AN
INVARIANT
OF P L U M B E D
Walter
If
closed
An
A
is a ring,
oriented
A-homology
a compact
that
the
Here,
bordism
SPHERES
D. N e u m a n n
a 3-dimensional
3-manifold
oriented
HOMOLOGY
M3
with
between
two
4-manifold
A-sphere
the
same
M!3
A-spheres
W4
with
~W
will
mean
A-homology
and
M. c___~ W
i n d u c e i s o m o r p h i s m s in
i
following,
+
means disjoint union,
in the
reversed
orientation,
compact,
and o r i e n t e d .
and
all m a n i f o l d s
Diffeomorphlsms
M3
2
= M I + (-M 2)
inclusions
and
a
as
S 3.
is
,
such
A-homology.
-
will
be a s s u m e d
should
preserve
means
smooth,
orient-
ation.
The
set of
A-spheres
We
forms
are m o s t
~/2-sphere
If
interested
and
M3
instance
A-homology
a group
every
is a
in
W4
with
of 3 - d i m e n s i o n a l
sum,
• .
which
Every
is a r a t i o n a l
then
the o p p o s i t e
its
we
~A
3"
is a
sphere.
~L-invariant
sign
denote
~-sphere
convention
(see
for
is used)
can
as
is any
and
its
= s i g n ( W 4)
simply
} W 4 = M 3 . This
invariant
A = ~/2 or
~/2-sphere
~I(M 3)
where
classes
connected
~/2-sphere,
[ 3 ] , where
be d e f i n e d
bordism
under
connected
invariant
range
(mod
,
parallelizable
is a
of v a l u e s
16)
~/2-homology
is s u m m e d
up
compact
manifold
bordism
in the
commutative
diagram
~3
.~ 8 ~ / ] 6~..
f
~3
The
because
result
groups
of
A
~3
are of
applications.
of G a l e w s k i
and
The
Stern
'
.~) 2 = / 1 6~-
interest
in t h e i r
most
important
[~J
and M a t u m o t o
own
right,
application
[ ~ ]
but
also
is the
that
for any
R e s e a r c h p a r t i a l l y s u p p o r t e d by the NSF. T h e h o s p i t a l i t y of the
Sonderforschungsbereich
40 in B o n n d u r i n g the p r e p a r a t i o n of
this p a p e r is also g r a t e f u l l y a c k n o w l e d g e d .
126
n ~ 5 ,
all
if and
closed
only
if
TOP
n-manifolds
contains
~3
are
simplicially
a subgroup
~/2
triangulable
on w h i c h
/~t
is an
isomorphism.
Another
group
on
application
C3 .
it
Given
to get
is
a knot
a homology
H I ( M 3 , ;~) = Z/p
P/q
is an
A-sphere.
to the
(S3,K)
lens
Thus
classical
if
The map
one
do a
of
the
same
and
the non
C3
main
plumbed
results
~ ( M 3)
(iii)
If
~
be
hand
might
For
following
an
possibly
yield
not
new
can be used
time-dependent
theorems
invariant
to
theorem
6.!
~ ( M 3) ~ ~
is
~
|6)
all
. . .k n. o .w n.
~
to 6.3,
of
with
the
for
to the
not
which
|979)
. . .
are
sense [IO])
invariant.
~/2-homolo
.
some
due
~-invariant
~
y bordi
of
sms
plumbed
lead
results
no
idea
described
If on
the
in-
in c o n s t r u c t -
bordism
are:
plumbed
last
together
(see
4.
of geosection.
~/2-spheres
reversing
latter
in s e c t i o n
/~
relation
in the
be p a s t e d
The
which
Moreover,
briefly
if
we
bordism
to an e x a m p l e
orientation
cannot
4-manifolds.
~
new
and
of course
problem.
positively.
this
invariant
then
Z~/2-homology
stronger
admit
surface
a
to some
might
discuss
of our
bordism
M3 ,
triangulation
to be
problem
we
of
be
then
under
the K u m m e r
a generalization
.
~/2-sphere
out
consequences
connected
(Ma~
. . .
YL/2-homology
triangulation
non-zero
type
a
hopefully
invariant
which
~ ( M 3)
any
answer
turns
.
=
M3
in W a l d h a u s e n ' s
is i n v a r i a n t .
in fact
significance;
morphisms;
for ~ / 2 - s D h e r e s
manifolds
diffeomorphism
(mod
bordisms
the
Some
simply
exists
.
it will
be
metric
on
t
a negative
such
with
~ ( M 3)
defined
variant,
ing
constructions
consequences,
( = graph
~/2-spheres,
solves
the
is d e f i n e d
manifolds
(ii).
other
are
There
is an o r i e n t e d
have
could
~roperties:
(i).
it can
though
M3
p/q
6 below.
followin$
such
similar
time-dependent
THEOREM.
and
Other
surgery
with
purpose.
Our
section
C3
3
= M p/q
is i n v e r t i b l e in
A
then
A
• ~3
given by
(S3,K) ~
[M3, (S3,K)] - [ e ( p , p - q ) ]
is well d e f i n e d
p/q
a homomorphism.
Thus i n f o r m a t i o n about ~ 3A
knowledge
concordance
p/q-Dehn
M p3/ q ( S 3 ,K)
space
p
can
knot
homeofrom
6.2)
two
depends
127
2. The
invariant.
For
graph
the
r
vertices,
weight
purpose
of
this
paper
to be a c o n n e c t e d
which
e.
we
The
label
graph
we
i = 1,2,
oriented
define
with
cycles,
,s
,
...
