Signature of branched fibrations

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signature of the branched fibration T (Dh+I,F) when 4[ (k + Z). Theorem 3.6 ... 1. ~:E + S represent a sum of the generators of HI(E;~ ) with orientations specified.
Signature

of B r a n c h e d

Fibrations

by Louis H. K a u f f m a n

I. I n t r o d u c t i o n A branched algebraic fibers

fibration

varieties

is a topological

that is p a r a m e t r i z e d

lying o v e r a c o d i m e n s i o n

in a l g e b r a i c

geometry,

topological

notion.

associated

with

degeneration

two s u b m a n i f o l d

b u t there

is a w i d e

W e have c h o s e n

isolated

(complex)

will be closely

analog of a d e g e n e r a t i n g

over a manifold V~

avenue

a definition hypersurface

associated

M

M

.

This

of choice

is a common

situation

for the c o r r e s p o n d i n g

that a b s t r a c t s

the m a i n features

singularities.

to a fibered

family of

, w i t h the d e g e n e r a t e

knot;

T h i s means

the k n o t plays

that the the role

of the link of the singularity. In s e c t i o n pairing.

a fibered k n o t values

2 we r e v i e w the d e f i n i t i o n s

Theorem

2.9 shows

is n o n - t r i v i a l

singularities. ([KN]). [KN].

3 branched

This

Theorem

K ® L = ~M F ~

where D k+l

construction

This signature

fibered

numbers,

k n o t and S e i f e r t

the S e i f e r t p a i r i n g

associated

~F = K .

fibration

T

with unit-length

of signatures

Brieskorn

singularities,

links

S3 .

by mimicking

of

eigen-

This leads

towards

branched require

knots

and

T (Dh+I,F)

K

fibrations. a m o r e general

coverings

concept

~ (K) T 4[ (k + Z)

This result (see

by a p u l l - b a c k

a more

general

to be a d i f f i c u l t

denote

general ([HI)

formula problem

fibration

0 ~ D2 .

L . the

Theorem

3.6

of the m o n o d r o m y generalizes

some

[N]).

cases

involving

and c o n c o r d a n c e

due to F. H i r z e b r u e h

of b r a n c h e d

[DK] and

to some special

coverings

of a m o r e

This seems

Let

when

L .

3.6 is a p p l i e d

along a s u b m a n i -

w i t h the fibered k n o t

of the e i g e n v a l u e s

and

cyclic b r a n c h e d

D k+l

knot has a In fact

T : D i+l + D 2 , b r a n c h e d o v e r

problem:

in terms

of b r a n c h e d

the q u e s t i o n

fibrations.

is o b t a i n e d

associated

of

and a fibered k n o t

of

fibration

construction

The p r o d u c t

fibration

forms of

a method

.

fibration

5 we show how to c o n s t r u c t

fibrations

(sk,K)

, a branched

to a s i g n a t u r e

4, T h e o r e m

K =

The b r a n c h e d

fibration

and the S e i f e r t

In s e c t i o n

to fibered

of the k n o t p r o d u c t

in terms of b r a n c h e d

for this s i g n a t u r e

In s e c t i o n

and r e l a t e d

(Sk+~+I,K @ L)

is d i r e c t l y

leads

of the b r a n c h e d

in

to a k n o t

K eL=

knot

M = T (Dk+~F)

situation

computations

are d e f i n e d

the m a i n p r o p e r t i e s

that is d e f i n e d

with

a formula L

fibrations

from a s i m p l e r b r a n c h e d

This b r a n c h e d

for

only on subspaces

associates

a new product

spanning m a n i f o l d

gives

of knot,

the c o m p l e x

is based on joint w o r k of the author and W a l t e r N e u m a n n

3.2 states

This c o n s t r u c t i o n

(SZ,L)

fold

over

of the monodromy. In s e c t i o n

i =

that,

invariants

of

class of b r a n c h e d for r a m i f i e d

for signatures and may,

or a change

covers. of

in fact,

in viewpoint.

204

In a n y and

to h a v e

theory,

case,

shown

and

some

diffeomorphism, Knots

a

fibered

section

knot

of

the

Definition

2.1.

S3

Definition

is

include

F C S

framework

connections

among

are

~

for

these

questions

singularities,

knot

A

smooth;

denotes

isormorphism

or

then

standard

over on

fact

the

notions

complex

subspaces will

(Sn,K)

is

be

in k n o t

numbers,

associated

of

use

a pair

for

consisting

of

oriented

the

is

to b e s ~ h e r i c a l .

spanning

said

is

intended

of disjoint surface

, embedded

in

for

sn

a knot

so

that

include

K =

.

eigen-

n-sphere

Sn

.

