William Browder. Princeton University ... and V will be a bundle neighborhood U of ~pS with handles D r à D q attached. (along S r-I à D q) with r ~ n .... The proof of this is standard as in Thom's proof of representability of homology by maps of ...
Complete Intersections
and the Kervaire Invariant
William Browder Princeton University
The topology of non-singular variety
V
of complex dimension
received considerable the early
Vn C
~pn+k ,
i :
V
--~
an isomorphism for
In particular
V ,
V ,
V
i. :
(image i.)2n
degree of
{pn+k
= di
so that
k
induces
polynomials
polynomials
i. :
is "concentrated
is
(image i.)
{pn+k)
has
determine the diffeomor-
n-connected,
~i(V)
where
--~ w i i = n .
so that the
(~pn+k)
which is
Thus, in some
in the middle dimension."
~ Hi (~pn+k)
degree of the
in
it follows that for a complete inter-
and an epimorphism for
Hi(V)
a non-singular
It was observed by Thom in
(~pn+k , V n)
d(H2n (~pn+k)) =
k
in recent years.
the pair
i < n ,
sense, the topology of
define
defined by
From the Lefschetz Theorem~
inclusion
and
n
(i.e.
intersections
1950's that the degrees of the
phism type. section
attention
complete
d
=
is injective for d I " ... " dk
i-th polynomial
P'l '
i # n ,
is the total
PI ' "'" ' Pk
is then completely determined by Polncare
duality. If and
V
s
is the largest integer less than
will be a bundle neighborhood
(along
S r-I × D q)
W U U'
where
W
with =
the attaching maps of the
of
r ~ n , r+q=2n.
U U (n-handles)
is along the boundaries.
U
and
The attaching maps of the
Research partly supported by an
~pS
we can embed
with handles
In fact, U'
The diffeomorphism n-handles,
n/2 ,
and
V
~pS D r × Dq
V
V , attached
can be described as
is another copy of type of
C
U ,
and union
is then determined by
the "gluing" map of
~(U')
to
n-handles will be closely connected to the middle
NSF Grant.
89
dimensional (as in
intersection
form.
[Kulkarni-Wood])
(see also
When
n
is even, close analysis
leads to interesting
[Wood] , [Libgober]
When
n
results on the topology of
, ~Libgober-Wood])
is odd, the intersection
form is skew symmetric and the analysis
basis for the middle dimensional homology
n ' S2i n Sli
spheres
SIj N S2k = [Sij}
using Whitney's
~
C
for
represent
V j ~ k
and
diffeomorphic
Sli N S2i
to
=
=
~ ,
.
any
If each
then a neighborhood
=
n > I ,
one point,
Hn(V)
(S n x S n - (2n-disk))
V
, when
of
j ,
n
odd ,
Sij
A can
by embedded
k
every
of
information.
embedding theorems,
S.mj N Sik
a symplectic base for
to have trivial normal bundle,
Hn(V)
and Haefliger's
with
'
V
.
the middle dimensional handles relies on more subtle homological
be represented,
of this form
,
i ,
so that
could be chosen
Sli U $2i
would be
and it would follow that
(U U U')
#
~
sn x S n
,
q the connected
sum of
the "twisted double" (n
=
2s + i) ,
(U U U') of
U ,
with i.e.
q
copies of
The question of finding a basis for
the Kervaire quadratic if
(S n X D n
C
x e Hn(V)
~
:
Hn(V)
) ~/2
is represented by a
defined,
and show that when
represented by When
~
Hn(V)
~
U
over
represented by embedded
V) ,
is ~LPs
,
This involves
such that
of
spheres
can be studied by the methods of
8n X D n C V .
of the degrees of the defining polynomials
U U U'
of the boundary.
invariant arising in surgery theory.
form
where
two copies of the disk bundle
glued by a diffeomorphism
with trivial normal bundle,
Sn × S n ,
V
cannot be defined,
~(x)
=
defining a 0
if and only
We give conditions for such a quadratic that any
x ~ Hn(V)
in terms form to be can be
Sn M D n . can be defined one can find the sought for basis if and only if
the Arf invariant
of
formula for computing
~
(called the Kervaire
it in these cases.
inva~iant)
is zero.
We give a
90
Our specific results are as follows: Let
V
dimension and let
C
{pn+k
n d
= =
2s + i ,
or
Suppose
are odd).
7 ,
exactly
then there exists
~
If
n
=
of degree
of the degrees
or
( s + i~ )
trivial
7
d I , ...
d I ,..., ~
, dk
are even
is odd, and
Sn C V
and every element
i , 3
of complex ,
V) .
coefficient
,
intersection
k-polynomials
a homologically
normal bundle
Sn X Dn C V .
complete
(= the degree of
If the binomial
(stably trivial) by
defined b y
d I .... d k
T h e o r e m A. k - f
be a non-singular
n # i , 3
with non-trivial
x e Hn(V)
every embedded
(so
can be represented
Sn ~ V
has trivial
n o r m a l bundle. This was originally proved b y and
[Wood, C!] Theorem
for complete
B.
