GENERALIZED
B.
A.
WITT
GROUPS
UDC 519.4
Dubrovin
In t h i s p a p e r we c o n s t r u c t a f u n c t o r f r o m t h e c a t e g o r y o f o n e - d i m e n s i o n a l c o m m u t a t i v e f o r m a l g r o u p s to t h e c a t e g o r y o f t o p o l o g i c a l A b e l i a n g r o u p s . F o r a m u l t i p l i c a t i v e f o r m a l g r o u p ,
t h i s f u n c t i o n is t h e u s u a l W i t t f u n c t o r . W e s t u d y c e r t a i n p r o p e r t i e s of the c o n s t r u c t e d tune=
t o r . T h i s f u n c t o r i s t h e n u s e d to d e s c r i b e m u l t i p l i c a t i v e o p e r a t i o n s in the t h e o r y of u n i t a r y
cobordisms.
1.
BASIC
DEFINITIONS
1. L e t R be a c o m m u t a t i v e a s s o c i a t i v e r i n g with u n i t y , R[[X, Y]] the r i n g of f o r m a l p o w e r s e r i e s in
two v a r i a b l e s , F(X, Y) ~ R[[X, Y]], a o n e - d i m e n s i o n a l c o m m u t a t i v e f o r m a l g r o u p o v e r R, I(X) 6 R[[X]] i t s
i n v e r s e e l e m e n t , and
~
o(x) = /lZ=op
Xi\
)dx
(1.1)
its c a n o n i c a l i n v a r i a n t d i f f e r e n t i a l (P0 = 1). If R is a Q - a l g e b r a t h e r e t h e n e x i s t s a s e r i e s
l(X)=
=oi+l
X i+l s u c h t h a t w(X) = dl(X) and
y ( x , Y) = z-I (1 ( x ) + t (Y))
(1.2)
( s e e [2]). T h e s y m b o l A w i l l d e n o t e the t o p o l o g i c a l s p a c e of f o r m a l p o w e r s e r i e s o v e r R w i t h o u t a f r e e
t e r m , with t h e u s u a l t o p o l o g y o f f o r m a l p o w e r s e r i e s . We i m p o s e on A t h e s t r u c t u r e of a n A b e l i a n t o p o l o g i c a l g r o u p A(R, F), s e t t i n g
(l + Fg)(X) = F (1 (X), g (X)).
I t is t r i v i a l to c h e c k t h a t t h e a x i o m s of a t o p o l o g i c a l g r o u p h o l d .
L E M M A 1.4.
Let
] ( x ) --~,~i~ a~x l .
(1.3)
L e t R not h a v e e l e m e n t s of finite o r d e r .
T h e n , t h e s e r i e s f ( X ) i s u n i q u e l y r e p r e s e n t e d in t h e f o r m
~e
] ( X ) = ~=1F:~iXi"
(1.5)
M o r e o v e r , if a m is t h e f i r s t n o n z e r o c o e f f i c i e n t i n f ( X ) t h e n oq = ai w h e n i -< m .
P r o o f . E x p r e s s i o n (1.5) is m e a n i n g f u l s i n c e its r i g h t s i d e c o n v e r g e s in g r o u p A{R, F). W e h a v e a n
equation over
( o~ a~X~ )
| e: z,Z=
=
~
l(:x~X~)"
From this we have
a,,,X "~ + o (X'?) =
~,,,X'" + o (X"~),
i.e., a m = am, which means that
cc
i
--
(1.6)
M. V. L o m o n o s o v M o s c o w S t a t e U n i v e r s i t y . T r a n s l a t e d f r o m M a t e m a t i c h e s k i e Z a m e t k i , V o l . 13,
No. 3, pp. 4 1 9 - 4 2 6 , M a r c h , 1973. O r i g i n a l a r t i c l e s u b m i t t e d O c t o b e r 22, 1971.