4-manifold
a connected
no
each
carries
P(F)
plumbing
of w h o s e
an
obtained
integer
by p l u m b i n g
i
according
each
t_~o ~
vertex
number
is then
i
e.
we
and
defined
in the
usual
way,
namely,
let
E.
be the
D 2 - b u n d l e over
I
p l u m b these t o g e t h e r a c c o r d i n g
then
S2
for
of e u l e r
to the
edges
l
of
~
(see
for
3-manifold
More
whose
call
M(~)
we a l l o w
disconnected
as above
by m a k i n g
are
is the d i s j o i n t
= P(PI ) ~ P(~2 )
M(F|)~M(~
2)
defining
P(~)
= D4
over
when
plumbing
could
graphs
include
sum),
also
~2
that
'
so
if
then
M(~)
the e m p t y
plumbing
graphs
for w h i c h
P(~)
is a p l u m b i n g
graph
to a l l o w
of h i g h e r
homology
all
oriented
graphs
convention
and
plumbing,
surfaces
yield
plumbing
~|
allow
the
=
graph
by
= S3
plumbing
If
never
We
M(@)
customary,
bundles
of
= ~P(~)
t~o
the
connected
sum).
and
at.cording
union
(boundary
(connected
It is more
and
We
generally,
components
cycles
[ 3 ]).
by ~lumbing
= ~l + ~2
P(~)
instance
obtained
genus,
spheres,
we want.
They
with
since
our more
such
restricted
are p r e c i s e l y
is simply
in our
graphs
but,
the
connected.
sense,
recall
that
the
matrix
A(~)
(aij)i,j=l'" . . . . .
a..
13
=
is the
intersection
namely
the basis
e.
I
l
if
i = j
=
if
i
=
0
otherwise
matrix
for
represented
s
is c o n n e c t e d
the
by
an edge
,
the n a t u r a l
by
j
to
basis
of
zero-sections
H2(P(r);~.)
of
the p l u m b e d
bundles.
THEOREM
is a
there
that
2.1.
~./2-sphere
exists
the
If
r
if and
a unique
following
i_~s ~ p l u m b i n g
only
subse t
condition
if
graph
detA(p)
SCVert(~)
holds
for
as above
is odd.
= If,
any
In
then
this
...
,s~
j = |,
...
such
,s
M(P)
case
:
128
(~)
~.
The
a..
S lj
~
then
only
tion
is
usual
Proof.
main
ingredient
is
that
an
that
(dot
d.x
We
assume
be
chosen
that
to
such
be
embedded
computed
as
~ x.x
It
is
easily
torsion
and
H2(W;~)
if
theorem,
then
seen
is
there
= 0
~"
is a W u
j
natural
b a s i s of
set
the
S
adjacent
of
~2R(M(p))
To
modulo
see
- d.d
of
~
= ~'j~S
can
,
be
ajj
,
16)
if
~
S
is
.
has
no
as
...
of
in
the
that
,~
is
property
s
of
d
this
is
the
that
It
be
such
~],
~(M(p))
a
can
elements
Vert(P)
so
can
.
of
implies
so
it
by
HI(W;~)
defining
in
that
~-invariant
theorem,
(~)
a class
representable
where
The
the
is
is
number):
moreover
if
SC
W4
x ~ H2(W;~)
multiples
P(P)
both
W4
(mod
exist
subset
Condition
d.d
is
the
particular,
for
of
set
no
follows
two
that
= signP(~)
d
- d.d
16.
~(M(p))
proposition.
In
(~)
all
it
type
the
where
for
and
even
16 r e d u c -
describe
intersection
does
to
= e.)
J
later.
class
for
H2(P(C);~)
theorem.
and
.
class
condition
vertices
spherical
d
up
modulo
M 3 = @W 4
Then
sign(W)
a..
JJ
diffeomorphism
]. We
it
and
is,
.
a unique
d = ~'j~S
to
need
exists
W
that
is
2)
d
in
unique
HI(W;~)
translates
(mod
=
Its
in L ~
Wu
(recall
.
oriented
we
, that
sphere
on
represents
a class
2~t(M 3)
even
the
integral
spherical
smoothly
not
7Z/2-sphere
such
- Z
a..
j ~ S jj
proved
since
a
Recall
d 6 H2(W;~)
ing
2)
~(M(P))
was
however,
M3
and
except
~(M(P))
Suppose
signA(P)
~-invariant
Everything
of
oriented.
=
o__n_n M(|TM)
depends
the
invarlance
is
(mod
inteser
~(M(T))
the
a..
3J
only
depends
on
M(~)
we
need
the
follow-
129
PROPOSITION
as d e f i n e d
obtained
2.2.
above,
from
If
then
~2
~ ]
M(PI)
and
F 2
are
~ M(F2)
by ~ s e q u e n c e
of
two p l u m b i n g
if and
only i f
th____~ef o l l o w i n g
graphs
Pl
moves
can be
and
their
inverses•
la.
Delete
with
a comFonent
weight
e+l
lb.
of
~
consisting
of an
isolated
vertex
~]
+I
" . /
e
•
e ±I
.-.--.~..~1
) .'.~
~_...~
±I
e2+l
0
e2
~..~l
e2_...~.
Ic.
2•
3
e
.---..___I
~
e.+e 2
---~.:
•
~'---'--2---/J~
_ ~
o
)r I +
. , .
(disjoint
+r t
union).
stand
Move
I is called
blowing
This
proposition
is p r o v e d
a forthcoming
contain
a proof.