~ :
is represented b y Theorem
C.
represented b y
A ,
Hn(V) --+
Sn × Dn C V
if
2Z/2
and
[Wood, HI
( ~ + ~ ) + i
such that
if and only if
With hypothesis embeddings
H]
as in
B ,
Sn × D n C V ,
is even, then a quadratic
x ~ Hn(V) ~(x)
Hn(V )
mutually
=
(i)
K(V)
zf all
K(V)
(ii)
=
If
0
,
d I , ... =
some
I~
d.'s I
(K(V)
~
, dk
~/2
)
(n
~
i , 3
or
0 .
has a symplectic basis intersecting
the intersection m a t r i x if and only if the A r f invariant invariant)
for hypersurfaees
intersections.
Notation as in
form is defined
[Morita,
of
~
exactly as in (the Kervaire
.
are odd ,
if
d
~
±i
(nod
8)
if
d
~
±3
(mod
8)
are even,
K(V)
=
1
if and only if
~
=
2 , ~Is
7)
91
and
8~d
.
In
C ,
imposes a condition and for
(n
~
=
=
=
~
Theorem
and
A ,
[Morita,
[Wood, CI]
the case where ,
V
H]
proved
B
[Wood, H] .
of T h e o r e m
B ,
For example s
$
~ i
is even,
Sq + 2
§i
we discuss the definition
sphere
(mod 2) in M
consider complete
S n C M 2n
(n
dimensional
intersections
V~ -I
(so
might equal
A ,
B
invariant
which will be proved elsewhere.
i .
form in a general context M2n ×
~ k C W)
there
which is h o m o l o g i c a l l y
(stably trivial) normal bundle.
and prove
invoke a t h e o r e m relating the Kervaire
K(V)
C(i)
first proved
Spin manifold,
(for framed
i , 3 , 7)
, with n o n - t r i v i a l
for hypersu~faces
[0chanine]
of the quadratic
~
C
The author first proved
gave another proof. is a
and
which is the only case of even degree when
is an embedded
section
(mod 2)
is even if and only if
and prove that when the form is not defined
trivial
0
which m a y be even.
s + 2
This case was also done in
at that time,
In
( s + 2I )
di
~
2s+l).
(k = i) .
s = 2q)
on the number of
2 ,
As well as
C(ii)
( s ++ i~ )
note that the condition
and of
C . V~
To prove
C
In §2, we we
and its hyperplane
g2
§i.
Quadratic
In
forms.
[Browder,
K] ,
or in framed manifolds
a definition
of the quadratic
was given using functional Steenrod
a geometrical version of this definition, for complete
form arising in surgery
intersections,
squares.
We give here
and then study when it can be defined
and its meaning.
First note:
(1. I)
Proposition.
NnCM
2n
with
For any i.[N]
=
x c H n (M 2n ; ~ / 2 )
, one can find an embedded
x
The proof of this is standard as in Thom's proof of r e p r e s e n t a b i l i t y h o m o l o g y b y maps of manifolds,
but using the additional
fact that for the
canonical
n-plane bundle
dimension
2n + i , so that there is no obstruction to finding a map
f :
T(v n)
M 2n
7 n , the first non-trivial
such that
duality, (1.2)
~/2)
Proposition.
M ; ~/2) V C W X
,
If
V
]Rq
N CM
x ,
W 2n+q
,
( [M]
with
class.
[V]
Further
is Poincare
connected, as in
representing V
S
occturs in
Similarly we get:
W ~y
k-invariant
meets
and
y c Hn+ I (W ,
(I.i)
,
y
[V]
W X 0
,
with ~
N
=
~V ,
Hn+ I (V ,
transversally
in
.
N o w the normal bundle M X
]Rq
restricted to
N .
of
N
in
q-frame on
V
is the space of orthogonal
n+q~q
0(n)
,
and is
(n-l)-connected.
get an element in = ~y
,
V C W X [0 , i)
Hn+ I
~
the last,
y
e
(V, ~ ~/2
W x 0
has a
The obstructions
normal
x
C
representing
connected,
the fundamental
=
is the Thom class).
M 2n ×
we can find
[0 , I) ,
N ; Z~/2) NCM
[M] @ f*(U)
of
~
lie in q-fames
q-frame given b y the product
to extending this frame to a H i+l (V , N ; 7ri(Vn+q,q))
in
]Rn+q ,
V
Hence all these obstructions (Vn+q, q
)) ~
~/2.
, and we would like to define
H n (W , M
so
; 2Z/2))
n+q,q
=
0(n+q)
/
are zero except
Evaluating ~(x)
=
where
~ ~[V]
on ,
[V] we (for
but we have made a number of choices
in this
93
process which depend on more than the homology class N
and the choice of
V , with
SV
=
Whitney class when reduced
Wn+ I , mod 2 .
q-frame in an
class
(n+q)
(see [Steenrod])
In the relative
the first
plane bundle is the Stiefel-
which becomes the ordinary
fact the relative Stiefel-Whitney
namely the choice of
N .