That

If

is,

K

a

circles.

(Sn,K)

~F = K

K C

links.

oriented

of

computations

an oriented

submanifold

to

embedded

main

pairing

unit-length

signature

closed,

knot

The

Seifert

with

the

compact,

definition

theory.

the

is a c o m p a c t

Here

the

oriented

symbol,

~

, denotes

boundary. is w o r t h

of this

Lemma

K =

two,

, this

F

some

only

a collection

2.2.

It proof

initial

homeomorphism.

that,

This

A knot

(n-l)-manifold oriented

recall

shows

n = 3

manifolds

denotes

will

2.9)

sphere,

When K C

an

Knots

a codimension

is a h o m o t o p y

link

given

interesting

all

is n o n t r i v i a l

III.

and

=

monodromy.

section

Sn

the

paper

Fibered

(Theorem

values of

the while

and

This result

of

to h a v e

signatures.

Throughout

II.

I hope

the

Proof.

If

K =

that

is s h o r t

argument

2.3. n

remarking

result

in

the

next

(Sn,K)

for

K

Let

E = Sn - N °

knots

and

always

it m o t i v a t e s

have

spanning

the

definition

surfaces. of

Since

fibered

knot,

the we

lemma.

is a n y

knot,

then

there

exists

a spanning

surface

. where

N

is a c l o s e d

tubular

neighborhood

of

K

.

Note

1 that

H

(E;~)

=

[E,S I]

where

[ , ]

denotes

homotopy

classes

of m a p s .

Let

1 ~:E

+

by

the

* e S

S

represent orientation

1

.

It

a sum of of

KC

is n o t

hard

corresponding

to

K x *

F C Sn

~F = K

with

Remark. be

It m a y

a smooth

definition Definition if t h e r e

.

see

generators We may

that

Thus,

by

N

of

HI(E;~

assume

that

e

)

with

is t r a n s v e r s e

is d i f f e o m o r p h i c

adding

a collar

to

orientations

d

to -i

K x (*)

specified

regular

D2

with

, one

to

~ -i(,)

obtains

.

happen

that

the

In

is as

follows:

2.4.

A knot

i) b - l ( 0 ) ii)

Sn

to

fibration.

is a s m o o t h

the

this

K =

mapping

= K ~

Sn

b / I I b l J:S n - K ÷

mapping

~:E

case

says

(Sn,K)

one

is

b:S n ÷ D 2

+

S1

described

that

fibered

~

with

, transverse

. S1

is a s m o o t h

fibration.

above

is a f i b e r e d

fibered to

can

structure

0 ~ D2

be

knot.

such

chosen The

b:S n ÷ D 2 that

to

formal

,

205

Here

llbl I (x)

fibered knot with

denotes

fibered

the d i s t a n c e

structure

b

from

will

b(x)

to the o r i g i n

sometimes

be i n d i c a t e d

in

jR2

A

by the n o t a t i o n

(Sn,K;b) The first example (sl,~;a) cation; fiber

Here a F

a-l(1)

=

mathematics

unfolds

The m a p is v a c u o u s l y

{i,~,~ 2 , .... ~ a-l}

=

from the empty

empty knots.

knot is the empty knot of d e g r e e

is d e f i n e d by the formula

is an integer).

is

these

of a fibered

a:S 1 ÷ S 1

This

comes

set,

a(x)

transverse

where

= xa to

0 e D2

~ = exp(2~i/a)

so do m a n y

, [a] =

and a typical

Just as all of

interesting

about b y the p r o d u c t

a

(complex m u l t i p l i -

knots

construction

come

from

discussed

in the

n e x t section. Another

construction

that gives

of the link of a singularity. that

f(0)

Vf =

= 0 .

g

f:~n ÷ ~

f

involves

the n o t i o n

be a complex polynomial

has an isolated

singularity

mapping

such

at 0 if the g r a d i e n t

(~f/Szl,$f/~z 2 ..... Sf/~z

neighborhood of

Let

One says that

rise to fibered knots

f

by

of

0 e ~n

L(f)

=

sufficiently

shows

that

mapping

) v a n i s h e s at 0 8 n a n d is n o n - z e r o in some d e l e t e d n U n d e r these c o n d i t i o n s one c a n d e f i n e a knot, the link

.

