
4,
Ann
rance
scien
o
t
6