From the theory of the Stiefel-Whitney obstruction to finding a
x ,
Stiefel-Whitney class
situation we are discussing,
class in the sense of
Wn+ I
this is in
[Kervaire].
Thus it is
homologically defined provided that this relative class does not depend on the chocie of
V .
W n+q X [0,i) ~n+l(X)
=
If
This will be true provided that any closed manifold admits a normal
0 , so that adding
~n+q
+I(X)
X
to
C
that is, its normal Stiefel-Whitney class V
is the normal bundle of
will not change the relative class of X
in
W × [0 , i) , the normal class
is given by the formula
n+ i the Thom class.
) ~(~) (1.3)
q-frame,
X n+l
u u
=
sq n+l u
u ~ ~+q
The natural collapsing map
has degree
l
c
(T(~) ; ~/2)
: EW
=
(W X [0,I]) / boundary
(mod 2) , and it follows that:
The following are equivalent:
(a)
w+ l(X)
(b)
Sq n+l :
(C)
Vn+ I (W)
=
o
for all
X~+ICW×
Hn+q-1 (W / ~W ; 25/2) =
0
(Vn+ I
=
the
[0, l)
> H2n+q (W / ~lW ; 25/2) Wu class)
is zero
.
Thus we get the condition: (1.4)
The obstruction to extending a
defines a quadratic ,
~(w;
25/2)
form
~] :
K
q-frame over
----e ZZ/2
i f and only i f
Vn+l ( w )
where =
Sq n+l
to
K
=
V
described above
kernel
H n (M ; 25/2)
0.
It is not difficult to translate this relative definition into the functional
N
definition of
Stiefel-Whitney class [Browder, K] ,
which shows
V.
94
it defines
a quadratic
One m a y prove
form.
(1.4)
directly as follows:
Since we have shown that the definition
is evaluation
W h i t n e y class it follows that the definition depends show it quadratic, C
S 2n+q
=
we first
~D 2n+q+l
function is additive
,
which may be done directly.
on t w o non-intersecting
N1 , N2
bordism of
N1
~i +I , ~2 +I
C
have even intersection to
N{
in
M ,
[0
number,
disjoint
from
~i ' ~2
, i)
S n × Sn ×
~q
C
M 2n ,
(which
•
then if N~ ~
To
It is clear that the
manifolds W ×
Stiefel-
only on h o m o l o g y class.
prove it in the special case of
then b o u n d non-intersecting If
of a relative
n > 1 , we m a y find a
(simply the first few lines
of the W h i t n e y process produces the cobordism from each pair of intersection points).
Take two intersection
joining them.
If
N2
to a connected
submanifold,
points
a , b e N 1 N N2
were not connected
would first take connected
if
M
and draw an arc on
we could first make a b o r d i s m of
were connected.
sum of its components~
If
M
without
N2 N2
were not connected~
we
changing the quadratic
forms. A neighborhood D I × D n-I × 0
is a neighborhood
which when added to intersection This If
(a)
But
(h)
Then
DI × 0 × Dn
a b o r d i s m of
= ~(x l) +~(x 2)
gl ' g2 ~(gl )
=
= ~l" x2
@ ( ( x I + gl ) + (x 2 + g2) )
(Xl + x2)
N2 .
D 1 × D n-I × D n
NI
to
N{
defines
where a handle
which has
2
less
N2 .
~ ( x I + x2)
so that
" (x2+g2)
in
produces
is odd, let
to the factors,
(x1+gl)
NI ,
points with
shows that
xI " x2
of this arc would be of the form
" (gl + g2 )
=
0
e" Hn( S n × S n)
?(gz) +
whenever
x I " x2
be the generators
: o , ~(gl + g2 ) =
gl " g2
is even,
i .
is even. corresponding Then
so that
= *(xl + gl ) + ~(Xl + g2 ) = *(xl) + *(x2 )" SO
~((~I + x2) + (gl + g2 )) = ~(xl+x2)
+~(gl +g2 ) = ¢(xi +x2)
+
i
95 Equating
(a)
and
(b)
we get
¢(x I + x 2) and
(x I • x 2) ~
(1.5)
Theorem.
i
mod
Suppose
1-connected,
(W,M)
an embedded
S n C M 2n
the normal bundle Hence
Sn
~
j :
X n+l C W
For
Vn+l(W)
Since
~q C
n-connected, and to
C
in
M 2n
trivial #
0
(Sq n+l x) [W]
0 #
there is a map
r :
Thom class in
W 2n+q
and hence
+
~(x 2)
Vn+l(W )
~q+!
(mod 2)
~
#
with
7
,
W
[]
is
Then there exists
=
but
does not admit a
8n
such that
[ + sI
is trivial.
of a closed manifold
q-frame in
x ~ H n+q-1 (W/~W ; ~/2)
; ~/2)
O-section will be our manifold
or
0 . ~U
is quadratic.
with non-trivial normal bundle.
By Thom's theorem,
~ (T(Tn+q_ I) , ~)
H n+q-I (T(Tn+q_l)
@
n # 0 , i , 3
is non-trivial,
q ~ 2 .