To
complete
that
~(M(~))
trivial
is
COROLLARY
of
invariant
2.3.
so we
The
Seifert
formulae
for
in
of
under
leave
formulae
,
its
and
the
2.1
we need
above
if one
are
does
not
and
will
only
moves.
up.
I underalso
verify
This
is a
reader.
[~ , t h e o r e m
which
blowing
3.2~
Siebenmann
it to the
of
inverse
, theorem
theorem
manifolds
~
and
~
of B o n a h o n
the proof
computation,
~-invariants
actually
paper
down,
6.2]
for
7£/2-spheres
reduce
modulo
the
are
16.
130
3. E x a m p l e s .
A quite
are
general
homology
method
null-bordant
LEMMA3.1.
Le___~t A
be
a connected
oriented
N3
such
M3
series
be
generating
given
by
a principal
4-ma.nifold
homology
the
ideal
with
be
an
o__n_n N
which
simple
domain
connected
= As
,
A-sphere
.
Then
H2(V;A)
obtained
M
= H3(V;A)
and
lemma.
let
V4
boundary
~V 4
= O
b__~yp e r f o r m i n g
represents
zero
s
index
2 sur-
A
--~3
in
~.:
SIX D 2
) N , i = 1, ...
i
on w h i c h the s u r g e r i e s are p e r f o r m e d .
Proof.
beddings
spheres
following
that
H|(V;A)
L.et
of
is
Let
,s
,
be
Then
~
the
em-
S1
1
i =
I,
...
H| (V;A)
ogy
,s
clearly
= A s , the
sequence
adding
,
s
for
represent
first
the
equality
pair
2-handles
an
to
A-basis
here
(V,N)
being
. Let
W4
by
be
along
the
~,
proving
the
lemma.
V
of
,
H|(N;A)
the
the
so
=
exact
homol-
result
~W
= M3
of
. Then
i
W
is
clearly
Applied
A-sphere
SI~S
2
Mazur
A-acyclic,
to
M3
V4
SIx
which
D3
results
represents
zero
in
this
by
~
lemma
a single
.
Such
shows
index
plumbing
3.2.
Let
2 surgery
manifolds
are
£|
~I
O ~
=
c
-
and
~2
be
b
cI
-
c
6 2 ___
the
following
~s
b2
b 2
b I
bt
0
al+c 1
a2
a
C
M(~l)
vice
two
..._~r
a~2 . . . . . . .
|
~ b l
if
on
called
graphs:
k
and
any
manifolds.
PROPOSITION
Then
that
results
versa.
an A - s p h e r e ,
Thus
by
8r
doing
if o n e
is a M a z u r
of
a single
them
manifold
index
2 surgery
is
sl~
S2
and
regresents
s
then
zero
o nn
M ( ~ 2)
the
other,
in
~
131
PROPOSITION
shown
to be
a Mazur
Before
hard
was
giving
to w r i t e
the most
matter
follows
reduced
moves)
fold
3.3. I f
by
many
productive
from
the m o v e s
Recall
we m a k e
graphs
from
some
one
Secondly
I found.
this
graph
P
with
it is e v e n
the
2.2
plumbing
.
it is not
3.2.
The
is a v e r y
if
above
easy
S ] K S2 :
~
it
can be
using
For
.
the
Seifert
inverse
mani-
graph
:2rl
. . . . . . . .
.
.
.
.
.
"~"
", sr
.
S
manifold
M(O; (l,-b), (~l'~l) . . . . .
(unnormalized
it
= O
alr
•
Seifert
O
~(M)
is
easier.
. . . . . .
the
of
(without
weight
which
Firstly
form
represents
all
yields
the
is so if and only
vertex
then
remarks.
of
[ ~ ] that
b
3.2
propositions
of p r o p o s i t i o n
to an i s o l a t e d
~/2-sphere
by p r o p o s i t i o n
if a p l u m b i n g
[ ~ ~ that
plumbing
is a p l u m b e d
manifold
proofs
down
to c h e c k
M
Seifert
~i/~i
invariants,
= Jail
as
in [ 6 ], [ ~ ] ) w i t h
air.]
. . . . .
(°<s'~s))
,
i
=
I.
. . . .
S
1
We
are
using
the
continued
Ix I .....
x r]
fraction
=
x I
-
notation
I
32
-
I
_
I
X
and we
assume
~.
# 0
for all
r
i .
i
This
Seifert
manifold
is
S l x S2
if and
only
if:
~.
= +I
i
for at
It
least
s-2
is thus
manifolds
which
of the
indices
i ,
a simple
but
are
to be M a z u r
shown
tedious
and
matter
b - Z~i/~i
= 0
to list
Seifert
manifolds
all
by p r o p o s i t i o n
0
132
3.2.
the
One
obtains
following
M(O;(l
Here
p,q,r
HI(M;ZO
(~,p)
they
all
ever,
and
~2
reduces
and H a r e r
[ 2 3 plus
manifolds:
of
Casson
checked
above
every
about
family
One w a y
c = -l
manifold
lO0
cases
of e x a m p l e s ,
four
exceedingly
case.
Seifert
so that
and
are
and H a r e r ' s
are
is to take
shaped
(-~,-#)).
there
a 3-parameter
calculations
to a star
integer
to m e a n
and
families
tedious
or
how-
of o b t a i n i n g
and
r
graph,
=
l
and
,
so
then
is the r e c i p [a=I + c l , a ~ . . . . .
a s]
S I ~ S ~ as the r e a d e r can c h e c k ,
M(P2)
patience.