9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
253
Now a p p l y i n g t h e s e r i e s f o r l to b o t h s i d e s of (1.6), we o b t a i n
c~
F (/(X), I (amXm)) = ~'i=m+l FaiX~"
(1.7)
But, t h e left s i d e of {1.7) i s a s e r i e s o v e r R w h i c h b e g i n s with a p o w e r g r e a t e r t h a n m .
the p r o o f is a n o b v i o u s i n d u c t i o n .
C O R O L L A R Y 1.8.
E x a m p l e 1.9.
A (R,
F) c A (R |
0,
T h e r e m a i n d e r of
F) is a s u b g r o u p .
L e t F m ( X , Y) = X + Y - XY be a m u l t i p l i c a t i v e f o r m a l g r o u p .
C o n s i d e r the m a p p i n g
: A-+R[[X]],
setting ]-+l--f
i X ) . T h e n , (/ + Fg)(X) = ] ( X ) + g ( X ) - - ] ( X ) g i X ) --> [l - - ] (X)l[i - g(X)], i . e . , k is a c o n t i n u o u s m o n o m o r p h i s m of g r o u p A ( R , F m) into the m u l t i p l i c a t i v e g r o u p of i n v e r t i b l e
e l e m e n t s of r i n g R[[X]].
2. D e f i n i t i o n 2.1. We c a l l the g e n e r a l i z e d W i t t g r o u p W(R, F) t h e s e t of i n f i n i t e v e c t o r s x = (x 1, x 2,
. . .), w h e r e xi ~ RVi, with the m a p p i n g s
w~(x) = ~ l ,
dp~_x~/d,
n = l, 2
(2.2)
d
d e f i n i n g a c o l l e c t i o n of h o m o m o r p h i s m s
in t h e a d d i t i v e g r o u p of r i n g R.
We s h a l l c a l l x 1, x 2, . . . the t r u e c o o r d i n a t e s of v e c t o r x, w h i l e w l ( x ) , w2(x), . . . a r e i l l u s o r y c o o r d i n a t e s . It f o l l o w s f r o m (2.2) t h a t the t r a n s i t i o n f r o m t r u e to i l l u s o r y c o o r d i n a t e s is i n v e r t i b l e o v e r t h e r i n g
R | Q, i . e . , a d d i t i o n in g r o u p W(R | Q, F) is u n i v o c a l l y d e f i n e d b y (2.1). W e now show t h a t W(R, F) is a s u b g r o u p in W(R | Q, F).
D e f i n i t i o n 2.3.
M a p p i n g E : W(R, F) --~ R[[X]] @ Q, d e f i n e d b y t h e f o r m u l a
(~
t
X,~)
(2.4)
is called the Artin-Hasse exponent.
T H E O R E M 2.5. T h e A r t i n - H a s s e e x p o n e n t d e f i n e s a n i s o m o r p h i s m of t h e g r o u p s E : W(R, F)
A ( R , F).
Proof.
F r o m (2.4) w e h a v e :
== l -t l1 (g (x)) + l (E (Y))I = E (x) + Pg (g).
(2.6)
Fu r t h e r m o r e :
Ix1 ~
/ t x7
%
.
,~la\
=
-
T h e t h e o r e m now f o l l o w s f r o m (2.6), (2.7), and L e m m a 1.4.
C O R O L L A R Y 2.8. T h e s t r u c t u r e of t h e g r o u p on W(R, F) i s u n i v o e a l l y d e f i n e d b y (2.1).
T h e p r o o f f o l l o w s d i r e c t l y f r o m (1.8) and (2.5).
E x a m p l e 2.9. F o r g r o u p F m o f e x a m p l e (1.9) we h a v e : Pi = 1 f o r i = 1, 2 . . . . .
i . e . , f o r m u l a s (2.2)
assume the form
w~ (x) = E.~/,dx~/d. C o n s e q u e n t l y , g r o u p W(R, F m) is the a d d i t i v e g r o u p of a W i t t r i n g
( s e e [1]). T h i s m o t i v a t e d o u r c h o i c e of n a m e f o r g r o u p W(R, F).
