This is, of course,a very special case of the Atiyah-Singer G-signature theorem. This approach has been used by Patrick Gilmer in [G]. It is also instructive to form.
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