We
follows
shall
by
First
note
~l
the
that
is
for
k > 0
orientations
the
and
proof,
by b l o w i n g
k-weighted
(after
0
which
3.2
.
the
though
The
case
case
k ~ 0
k = 0
it can be
then
is g i v e n
seen m u c h
directly.
the r i g h ~ of
to
prove
interpreting
easily
graph
just
reversing
suitably
more
not
mentioned
of an
Casson
of S e i f e r t
although
yields
one
The
the w e i g h t s
rocal
by
to give
I have
is t a k e n
that,
of w h i c h
family.
family
with
negative
each
the
choos~
~
seem
of
family
i n t e g e r s w h o s e p r o d u c t is not -l
2
(by c o n v e n t i o n a S e i f e r t p a i r
(pqr+l)
is r e m a r k a b l e
above
examples
arbitrary
order
with
to check,
the
the
parameter
, l),(pq-q+l,q),(qr-r+l,r),(rp-p+l,p))
are
has
It
all
three
equivalent,
-2
.1~
k
such
-1
-2
by
move
-
.
.
.
up
(-l)-vertices
vertex
one
directly
the
2 of
.
.
.
c-l
2.2,
to
-2
c-1
c
--
I
. . . . . .
~t
v
e l
the
c I
-
graph
.
.
.
.
to
equivalent
blow-ups)
-2
.
obtains
--4
c
r
133
This
can be r e p r e s e n t e d
framed
in
Kirby's
link
calculus
[4-]
by t h e
link
, L
,~
~L.r-----.~
o"
"
•
--JO %
By
sliding
handle
the
over
handles
this
which
handle
we
are
linked
with
the
(-|)-framed
obtain
÷
~.~--"~
.. /
0"
"~-J'~".
--~-~'
/
~ -a~k-2
"
1
o
@o.
and
by
iterating
link
on
over
the
page.
the
next
this
page.
el-framed
procedure,
We
handle
then
we
eventually
obtain
slide
the
(a|+k)-framed
to o b t a i n
the
seeon4
link
the
first
handle
on
the
next
134
+ O'I+.
. ..
+" 0 - l
k
~J
J
We now perform a single index 2 surgery by deleting the handle
with framing
c|
,
giving us the next
link (next page). Then by
reversing our initial procedure which produced
can unwind
them again,
argument which
showed
is equivalent
to
k
twists we
to end up with a framed link represented
by the graph on the next page,
~
the
called
that
~!
~2
completing
'
I~
is equivalent
Finally,
to
the proof.
P!
the same
shows that
135
-I
"|
,O,...÷O
J
k
~_,,"~,..._rJ, } Y " - r / ' m
..,
t
"-J~aS
b~~
tJ
b•t
~2
bl
-1
al+c 1
a2
~-2
c2
i
For
4. A
the p r o o f
for
The
ed
spin
~-invariant
M3
a unique
spin
a spin
Let
M(P)
ure
P
without
spin
X
one
over
H2(P(P);~/2)
of
this
spin
A class
structure
defined
--
of our
invariant.
graph
has
as
an o b s t r u c t i o n
.
The
obstruction
w.x
M(C)
=
x.x
in
Poincar~
will
wE H2(P(F);~/2)
on
Then
is
be
for
an
spin
2 but
to any
called
all
if
M
~/2admits
structures
).
suppose
spin
struct-
H2(p(P),M(~);~/2)
dual
w~
the h o m o l o g y
the h o m o l o g y
if and o n l y
orient-
for a
HI (X;~/2)
in s e c t i o n
~/2-sphere.
P(P)
such
the v a r i o u s
by
closed
defined
because
classified
a
for any
It is w e l l
structure
are
necessarily
~
class,
--
an e x t e n s i o n
(in g e n e r a l
be a p l u m b i n g
M(r)
some
manifold.
structure
to e x t e n d i n g
Wu
on
need
is a c t u a l l y
spin
manifold
is not
Y
3.3 we
k-
structures.
3-dimensional
-sphere
on
of
a
w
Wu
class
satisfies
x ~ H2(P([~);~/2)
of
136
The
basis
Wu
class
~1,
...
w~
is
of
P ~S
a
linear
w~
for
some
of
~
well
We
subset
call
Sc
M(P)
if
so w e
[|,
and
call
In
defined
only
such
the
,s~
if
a
analogy
Wu
is
it
the
so
completely
the
vertex
set
for
the
sPi n
Wu
set
simply
for
THEOREM
4.1.
diffeomorphism
reduction
a Wu
This
spin
was
used
to
We
shall
space
prove
over
~
desenerate
is
= 0
vector
If
and
V
VIE
4.3.
If
~/2
V
and
if a n d
only
class
is
the
Proofs.
We
prove
means
V ~1 C V l
we
have
then
wE
if
unique
V!
(Vt)m
V = V 1 ~
V1
do
Wu
and
on
and
~
=
is
[0,x~
= V1
and
class
A
on
2.1,
4.2
to
the
spin
modulo
16
(M(F),~).
include
same
the
method
as
that
is
spaces,
bilinear
e symmetric
1 ,
if
then
sign
I ,
of
w.x
to
V1
V
if
for
reader.
.
V 1
= w ( V I) ~
i__~s
V1
all
V1
Since
is
V 1
in
complement.
x
V!
bilinear
is
= x.x
is
= sign
then
V
the
forms.
bilinear
symmetric
some
= w(V)
oriented
Its
with
orthogonal
for
shows:
in [ ~ ] .
w
with
2.1
manifold
bilinear
only
Conversely
w
.
define
the
2.2
codimension
leave
y
theorem
~
by
codimension
if
of
spin
structure
element
where
so
to
equipped
has
the
4.3
on
a non-degenerate
V 1CV
,s~
...
e.