(W , ~W)
,
M 2n ×
means there is an 0 ,
¢(x l)
we can find an embedding
whose normal bundle #
:
and suppose
D n+l Sn
Vn+l(W )
i
which completes the proof that
M 2n ×
is homologically
Proof:
the
2 ,
+
,
such that
and the transverse
X n+l ,
W × [0,i) .
such that
since r U
n + q - I > n, =
x
(U
the
inverse image of
which we may assume connected,
by
choosing a component with the above property. Let X0
X0
=
X - (int D n+l)
has the homotopy type of a
n-connected,
so that n
~X 0
=
dimensional
it follows that there is a map
Sn .
Since
complex.
f :
X0
Since
~
M
X
was connected, (W , M)
is
such that
f XO
~
n
M
ni W
Let Since
g
be the composite M
is
an embedding
Sn
1-connected, (again called
embedded sphere
g(S n)
~X 0 C
commutes
XO
~
M ,
up to homotopy.
so that
g,[S n]
=
0
the Whitney process will produce a homotopy of
g
g) , and we wish to show the normal bundle
is non-trivial.
Using the Whitney general position embedding theorem,
we may deform
~n
.
to to this
96
X0
f
,
M ×
~ q X (-i , O]
extending
g
which meets
M
~q
x 0 .
C
M X
to an embedding
M x
]Rq X 0
transversally
On the other side, the embedding general position) g(S n)
:
go :
XO
g :
g(SD n+l) C C
embedding
M X
X 0 U D n+l
general position)
=
.n+q C1
let
e
~
of
g(S n)
X---~
~ q X (-i , O]
in
=
go(SXo)
W X 0
W x (-i , i)
]Rq ~ W X 0
~(D n+I) c w x
(1.6)
If
n ~ 1 , 3
C
(using
in
The two embeddings
D n+l C W X [0 , i)
j :
g(S n)
transversally
together define an
which is isotopic X
(by
C W C W X (-i , i) .
defines a
q-frame in the normal ~n + gq
(so that
~+q)
=
be the obstruction to extending this
of
and
k-frame over the
[0 , l) or
7 , the obstruction
~
=
0
if and only if
is trivial. Proofs of
(1.6)
We assume
~n
D n+l X 0 X 0 ]Rq X 0 =
C
c
D n+l x D n X
~(D n+l)
and
W X [0 , I) .
M X [ - i , O]
gl(x)
can be found in
U
]Rq
C
or
[Browder, S ; (IV 4.2)] .
W × [0 , i)
Sn X D n X 0
Let
V
In that case we can
(using
a neighborhood
be the
V X
(1.6))
of
cobordism of
(D n+l X D n) , so that
g(S n) M
with ~M
C
M
×
defined by
] R q C W X [-i , l] , and
(intv) x m q
Hence we have a factorization ~(W X [-i , i]))a-a--~ E q V/~V
~n+q (y ~ ~/2)
and
H n+q (T(~) ; ~ / 2 ) so that
[Wall]
is trivial and produce a contradiction.
find a framed handle
V
meeting
M X]R q C W x 0
normalbundle
~n
g :
X0 = ~ M X
extends to an embedding
mk X 0 C W X 0 . ,
M x
C
Wn+ 1 (Vn+q, q)
Lemma.
Sn C M
to our original embedding
The product structure bundle
X
~ q X (-i , O]
gl :
g :
D n+l C W X [0 , I)
M X 0 CM
go :
b )
of the collapsing map T(~ + g I)
(Sqn+l(x)) [W] the Thom class.
Sq n+l ( Z -q (b'U))
[V]
~
=
so that
where
=
W X [-i , i] /
(ba)*(U)
(Sq n+l (Zx)) [y]
It follows that 0 ,
Y
#
=
0
Zx
, ~
(Sq n+l (b'U))
E -q (b'U)
e
e
(zq[v])
Hn(v/~v ; ~ / 2 )
0 , ,
97
which leads to sought dimension
n .
after contradiction
This completes
since
the proof of
Sq
n+l
annihilates
cohomology
of
(1.5). []
On the other hand we have: (1.7)
Proposition.
n odd , bundle
and of
quadratic (1.7)
If
~ :
@(S n)
Vn+l(W)
S n C M 2n is trivial
form of follows
(1.4)
=
with
0 , ~
(W , M)
nullhomotopic
if and only
~(~.[sn])
n-connected, in =
W . 0
n # i , 3 , 7 , Then the normal
where
~
.
easily from
(1.6)
and the definition
of
%~ .
is the
98
§2.
Complete
intersections,
In this paragraph,
complete i n t e r s e c t i o n s form of
§i
their normal bundles and the quadratic
we apply the results of
vnc
~pn+; ,
to be defined,
§i
form.
to the case of non-singular
give the conditions f o r the quadratic
and calculate the Kervaire invariant when it is
defined. Recall that a submanifold if
V
dim V
is the locus of zeros of
k
=
and
2n
(real dinension)
completely determine differential
=
V
n ,
is a non-singular
homogeneous codim V
and
V
2k
PI 7--., Pk
The degrees
•
d.1
where of
P.l
Thus any question we ask in
must have an answer in the form of a formula
d I ,..., ~
,
and we will use the notation
V ~ (d I .... , ~ ) . From/the topological
point of view it is convenient to view
transversal
inverse image, to have its normal bundle in
may assume
Pi(! , 0 ,..., O)
~. :
di zI
complete intersection
polynomials =
up to diffeomorphism.
topology about
involving only V
V C GLPn+k
~pn+k
, ~pn+k
di Zn+ k
,...,
) ,
#
by
for all
~i (z0
7~l
so
0
(z 0 = O)
i .