In W(R, F) w e i n t r o d u c e t h e t o p o l o g y i n d u c e d b y the n a t u r a l v a l u a t i o n : v(x) = n, if x n is t h e f i r s t n o n z e r o t r u e c o o r d i n a t e . T h e n , E b e c o m e s an i s o m o r p h i s m of t o p o l o g i c a l g r o u p s .
L e t F be t h e c a t e g o r y w h o s e o b j e c t s a r e t h e p a i r s (R, F), with F b e i n g a f o r m a l g r o u p o v e r r i n g R.
A m o r p h i s m f of c a t e g o r y F is a r i n g h o m o m o r p h i s m f :
R1 -~ R2, w h e r e F 2 = F ( , i . e . , F 2 i s o b t a i n e d b y
a p p l y i n g f to t h e c o e f f i c i e n t s of F I. We note t h a t then p~2 = f ( P i F i ) w h e r e P i i s a e f i n e d b y (1.1). W e d e f i n e
W ( f ) : W(R I, F 1) ~ W(R 2, F2), s e t t i n g W ( f ) (x 1, x 2. . . . )t= ( f ( x l ) , f ( x 2 ) . . . . ). T h i s , o b v i o u s l y , i s a c o n t i n u ous i s o m o r p h i s m of t h e g r o u p s .
254
T h e r e is a u n i v e r s a l o b j e c t ( s e e [3]) in c a t e g o r y F . W e d e n o t e it b y t h e s y m b o l (L, U). S i n c e L is a
t o r s i o n - f r e e r i n g we h a v e t h e n d e f i n e d t h e g r o u p W ( L , U). If, now, (R, F) is a n o b j e c t of F and f : (L, U) --~
(R, F) is t h e c a n o n i c a l h o m o m o r p h i s m , t h e n W ( R , F) is d e f i n e d a s the i m a g e o f g r o u p W ( L , U) u n d e r the
h o m o m o r p h i s m of W ( f ) into t h e s e t of a l l v e c t o r s w i t h c o o r d i n a t e s i n R . T h u s , W b e c o m e s a f u n c t o r f r o m
c a t e g o r y F to t h e c a t e g o r y of t o p o l o g i c a l A b e l i a n g r o u p s and t h e i r c o n t i n u o u s h o m o m o r p h i s m s .
2.
ENDOMORPHISMS
OF
FUNCTOR
W
1. D e f i n i t i o n 1.1. We d e f i n e t h e f a m i l y of m a p p i n g s b y s h i f t Vn: W ~ W, s e t t i n g
Xrn ,
(v~ (x))~
=
if
0
LEMMA 12.
Proof.
Vnisamonomorphismforn
nlrn ,
(1.2)
w
otherwise
= 1, 2, . . . .
W e c o m p u t e the a c t i o n of Vn on t h e i l l u s o r y c o o r d i n a t e s .
wra (Vn (x)) = ~-~Jdl,n dP rn
- 1 a'ct/n
~/~
We have:
)
d
i .e.,
Wr~ (Vn (x)) :
I~k'=rn/n
nh'p~-lx~ = nWm/n (x),
[ 0
if n Im,
otherwise
(1.4)
T h e a s s e r t i o n o f t h e [ e m m a t h e n f o l l o w s f r o m t h e o b v i o u s i n j e c t i v e n e s s of m a p p i n g (1.2) and the a d d i t i v i t y
o f e x p r e s s i o n (1.4).
D e f i n i t i o n 1.5. We d e f i n e t h e f a m i l y of F r o b e n i u s m a p p i n g s Fn: W ~ W b y t h e i r a c t i o n on i s o m o r p h i c g r o u p A . L e t ~l . . . . .
~n be t h e f o r m a l n - t h r o o t s o f X. We s e t
F,~ (1 (X)) = ~ 1
F/(~0"
(1.6)
T h e r i g h t s i d e o f (1.6), b e i n g s y m m e t r i c in t h e ~i b y v i r t u e o f the c o m m u t a t i v i t y of g r o u p F, is t h e k e r n e l
o f R[[XJ]. T h e a d d i t i v i t y of m a p p i n g F n i s o b v i o u s .