J
on
the
n on-degenerat
has
is
not
of
lemmas
form)
~1,
structure
theorem
proposition
spaces
V
of
depends
hard
spin
spin
2 we
- ~.
proof
and
not
is ~
over
l-dimensional,
is
simple
LEMMA
Wu
crate
two
induced
desenerate
erate
(M0~),~)
(with
space
the
~-invariant
this
as
structure
(~)
section
A(r)
onlz
it w i t h o u t
need
4.2.
to
strengthening
but
dimensional
LEMMA
of
usual
involves
structure,
finite
The
the
natural
written
set.
in
sign
~(M(P),~)
type
is
be
set
some
condition
definition
parallel
the
can
of
7rS~
A proof
of
it
S~
the
~(M(£'),~)
,
~ . 04.
j~S~ J
satisfies
subset
to
=
subset
S~
...
combination
H2(P(p);~/2)
xeV
degen-
Now
w.x
V ~I
= x.x
non-degen-
w(V~)
is
not
in
137
VI
since
We
and
its
w(V~)
can now
proof
# O
prove
and we
S|~ S 2
P(~])
by a d d i n g
2-handles
first
M(~I)
3.3.
assume
to
do not
of
the
in the
form
alter
link.
V|
This
! subspace
V!
= sign
also
the Wu
set
either
by
follows
one
one
both
sees
S
which
the
Wu
to
of
the
condition
sets
and
that
sign
2.1
what
,
and
the
with
k
.
Thus
of
also
~j~S
A(~),
to
3.2.
and
c|
set
~2
the
But
O-weighted
are
of
is a Wu
the v e r t i c e s
vertices
in n e i t h e r )
happens
a|
there
degen-
= sign
proof
weighted
us
is a
Thus
A(F])
the
3.2
gives
has
A(P 2)
that
O
at
and
follow
S C V e r t ( P I)
weighted
theorem
or
and
of
a component
H2(V2;~)
= sign
vertices
not
)V I =
proof
= SI~ S 2
through
precisely
in
k-weighted
S2
to
non-degenerate
H2(V2;~)
is,
A(P|)
4.3
goes
the
contains
of
that
sign
as one
vertices
the
4.2,
or both
But
S 3 = ~D 4
delete
Thus
=
obtained
since
a handle,
~V 2 = M(P2)
lemma
vertices
(~)
that
S
lemma
2)
in the
we
P(~)
H2(VI ;~)
since
easily:
in
$ 2 C V e r t ( ~ 2)
sees
of
link
where
to r e m o v i n g
to
has
modulo
the
in
3.2
M ( C 2)
VI
link
H2(V|;~)
point
and
4-manifold
framed
true
alter
the
can use
correspond
exception
As we
S C Vert(C|)
In p a r t i c u l a r
then
is
form,
V2
Similarly
ing
3.2.
diffeomorphic
intersection
whence
are
of
this
corresponds
codimension
the
fact
until
V2
sign
along
of p r o p o s i t i o n
is a ~ / 2 - s p h e r e
to the
proof
(in
a manifold
erate
the n o t a t i o n
~I(~i)
D4
is a ~ / 2 - s p h e r e .
we
We use
is d i f f e o m o r p h i c
pictured
intersection
.
which
possible
by
consider-
vertex
either
one
in both
ej = ~ J e S 2 e
j
,
so
~(M(f~|))
Thus
~(M(PI) )
M(~ 2)
ting
on
equals
= $I ~ S 2
But
of a single
point
S|~ S 2
have
~
= O
=
the
sign
~-invariant
SI~ S 2
with
.
A(f' 2)
can be
weight
Thus
this case. The case that
M ( P 2)
S I S2
is c o m p l e t e l y a n a l o g o u s .
the
O
- ~.J6S2e j
of
some
given
,
proof
by
spin
the
so both
of
3.3
is a ZL/2-sphere
structure
graph
spin
on
consis-
structures
is c o m p l e t e
for
and
is
M ( ~ I)
138
5.
Seifert
We
spheres.
recall
some
facts
coprime
integers,
each
Seifert
manifold
whose
(0;(~I'~I)
....
manifold
-sphere
and
' ( ~ n ' ~ n ))
will
every
diffeomorphic
necessary.
given
and
to
a manifold
if
and
We
define
after
instance
only
order
~
if
No
to
the
computing
it
that
~ .l
formula
formula
(see
~)
5.1.
=
]
is
for
in
are
l
~n )
is
It
a
is
a
~-
~-sphere
reversing
is
orientation
of
these
manifolds
as
links
of
a permutation
if
are
certain
complex
are
the
(i).
If
the
~(~(~I
If
~"
= -o<"
n
n
mod
A<Z(~, . . . . .
Proof.
(i).
+ 2 ~ l . . . o#
n-I
It
2.3
2C~'l
~n )
definition.
.....
6.~
above).
~"n
~n
"#n ))
is
seems
the
The
mod
only
known
and
fastest
general
2 ~ I ... c~ n-l
to
prove
(i)
for
then
~n-1'~n )) "
~"
n
=
of
formulae
then
= -~(Z<~, . . . . .
in
way
° ~ (Z<~I . . . . . an-1 '~n))
" "" ~ n - 1
%))
suffices
Let
(~l . . . . .
following.
&cZ(~1 . . . . . ~))
--
of
Z(~] . . . . . % )
of [ ~ , t h e o r e m
corollary
I know
THEOREM
0¢
n
pairwise
'~n )
.....
permit
closed
practice
(ii) .
unique
that
~(~i . . . . . %,i)
it
=
which
descriptions
~(~I ....
for
be
a
singularities.