~pn+k
V
as a
evident.
We
Define maps
Zn+k)
=
(Pi (z0
is a hypersurface
Zn+k)
of degree
di .
k
Define
g~pn+k
7:
'
k V
:
7 "l (
)
,
where
=
z
of
of the coefficients
will be a non-singular
Proposition.
the vanishing dx n
by
A i.
Then
is the
0-th coordinate
in the
~pn+k .
Small perturbation transversal
gpn+k
M i=i
The non-singular PI
,.
""' Pk
(~pn+k), and
bundle map into the bundle
on
(if necessary)
will make
manifold and we get:
complete intersection
~pn+k
Vn
defined by
represents the homology class
the normal bundle
~
of
( dl + (~d2 + ... + ~ dk)
V C ~pn+k over
has a natural
~.pn+k
where
x n
99
is the generator bundle
dual to
over
{pn+k
(Jl
=
dk)
.
(~l+... + dk) (2.2)
_(J1
in
E
+ ...
,
=
dI
+
Y
V ×
~q
C
7
~
where
the canonical
of
of
V
§i
by embedding
which is stably inverse of
V C E
Vn C
E = E(7)
to
has a bundle map into
trivialization.
intersection
E ,
(~) ,
into the situation
which has a natural
Hence
{pn+k ,
7
has a natural
a representative
of
C K({P n+k) .
Note that the framing
is determined
intersection,
and the polynomials
structure
V .
To apply
c
the total degree
space of a bundle
The complete
+ yk)
of
dk
.-°
Then the normal bundle
Proposition.
framing
,
this situation
the total
+ ... +
¢
d
,
We may transform [pn+k C E
k
by the structure
PI " ' ' '
§l , we need to calculate
form is well defined.
Note that if
n
Pk '
of
V
as a complete
not simply by the differentiable
Vn+ I (E(Z)) is even, Vn+ I
,
to see if the quadratic =
0
since it lies in
a zero group.
(2.3)
Theorem.
= the number Proof:
Vn+l(E)
#
of even integers
(I + x) n+k+l
n (l+
i=l =
W(T
and o n l y i f
( ~ + 1~ ) @
among the degrees
0(mod 2)
d I ,..., d k ,
,
where
n = 2s + i .
class
= W((n+k+l)~-(~l+...+J~))
=
di~)
k
W(E)
if
The Stiefel-Whitney
W(S) = W(~pn+ k + ~ )
Hence
0
(i + x) n+k+l
(1+
(i + x) n+~+l
x)k_~
n+~) in dimensions
where both cohomologies
agree,
and hence
CP
h+l(E)
Vn+l (¢pn+~)
But
; 9Z/2)
~ ~n+2~
H ~+2~-I
(~+$
Hn+2~-i
({pn+ ~
H2 ({pn+~ ;
; Z~/2)
2Z/2)
Vn+ l
({pn+~
is generated
(since
(¢pn+~)# 0
by
n = 2s + 1) ,
; 7z/2) x
s+~
so
,
if and only if is non-zero.
where
x
Sqn + l (x s+~)
Sq n+l The group
generates
=
Sq 2s+2 (x s+~)
100
=
s + ~ ( s + i ) In
n+~ x
E ,
,
x s+l
which completes the proof. O is represented by
(1.5)
may be taken oriented,
(2.4)
Corollary.
~pn+k
is odd ,
2
embedded
Sn C V
bundle,
=
PI ''''' Pk
~pS+l , so
If
n
=
2s + i ,
of degree
which is homo!ogically n ~ I , 3 or
~
in the proof of
V
non-singular and if
d.'s , z
trivial,
in
s + ~ ) ( s + i
then there is an
and has a non-trivial normal
7).
To calculate the Kervaire invariant (where
X0
d I ,..., d k ,
number of even degrees among the
(provided
other cases
=
and we get:
(Morita, Wood)
defined by
X
is well defined)
(the Arf invariant
of ~ )
in the
we use the following theorem.
The
proof will be given in another paper, and it follows from a combination additivity theorem for the Kervaire on index) (2.5) let
(analogous to Novikov's theorem
and the product formula for the Kervaire
Theorem.
Let
~0 -I C ~pn+k-i
Vn
C
~pn+k
Vn
K(V)
=
invariant.
be a non-singular
be a non-singular
forms are defined for both equal,
invariant
and
of an
hyperplane
complete
section.
intersection,
and
If the quadratic
~0_ 1 , then their Kervaire
invariants
are
K(V O) .