(s2n-l,L(f))

small,

[(f)

L(f)

where

f/l[fl I:S~n-l^ - L(f)

÷ S1 .

z0a 0 + Zla 1 + "'" + zann p r o v i d e

of a s p a n n i n g Definition

2.5.

Let

Seifert pairing Sn - F

numbers

normal in

obtained

denotes

where

homology

group.

spanning

surface

T,(X) C

While F

boundary

Let

embedded

S e i f e r t pairing, Then,

for

is to c o n s i d e r of these

6K(x,y)

x,y

H,(X)

(see

is d e f i n e d

Milnor

is given by the [B])

f(z)

=

for the e m b e d d i n g

pairing:

surface

F~

as follows:

into its c o m p l e m e n t

= i(i,x,y)

([M])

For

singularities.

invariants

is the S e i f e r t

F

is the t o r s i o n pairing

, w e h a v e chosen

between

2.6.

polynomials

by p u s h i n g

the S e i f e r t

where

i

F

F2nc in

Sn Let

.

The

i:F ÷

along the

denotes

linking

S 2n+l

(F)

dimension,

.

be a c o m p a c t Let

:Hn(F) ,

=

integral

upon the choice

of

in the notation.

then there

oriented

Q:Hn(F)

× Hn(F) + ~

+

H,(X)

is a w e l l - k n o w n

form on

F

.

This

([LI]) .

× Hn(F)

= @(x,y)

T h a t is,

of the r e d u c e d

depends

and the i n t e r s e c t i o n

t h e o r e m of J . L e v i n e

and let e H

has a m i d d l e

S 2n+l

subgroup

actually

to omit this d e p e n d e n c e

the S e i f e r t p a i r i n g

is given by the f o l l o w i n g Theorem

In

The Brieskorn

t h e free p a r t of the r e d u c e d homology.

If the s u b m a n i f o l d relationship

s .

of the c o m p l e m e n t

be a k n o t w i t h s p a n n i n g

Then

of

0 < ¢

corresponds

.

We shall

by induction Note that

on

such that

let

0

Hence

are a s s u m i n g

I ~ 1 , this

= 1

subspace

I I~II = 1 .

Hence, I I~I]=

T h u s we m a y a s s u m e on

0 ~

s

k < s

at

while

induction,

suppose

that

computation

This completes

since

completes

.

Hence

.

[]~II = 1 with

= -l

.

positive

Bs_ 1

= exp(2~i/a)

if

or

Al

@IBm_ 1 7 0

@(ek, e s) ~ 0



The same a r g u m e n t

that

I Ill I = 1 .

~ 0

+

(l-h)

.

denote

the

then we now make

for any

= @ ( e s , ( I - h ) e0) =

=

(l-i)O(e0,es)

as in t h e f i r s t i n d u c t i o n To c o m p l e t e

for or

this s e c o n d

0 ~ k < £ ~ s - 1 .

@ ( e l , e s) ~ 0 =>

@ ( e i , e j) = 0

But

for

there

~

as b e f o r e

the p r o o f

Then

Ilkl I = 1 .

i ~ s

that

Jordan blocks

and

that

=

IlIIl = 1 .

OlAf 7 0 => induction

corresponding

j ~ s .

- as-l)

This

I I~II = 1 . arguments.

The

to the s a m e e i g e n v a l u e

@ . is a d i r e c t Suppose (l-l)e

= (-1)n+l~

relationship that =

.

@(x,x)

(-l)n(l-i)e For example, =

~.

If

between = o~

0

.

If I

if ~ =

the e i g e n v a l u e and t h a t

~ ~ a-i

surface

with basis

[a] =

~ 1 , then

~ = 1 - w with (l-~o)...(l~

(sl,~;a)~

and

hx =

w ~ i,

n)

I = W0Wl...w--n.

the e m p t y k n o t of d e g r e e a,

~0(F)

shows

by very similar

~ =-(i-~)/(i-~)

then

for

< e s s ,e > = @ ( e s , ( l - l ) e s

This knot has spanning Thus

(I-~

I III I = 1 .

and hence

pairing.

= 1

and

s = 0 .

U n d e r this a s s u m p t i o n

~ 0

note that

is o b t a i n e d

then

Consider

integer.

=

for

[ e 0 , e l , . . . , e s _ I] ~

@(es,ek)

@(es,ez)

Different

Hence

n = 0

w i ~ 1 ,II~ill

An example.

= 0).