Note
in
'~n
invariants
Z(~I .....
[ @ ] , for
"'"
exists
Seifert
manifold
other
~l'
there
satisfy
denoted
such
Various
Then
"'" a~n(~-~'~i/a<i)
Seifert
in [ ~ ] a n d
surface
be
[~ ] . Let
~ 2
1
unnormalized
~l
This
from
~.
"
139
-bl]
r'
-'°12
the n o r m a l
the
form p l u m b i n g
canonical
graph
characterized,
that
b..
be
lj
is n e g a t i v e
normal
~2
.
As
in the
for
plumbing
~
graphs
shown
efficient
a simple
form
graph
sense
of
[ ~ ] (called
= ~(~I . . . . .
~¢n )
for
the
~
,
by
in [@ ] , its
It is
condition
intersection
form
definite.
A reasonably
in [@ ] , and
the
1
__nSn
in [ ~ ])
among
all
A(U)
)
=
"X2 n
be
-bls
--
algorithm
computation
plumbing
graph
for
with
~"
computing
that
for
~
algorithm
E"
is given
shows
= ~(~| . . . . .
that
~n-l'~n )
is
]bl!
p"
=
-b12
-b|s
b21 -b22
'''-~
....
-b~
-b2s 2
--_~
-b
n2
with
k = [~I
Now
set
of
S
sign
be
Vert(~')
by
one
vertex
differ
at most
2 ,
we
4 .
they
are
But
It
2~I
above,
duced
then
by
The
for
- ~'n
proceeds
~n
so the
like
the
!
set
sees
of
2
-2
~
,
above
{i)
~"
= I .
theorem
proof
For e x a m p l e ,
implies
Since
for
then
with
~
n
A(P)
one
for
with
I~"
as a subS
plus
,
and
differ
differ
by
are
apply
signs
E(~l,~2,1)
~(~(~i,~2,2m~i~2±I)
if
deleted
~(~I . . . . .
suitable
and
(i),
that,
~"
=
n
argu-
The
~
is as
and
k
re-
DCn-l'~n)
interpretation,
is always
= O
by
at most
equal.
< 2~¢I "'" ~ n - I
can
the
and
~s'ej
so they
verifying
minus
graph
sign
8 ,
S
or
for
~(~')
by
after
itself
S"
e.j_ and
and
(ii)
is valid,
S
set
~jGS
because
is a p l u m b i n g
above
set.
divisible
case
Considering
either
~(E)
to prove
~.
is a Wu
and
that
both
for
that
I~"
the Wu
by
see
is e n o u g h
"'" ~ n - I
ment
the Wu
uniqueness,
A(~')
-k
"'" ~ n - I / ~ n ] + 2
let
(-2)-weighted
hence,
,,o. n S n
the
also
3-sphere,
for all
m> O
o
140
Until
-spheres
recently,
which
and H a r e r ' s
lists,
ion 3.3,
while
Recently
R.
are
of
the C a s s o n
above
6.
remaining
has
acyclic
announced
lists
1. Is
does
results
also
show,
hence
of p l u m b e d
included
were
could
zero
be
have
in C a s s o n
by hand.
examples.
certain
~
~/2-
by p r o p o s i t -
checked
additional
for
that
the
by
have
They
~(~i,~2,~3)
equal
to zero
by
in
the
for and
~I(M)
only
to date
there
does
of
--~ -~I(W)
chosen
contractible.
of
the
for
such
a relation.
bordism
homology
even
M3 ,
construction
might
One was
a
~/2-
M
so
can
is a
again
restricted
far
bordism
out
sometimes
some,
is that
the
be c h o s e n
W
~/2such
that
can even
on
be
restricted
W
lead
to
are
still
of
might
still
be p o s i t i v e
respect
to
be
detected
~-invariant
invariant,
I
~/2-spheres.
the p r o b a b l e
which
with
3.3
to y i e l d
conditions
I, if n e g a t i v e ,
the
in t h e i r
and no
~-sphere,
these
they
them,
obstruction
of
certain
always
points
of
a nontrivial
the p r o o f
answer,
side,
for most
spheres
found
However
can
a positive
the W i t t
and
class
this
by
the
of
interest
stronger
the
only
the
Casson
other
torsion
vanish.
If a c o u n t e r e x a m p l e
for o b j e c t s .
quite
seemed
~W = M
when
bordism
f o r m of
first
If
Equiv-
bounds
the p o s i t i v e
to be
Non-triviality
relation
on
homology
of h o m o l o g y
to p r o b l e m
invariant,
linking
This
examples.
concepts
answer
homology
Gordon
these
at
with
is onto.
are
seem
the e x a m p l e s
W
nature
stronger
which
against
by a n a l y z i n g
rational
invariant?
which
construction
found
gave
manifold
and
a uniform
counterexamples:
M3
= 0 ?
examples
But
A property
acyclic
~(M)
both
giving
bordism
~/2-sphere
many
constructions
they
~/2-homology
give
examples
finding
a
plumbed
weigh
they
construction.
found
~
every
4-manifold
Our
its
some
Harer
although
known
were
examples
I(~i,~2,2~i~2~3)
and
examples
~3
/2
~-invariants
form
PROBLEM
but
their
in
comments.
alently,
to
all k n o w n
zero
theorem.