Note that the definition
of
K(V O)
may have some extra subtlety as we
will see in the calculation. We can immediately derive the formula for is odd.
In that case
(2.3)
sections
implies
K(V O)
for the zero dimensional complete d
K(VnO_I ) ,
-i
D
d
=
d I ...
forms are defined ...
D
von-i
so that
and we are left with the problem of computing
(2.5)
for
=
when
implies that the quadratic
for all the iterated hyperplane K(V n)
K(V)
intersection
of degree
d ,
that is,
similarly oriented points.
This calculation
is a special case of that of
actually equivalent to it using a product formula.
[Browder, FI~K] and is We do it explicitly as
follows. Suppose
vO(d)
=
d
disjoint points,
embedded in
W .
A framiD~ of
101
vO(d)
is simply an orientation
vI(W)
=
0
means that
W
on a neighborhood
is orientable.
of each point and the condition W
Suppose
and the orientations
at all the points are the same.
(2.6)
K(V0(d))
Proposition.
Proof:
Wm
Since
=
is connected,
I 0
if
d
~
±i
mod 8
i
if
d
~
±3
rood 8 .
is connected we may assume that
that the symmetric group
Zd
acts on
V0(d)
,
d
vO(d) C ~m C W
is odd,
and
preserving the framed embedding,
so that
#](C-X)
x c K0
Now
K0
has a basis
are
x 0 ,..., X2s
(2.7)
=
#](X)
~ ~ Ed ,
for
ker H 0 (V0(d) ; Z~/2)
,
d
=
=
Further the intersection
2s + i .
Since
#](xj + x O)
Zd
for all
I° 1
Define a module A
As
,
with quadratic
#](ai) = 0
S
for all
(ai ~ a J ) =
I~
(the opposite of an orthonormal basis). form
( , )
is non-singular
Similarly, all
i ,
and
H 0 (DTM ; ~/2)
define
where the
acts transitively
.
d
points
on this basis,
i , j .
product
(~i +~o ) " (xj +~o ) =
be a basis for
~
{x I + x 0 , x 2 + x 0 ,..., X2s + x 0} ,
#](xi + x O)
(2.8)
=
Bs ,
on ~
As
form i ,
i : j i ~ j
#] by letting
a I , a 2 7..., a2s
and
if i if = j ~ j
It is easy to check that the bilinear
and is the associated bilinear form to
by the basis
b I ,..., b2s
,
~(bi)
=
#] .
i ,
102
(bi , hj)
(2.9)
Lemma.
(as modules Proof:
Then
AI + B s-i
Bs
~
B I + As_ I
a new basis
be generated b y AI
±
B's_l
Similary, B I C Bs Then
~
with quadratic
Define
A I CAs
As
'
Since
~
=
j
1
i
~
j
forms). for
As
a~m
by
aI , a2 ,
B's_l
define a new basis for by
BI + A's_l
Arf
i
(A I)
=
=
C
ai + aI + a2 As
bI , b2 ,
Arf
l b'.
by
A's_l C B s
as orthogonal
0 ,
Bs
direct
(B I)
i
B's_l =
generated
we get
~
and
let t
by
Bs-I
A's_l
!
2s
" let
b3v ' . . . '
by
a
a3 "''~
b i + bI + b2 ,
sum, and
,
,
be generated
and it is easy to cheek that
be generated
Bs
0
Z- As_l
b
st
•
.
:
A~f (As> : Arf (Bs_l) so that
Arf (B s)
=
i + Arf (As_i)
Arf
(A s )
=
I + Arf
(As_ 2)
Arf
(B s)
=
i + Arf
(Bs_ 2) .
Hence:
f
(2.10)
Proposition.
But in
(2.6)
Amf (B s)
,
(2.6)
and the calculation
odd ,
i.e.
Theorem
C(i)
of
defined
v
in
we need:
V~u °I
n
if
s - i
or
2
nod 4
0
if
s - 3
or
4
nod 4 .
=
2s + i
which completes
K(V n (d I ,..., dk))
d
is more difficult,
for the iterated hyperplane
for the first hyperplane
Wu class
d
)!
when
d
=
the proof of d I ... d k
is
.
The case of even degree m a y not be defined
if
K0 ~ B s
=
section
lies in a zero group when
since the quadratic
sections.
~0 -I C V n , n
is odd).
However,
form
it is always
(since the appropriate To make the calculation
•
103
(2.11)
Proposition.
oriented. x2
~
If
x
Let
~
H
H m (M ; EZ/2>
M 2m x
(M ;
m
]Rk C W ~
7z) ,
Vm+ I (W)
i. (x 2)
the reduction of
=
=
0 ,
x nod 2
0
i :
then
,
m
M---~
~(x2>
even, and W
-
M
inclusion, x • x
nod 2
2
We sketch a proof,
(compare
First note that if i.(x 2) Vm(W) =
(i.x 2)
(Vm(M) U y)
x 2) •
Hence
quadratic (2.12) (K
=
=
[M]
so
=
x - x
Lemma.
x • x
Vm(M) is even,
Let
~
0 , -
then
x 2 • x 2 mod 2 ,
([M] n
y)
=
so
9(x)
vm
x - x 2
(M 2m ; ZZ/2)
M(x)
=
' ) H m (W ; ZZ/2))
form defined
0 .