(as in the p r o o f above)

and

(-l)n@(y,x)

( i - ~ (l-h)

the proof

@IBm_ 1 z 0 .

that

assume

orthogonal

For e i g e n v e c t o r s

(l-l)/(l-~)

( 1 - 1 ) @ ( e 0 , e 0) = +

induction.

the f i r s t i n d u c t i o n

Then

Therefore

By i n d u c t i o n ,

the same calculuation

the v a l u e o f the S e i f e r t lx

< e 0 , e 0 > = @(e, (I-h) e0) =

( l - l ) @ ( e 0 , e 0) = @ ( e 0 , e 0) +

@ ( e s ,e k) = @ ( e k , e s) = 0

the s e c o n d

in fact,

Remark.

that

shows that

details will be omitted. are,

Let

@ ~ e 0 , e s) ~ 0 =>

T h e r e s t of the p r o o f

I

s .

@ 7 0 , @ ( e s , e s) ~ O .

( l - l ) @ ( e s , e s)

.

This completes

induction,

We n o w m a y t h e r e f o r e Thus,

Al

The proof will

< e 0 , e s > = @ ( e 0 , ( I - h ) es) = @(e0, (l-l)e s - as_l)

n o w shows t h a t

a similar

for

w e a s s u m e t h a t the r e s u l t h a s b e e n s h o w n for all

implies

@IBm_ 1 ~ 0 , @(e0,es_l)

s = 0

that

= @(x,y)

that

to s h o w t h a t

To s t a r t this s e c o n d ( l - k ) @ ( e s , e 0)

that I # 1

he k = lek + ek_ 1

@ ( e 0 , e 0) ~ O , t h e n

1 .

by induction,

induction

to a s s u m e

T h a t is, w e a s s u m e

and

Thus we have

implies if

of s i z e l e s s t h a n

satisfying

(since

.

s p a n n e d by t h e s e b a s i s v e c t o r s .

a second k

. and

Continuing Jordan blocks

suffice

w i t h this b a s i s .

S i n c e we a l s o k n o w t h a t

[(l-l)/(l-~]@(e0,e0) l~

01A l

< e 0 , e 0 > = ( l - l ) @ ( e 0 , e 0)

"

Therefore

he 0 = le O

denote

(-l)n

(-l)n(l-l)@(eo'eo) that

it w i l l

s .

=

0(e0, (l-l)e 0)

IIkIl = 1

to a s i n g l e J o r d a n b l o c k .

{e0,el,...,es}

k = 1,2,...,s proceed

that

where

a

F = {l,~,w 2 ,. .., a-l} {e0,el, .... Ca_ 2}

where

is a where

208

ek =

[ k] _

p e F

.

[ k+l]

and

The monodromy

therefore

[p]

denotes

the

acts via rotation

he k = e k + 1

(but n o t e

that

integral

homology

class

of the p o i n t

by

, hence

h[~ k]

=

2~/a

1 + ~ + w2 +

... + w a - i

[ k~l]

= 0

and

and

e a _ 1 = - ( e 0 + e I + ... + e a 2 ) ) . L e t A = H 0 ( F ; ~ ) . T h e e i g e n v a l u e s of t h e --.2 -a-i are ~,~ ,...,~ A c o r r e s p o n d i n g b a s i s of e i g e n v e c t o r s is g i v e n b y k .2k (a-l)k Ek = e 0 + ~ eI + ~ e 2 + ... + ~ ea_ 1 . T h e i n t e g r a l S e i f e r t p a i r i n g h a s

mondromy

matrix

ia

with

respect

a

to the b a s i s

~--~

is an

{ e 0 ..... ea_ ~}

(a-l)

×

(a-l)

,

where

matrix.

1

.

1

1

It is an e a s y Its m a t r i x

calculation

Aa

the diagonal

' with

to see

respect

that

the S e i f e r t

to the b a s i s

pairing

over

{ E I , E 2 , . • . , E a _ I}



is d i a g o n a l .

is g i v e n b y

matrix

A

=

a

1 _

2

a_

III.

Branched

Fibrations

In this

section

to s i n g u l a r i t i e s , 3.1.

such that

T-I(0)

i)

T

Any

has only

< CL

must

satisfy

knot

values

be the result

of smoothing

so t h a t

This knots (~,0)

TIS n

is an a b s t r a c t i o n

rise

2

explain

their

relationship

interior

and

for

(Sn,L)

is a s m o o t h

of

fibration

D2 .

0 < r