Final
for
so
the
Stern
the
almost
represent
be
to p r o b l e m
| exists,
applicable
mentioned
in
the
the n e w
to c o n s t r u c t
idea
other
introduction.
in
sought
Another
might
141
be a c l o s e d
inite
simply
connected
intersection
In
seems
this
with
nontrivial
even
def-
form.
context
unlikely
4-manifold
we m e n t i o n
the
following
answer,
problem,
to h a v e
a positive
though
If M
is a p l u m b e d
~./2-sphere
and
definite
which
it w o u l d
be n i c e
if it did.
PROBLEM
V is
simply
sign
V ?
2.
connected
What
A positive
for
of
such
~
M
answer
and
even
to the
suggest
form,
simply
connected
second
part
a faint
hope
M = ~V 4
is then ~ ( M )
to
would
~/2-spheres,
itive
for
3.
Can one
keeping
If one
just
is
extend
its
wants
connected
16 r e d u c t i o n
~
sum,
,
HI(V;~/2)
answer
for a g e n e r a l
to t h e o r e m
though
the
answer
Indeed
the L i e - i n v a r i a n t
~(T3,~)
extend
= 8 (mod
One
might
Z/2-spheres
hope
problem
!
definition
~(-M)
answer
below
holding
question
spin
structure
(r3,~)
they
are)?
= -~(M)
for any
for
to a r b i t r a r y
invariant,
is p r o b a b l y
same
and
~
diffeomorphism
satisfying
the
of
(whatever
,
whose
yes
(it
on
that
for
arbitrary
is a f o r m u l a
fail.
, so 6.1
does
Indeed,
like
the
framed
the one
links
of
, all
a.
i
rather
than
-8
it
determined
gives
, which
the
but
odd,
,
= - ( a l + a 2 + a 3 + a 4 ) - 4 + sign
form
2.1,
link
which
a2 = a4 = 3
not
representing
theorem
manifold
intersection
al-
is no.
satisfies
a.
is
-I , r e p r e s e n t s the S e i f e r t
I
M(O;(l,-l),(ai+l+2,1),(ai_l+2,1),(ai+2+l,l))
~(M)
mod
~/2-sphere),
T3
if some
has
add-
is
spin m a n i f o l d s
~ -(T3,~)
~t ~ f I ~ ~ ) ~ 3
and
=
= O ?
spin m a n i f o l d s .
hope
there
must
6.1
the
16)
to a r b i t r a r y
definition
properties
then
to
the
an o r i e n t e d
equivalent
the
where
.
PROBLEM
this
has
if one w e a k e n s
and
by
I(2,3,7)
above
A
the
link.
,
whose
formula
(indices
where
But
if
M =
mod
4)
A
is
a 1 = a3 = 1
~-invariant
would
give.
,
is
+8
142
PROBLEM
involution
The
orbit
4.
answer
space
the
Browder
Livesay
Z£/2-sphere
is yes
is e i t h e r
Despite
have
Is
on a p l u m b e d
for
M
Seifert
questions
large
about
of a free
equal
manifolds,
sufficiently
the m a n y
invariant
always
at
to
least
or a S e i f e r t
~
,
the
~(M)
if
?
the
manifold.
invariant
does
applications.
THEOREM
plumbed
6.1.
If
4-manifold,
~(M,~)
M3
is the b o u n d a r y
then ~ 9 ~
nontrivial,
(M,~)
an[
spin
admits
of a s i m p l y
structure
~
no o r i e n t a t i o n
connected
on
M
reversing
with
spin
homeomorphism.
Proof.
Clearly
~(M,~)
# 0
,
so
~(-(M,~))
=-~(M,~)
~(M,~)
As
already
dimensional
Siebenmann
announced
3-dimensional
the
spin
this
that
THEOREM
In
is u n i q u e
6.2.
If
the
a__~sthe unio~n
4-manifolds,
theorem
However,
this
and
case,
all
along
of
for m o r e g e n e r a l
conference,
be
their
large
~/2-spheres,
disregarded.
spin
two
for
3-
L.
sufficiently
indeed
thus
oriented
W] U (-W 2)
pasted
this
for
can
closed
fails
at
it is true
~-spheres.
structure
be written
plumbed
remarked,
spin m a n i f o l d s .
manifold
simply
W4
can
connected
bo~daries,
then
sign W 4 = O .
Proof.
the
spin
~(M,~)
-~(M,~)
Let
Z
Let
as
~
singularity
~!
by
and
~3
the
M
.
by
theorem
~Wn)
and
was
for w h i c h
then
following
also
was
this
as
be
the
in s e c t i o n
V
Z
with
as
restriction
given
V
for
has
to be
5
We
negative
the
the m i n i m a l
restricted
theorem.
~
W = sign W l - sign
of
W2 =
4. ;.
be
taking
let
sign
considering
In [ ~ ~ a list
~2
and
Then
of a 4 - m a n i f o l d
f o r m by
singularity.
that
= O
the b o u n d a r y
surface
= ~W 2 = M
to
= ~(~I .....
intersection
of
~W]
structure
link
can w r i t e
definite
of a c o m p l e x
resolution
of
n = 3
low v a l u e s
even
and
form.
small.
It
This
this
turned
out
is e x p l a i n e d
143
THEOREM
~I~2
The
above
V
never
has
even
form
has
just
sketch
even
form.
good
with
in the
~
as
the
Let
the m i n i m a l
proof.
~
be
resolution.
proof
let
manifolds
P
be
the
Then
~ ~ P(f")
of 5. I.
Let
V
as
and
smooth
manifolds
for
~m
= ~(~l
~
the
corresponding
•
"
the p r o o f
of
the p r o o f
of 5. |).