Then
(p(A) = i = @(Z~) . arbitrary
× 1
C
=
0
mod 2
=
(y U y)[M]
(where
[M] R y =
is a well defined
x c K
% :
as in
K ~
§l ,
in these circumstances
ZZ/2 and
q0 :
such that
S ; (IV. 4.7)]
f~ e H m ( S m × S TM ;
T h e n b y adding
Sm × S m
if n e c e s s a r y we get
K
~(x) = 0
to
M(x)
Z~/2) M 2m
=
(for and
~(x)
•
For the condition
S TM X S m
adding
C
Z~
S 2re+l) ,
to an
(compare the proof of
.
Thus to prove
bordism
x2 • x2
=
~ = ~ .
implies that on the diagonal
(2.13)
i*(Vm(W))(x2)
and
form
The proof is similar to that of [Browder,
(l.4))
=
(M)(x 2)
"
.
Vm(M)(x2)
be our usual quadratic
another quadratic
implies
=
P] , [Brown])
form.
ker %
7Z/2
0 ,
[Morita,
Lemma.
(2.11)
If
U 2m+l x
k
(M ; =12)
is large,
]Rk C
W × {0 , i}
it suffices ~(x)
W × [0 , i]
,
and
to show =
,
Vm + l C U
0
8(U X ,
~V
@(x)
=
0
implies
q0(x)
=
0 .
implies there exists a framed ]Rk) =
=
N TM C M
M ×
]Rk X 0
with
U
INm]
M' =
X
]Rk
x2 e
.
m
Proof:
As ill
§i ~ we carl find
N TM C M 2m
representing
x 6 }{
(M ~ 2Z/2)
m
and
V m + l C W x [0 , l)
the normal bundle to the framing of
M
in
U
with
~V
admits a W).
=
N CM
x 0 .
Then
9(x) = 0
k-frame extending that of
The complement
of this frame is a
N
implies that (coming from
Dm
bundle
over
104
V
which meets
M 2m
disk bundle to required isotopy
in the normal disk bundle
M × [0 , s]
except for the condition if
k
~(x)
Y~+I(u'MUM' , i' y Now
=
x • x
0
i :
%
(i*@
i' (~)
x
x~
mod 4 )
Since (j : ~ / 2
of the type
but this may be achieved b y an
x ,
since
byPoinear~duality,~2
=
iy
, ~m(~;
U ,
,
,
0 ,
=(~(i
:
i
~
---~ ZS/4).
(nod 4) (see
U i'
0 But
=
~
U
~
[Morita,
:
0
~/2
H m (M ; ~ / 2 )
P])
.
Now
~
~
(~2)[M]
x2 • x2 ~
x 2 = ~j ,
the inclusions.
where
so that
0
it follows
~
M' ---~
~(~)(i.[M])
where
[~]
:
i' :
square
=
= (T~(6~(~))
mod 2
•.j
~
•
=
and hence
~/4
) ~/2
~
0 .
that
y) +
ff(±'*~([M]
(f(~)
(Z.[~])
M U M'
= ~U------>
mod 2
it follows
,~(~)
e
- [M'])
U .
that
(~,[~U]) 4
j. H 2m (U ; ~ / 2 )
(we get a factor of
2
from each
e
and j.
J.H2m
j. H 2m
(U ; ~ / 4 )
and
which multiply
J.H2m to
0 mod 4) x • x
-
We n o w proceed even.
Recall ~
0
~
=
defined),
The quadratic (~pn+k-i
mod 4
so
~p(x) = 0 , which complete
to the calculation
K(Vo -I (d ! .... , ~ ) )
Hn_l
and adding this
=
H2m (U ; ~ / 2 )
@ ( i y)[M]
~. [~U]
Hence
(to have
M' C W × i ,
@(~2)[M]
(~))[M]
=
are paired to zero become
,
e
(y) e j. H 2m (U ; ~ / 2 )
(all
M ,
[M] N ~
is the Pontryagin
=
Since
in
a framed cobordism
we note that if
M---~
--2 x [M]
=
H2m (M ; ~ / 2 )
(i*~)[M]
=
; m/2)
0 ,
H 2m (M ; ~ / 4 ) i. :
Nm
is large.
Now to show
~/2)
clearly defines
of
form
; 2~/2))
number where
of
of even
K(Vn(dl di's
n = 2s + i .
By
but we must make this ~
is defined
and since
on
n - i
and
,..., ~ ) )
(2.5)
=
for
( s + i~ )
statement L
the proof of
ker
is even and
,
_
d 0
=
(2.11) 17. d I ...
nod 2
K(Vn(d I ,..., dk))
=
more precise. (%-i d
(Vo ~ ZZ/2) is even,the
associated
105
bilinear
form is singular
be defined then
~(r)
(2.5)
.
on
it is necessary =
0
(see
L .
that if
[Browder,
We will now study the middle using coefficients
in
Thus,
ZZ(2 )
for the Arf invariant
r e L FPK])
and
, but this
dimensional
(i.e.