V
that
of
'~n-l'~Cn+2m~'1
~
is o b t a i n e d
graph.
set
(-2)-weighted
our
from
manifolds)
the W u
2m
5. l (note
is o b t a i n e d
complex
some
S
assumption
~
~
curves
on ~
consists
which
are
only
the v e r t e x
weighted
-b
in p a s s i n g
from
to
~
would
contradict
Vm
for
m
~(Im)
on
~3_
C3
by
{03
and
f(x)y,z)
the
on
fibers
down
,
in
,
of
blow
it
in
of
follows
in
process.
S ,
for
chain
if
of
the
and
this
intersection
form
proof
Wahl
bound
down,
of
has
5.]
= (~x,~-Py,~Pqz)
now
and
the p o l y n o m i a l
shows
to 5.l.
obtained
rational
singularities.
invariant
in
correspond-
the w h o l e
examples
homology
Namely
~ = exp(2~i/(pqr+l))
~(x,y,z)
sense
and
this
is not
would
The
Jonathon
smoothing
leaves
~
is a c o n t r a d i c t i o n
which
and
(in the
as
k = 2
let
Then
this
balls
of
as a
p,q,r
G = ~/(pqr+l)
action
f: C3---~ C
is free
given
= x P y + y q z + z r x . Thus
f
induces
g: ~3/G----) C
and
-I
g
(t) , t~O , are s m o o t h i n g s of the i s o l a t e d s u r f a c e
singularity
(g-| (O),0)
W 4 = g-I(t) O{D/G)
(D
satisfies
H| (W)
satisfies
IH| (M)I =
fiber
2-spheres,
|979:
integers
large.
which
spheres
of w o r k
by
Milnor
n-I
m
curves
definiteness
,
September
be p o s i t i v e
acts
negative
sufficiently
homology
by-product
on
exceptional
=~(E)-mm
Added
rational
"'" c,~
from
/~ ,
of v e r t i c e s
ns n
V
m
(-2)-weighted
n
of
blown
it w e r e ,
to
implies
down
curves
In p a r t i c u l a r ,
then
vertices
by b l o w i n g
exceptional
for
ing to e x c e p t i o n a l
new
be
m
""
~n
a chain
that
n
~" ~ 04
. ~<
and
n
! ""
n-I
c o n s t r u c t e d like
V , but u s i n g
m
by a d d i n g
of
~
Suppose
m
corresponding
and
if
"'" ~ n - I
We
V
6.3.
and
In fact,
V4
Wahl
the u n i t
= G , H2(W)
showed
disc
in
of
f
using
C3
= H3(W ) = O
(pqr+l) 2 , H2(M)
W = V/G
that
= 0
is h o m o t o p y
,
the
"Milnor
and
O<[tt
whence
fiber"
small)
also
~W 4 = M 3
. (This
is b e c a u s e
the
equivalent
to a w e d g e
of
{pqr)
.)
the m e t h o d s
of
[9]
one
can v e r i f y
that
M3
is
144
the Seifert
mentioned
manifold
M(O;(I,I), (pq-q+|,q), ( q r - r + l , r ) , ( r p - p + l , p ) )
in section
3 . It is surprizing,
Wahl's
construction
from earlier
"new",
but
approach
the
same
ones,
given
that
could well
the difference
his examples
give many
other
of
are not
examples.
References
1 .
A. Casson,
J. Harer,
homology
2.
D.E.
balls,
Galewski,
ulations
R.J.
of
82(1976),
3.
4.
W.D.
Forms,
No.
Dekker,
4 (Marcel
A calculus
W.D.
Neumann,
7.
W.D.
Neumann,
Invent.
]0.
bound
Amer.
triang-
Math.
Soc.
Differentiable
in Pure
Manifolds
and Appl.
Math.
1971).
in S 3 )
Invent.
Math.
45
44
(Bonn
et vari~t~s
Paris
of
1976).
their
involu-
1970).
intersections
and a u t o m o r p h i c
285-293.
for p l u m b i n g
singularities
applied
to the
topology
and d e g e n e r a t i n g
com-
To appear.
Seifert
and o r i e n t a t i o n
F. Waldhausen,
(Univ.
the ~ - i n v a r i a n t
42(1977),
F. Raymond,
Proc.
(1978),
d'homologie
Thesis
Schriften
Math.
surface
curves,
keiten,
Koh,
links
complete
A calculus
Topology,
664
and
Math.
Brieskorn
Neumann,
ariant
which
of simplicial
Bull.
Notes
New York,
m&trisables,
Bonner
Neumann,
plex
W.D.
S.S.
Lecture
for framed
S;-actions
of complex
9.
manifolds,
Vari~t&~ simpliciales
6.
forms,
spaces
35-56.
T. Matumoto,
tions,
lens
Classification
Neumann,
and Quadratic
R. Kirby,
W.D.
Stern,
topological
topologiques
8.
homology
916-918.
F. Hirzebruch,
(1978),
5.
Some
Preprint.
Santa
manifolds,
reversing
Barbara
maps,
1977,
plumbing, ~ - i n v Alg.
Springer
and Geom.
Lect.
Notes
163-196.
Eine
Invent.
Klasse
Math.
von
3-dimensionalen
3(1967),
308-333,
Mannigfaltig-
and 4(1967),
87-117.
October:
L.C. Siebenmann,
in "On vanishin~ of the Rohlin invariant and
nonfinitely amphicheiral homology 5-spheres")
these Proceedings,
M ~ -M @24(M) = 0
(~/2)-spheres.
for a very large c!~ss of
a different definition of ~
and interesting e}:amples.
:ho~,is
He also gives