,. ~n+k - I , ~ ,V0)
Hn_l
Hn-i
(~pn+k-1)
9z
also splits
({pn+k-1)
.
Lemma.
of
The Poincare
dual of
~
h = i@(g)
L
=
~
on
L
for all
included
in
~-l
form on
to
x e L ,
all odd denominators)
and
we will
can be easily computed. i. :
and therefore splits.
and we let
0
is implicitly
(n - i) - connected,
is onto,
Hn-l(v O)
(2.14)
(
Z
is
=
intersection
introducing
put it in a form in which the Arf invariant Since
(r , x)
of
,
Hn_ 1 (V 0) i* :
Hence
where
)
g
Hn-1
(¢pn+k-l)
generates
the annihilator
of
h([Vo] N x)
(h
h
under
, ). g(i.x)
Proof. (h , ~)
.
= (g
h U
e
H n-I (V O)
is indivisible.)
g) (d[¢pn-1])
generated
i*(g)(x)
: h(x)
=
=
U
x)[Vo] )
=
[]
Now let since
=
by
h ,
=
be such that
Now d .
(h,h)
(h , ~) = i , (which is possible
=
(i*gU i'g) [vo]
Hence the quadratic
form on
A
=
(gUg)
=
i.[V0]
the submodule
, has the matrix
(: :) and hence has odd determinant Hence,
over
that the matrix
ZZ(2 ) , for
H n-I
ad - ! , since
d
is even.
we can find a complementary (V 0 ; ZZ(2))
=
A + B
summand
becomes
B
to
A
so
106
Since
B ± h ,
and
B
is the largest submodule of
which the bilinear form is non-singular is the
Arf invariant of the quadratic
is the matrix for this intersection (2.15)
Proposition.
=
See for example ZZ(2 ) ~
x 2• x
form
(mod 2)
on
on
Arf ~ = K(V 0) B ,
and
T
form.
IO
if
dot T
-
±i
mod 8
i
if
det T
~
±3
mod 8 .
[Hirzebruch-Mayer a matrix
T
=
; (9.3)] •
We sketch the proof here.
~ith even diagonal entries and odd determinant
may be put in form of the sum of
STS t
(mod 2) , it follows that
(h))
The Arf invariant
Arf (B)
Over
(annihilator
iI
2 X 2
blocks
bI
I(a i ) a l) 0 1 (12b2 l) 0
For given a generator that
(g , g')
g
of
B ,
(over
For if both
a ~
quadratic
~(2))
( a i ) ib b and ~ residue
multipl~ng
since
is odd so that over
whose matrix may be made into module
( it
det T
2(2 )
( a i ) ib
dot T
(a
b even)
are odd. mod 8) .
dete~inants
defining
~)
=
and
g'
g' ~ B
such
generate a submodule
and we may then split off this
clearly the Arf i n v a r i ~ t
Then me
of the
(ad - i) -I . ~plies
is odd, there is
and proceed by induction.
ab - i
~ ~ 2
~
2 × 2
i ~
addingArf
if and only - I
is not a
invari~ts
~d
blocks. on all of
The condition so that
is
±3 mod 8 (i.e.
result then f o ~ o w s ~
Since the bilinear form is ~ o d ~ a r that
g
bt
Hn-I(Vo ) ,
( s + i~ )
4 I d ,
(~
=
~
0
it follows
mod 2
number of even
(for di's) .
107
Hence
-(ad - i) -I
(2.15),
a
=
Arf B
i + ad + a2d 2 + ...
=
0
if
(~ , ~)
=
(Vn_l(V O) , ~)
(~ But
=
8 I d
or if
a
e
i + ad
is even.
Hence, by
It remains to calculate
~) mod 2
so
Vn_l(V 0)
is nonzero and equal to
h (mod 2) .
where
=
,
E
(mod 8) .
~pn+k-I ,
As in
E ( - ( J I + ... + ~dk))
~o
a
(2.3)
(~ , ~)
a But
Vn_l(E)
The latter happens if and only if
Vn_l(V O)
=
i* (Vn_l(E))
the total space of this stable bundle over
is odd if and onlyif
we get that
is odd if and only if
#
0
Vn_l(~)
if and only if
( s + ~ )
@
0
mod 2
#
O. ~_~n-l~ vn_ik~ )
#
which completes the
s
proof of Theorem C .
[]
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[K] ,
The Kervaire invariant of framed manifolds and its generaliza-
tion, Annals Of Math
90
(1969) , 157-186.
[FPK], Cobordism invariants, free involutions, IS] ,
Trans. A.M.S.
the Kervaire invariant and fixed point
178 (1973), 193-225.
Surgery o__nnsimply-connected manifolds,
Springer Verlag, Berlin
1973. E. H. Brown, Generalizations (1972)
of the Kervaire invariant, Annals of Math. 95
368-383.
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M. Kervaire, Relative characteristic
exotische Spharen und
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classes, Amer. J. Math. 79 (1957), 517-558.
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63
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