L E M M A 1.7.
w,~
( F . (x)) = w~n (5).
(1.S)
P r o o f . Due to the s i m p l i c i t y o f t h e c o m p u t a t i o n s w e v e r i f y o u r a s s e r t i o n on e l e m e n t E((1, 0 . . . .
X. We h a v e :
F . ( X ) = Z-1(t(~1) + . . .
Pl
+ --~ ( ~ + 9
+ Z(~.)) = Z- 1 ( ( ~ + . . . +
+ ~'~
..) + ...)
) = Pm-l"
T H E O R E M 1.9.
~.)+
\
= l -a (p,~_~X + p ~_~ X 2 +
I
But, Wm(1, 0, 0 . . . .
)) =
. .).
2
Now, (1.8) f o l l o w s f r o m t h e p r e v i o u s c o m p u t a t i o n s and (1.2.4).
V n and F n h a v e t h e f o l l o w i n g p r o p e r t i e s :
(1) V m ~ Vn = V m n ,
(2) F m ~ Fn = F m n ,
(3) Fn ~ V m = V m ~ Fn w h e n (m, n) = 1,
(4) F n o V n is m u l t i p l i c a t i o n b y n in Z - m o d u l e W,
(5) ( l / n ) V n o F n i s t h e p r o j e c t o r of w ( R @ Q, F) on v e c t o r x s u c h t h a t wm(x) = 0 w h e n
(6) ~ - _ ~ , (~,p)=l ~(n)
~- V~ol~'n is t h e p r o j e c t o r of W(R |
n ~m,
Zp, F) on v e c t o r x s u c h t h a t Wm(X) = 0 w h e n
m ~ ph (p is the MSbius function).
Proof. By virtue of (1.4) and (1.8), the action of Fn and Vn on illusory components in generalized
Witt groups is identical to the action of the shift and Frobenius homomorphisms in ordinary Witt groups.
255
With this in mind, we r e d u c e a s s 3 r t i o n s (1)-(6) of the t h e o r e m to the c o r r e s p o n d i n g a s s e r t i o n s about o r d i n a r y Wttt g r o u p s , p r o v e n by c o m p u t a t i o n s on the i l l u s o r y c o m p o n e n t s {see, for e x a m p l e , [1]).
Now, let r i n g R be an a l g e b r a o v e r the r i n g of p - a d i c i n t e g e r Zp.
Definition 1.10.
F o r m a l group F is p - t y p i c a l if, in group W(R, F), win(x) = 0 when m ~ ph.
T h e e q u i v a l e n c e of this definition with that given by C a r t i e r (see [4]) follows i m m e d i a t e l y f r o m (1.8).
T H E O R E M 1.11.
L e t / p ( X ) = E(Trp(1)). Then, f o r m a l group Gp(X, Y) = f p i F ( f p ( X ) , f p ( Y ) ) is p - t y p i -
cal.
Proof.
Follows d i r e c t l y f r o m a s s e r t i o n (6) of T h e o r e m (1.9).
3.
CONNECTION
WITH
I. Let F(u, v) be the formal group of geometric
g (~) =
TOPOLOGY
cobordisms,
and
y;(_o
(1.11
its l o g a r i t h m (Mishchenko s e r i e s ) (see [6]). H e r e , u E U*(CP ~176= ~2U [[u]] is a u n i v e r s a l e l e m e n t , i.e., in
this c a s e the c o e f f i c i e n t s of the i n v a r i a n t d i f f e r e n t i a l (1.1.1) have the f o r m Pi = [ c p i ] , and f o r m u l a s (1.2.2)
take the f o r m
w~ (x) = ~'dl,~d tOP'S/d-11 z~ z.
(1.2)
LEMMA 1.3. Let q E A U @ Q be the m u l t i p l i c a t i v e o p e r a t i o n in the t h e o r y of u n i t a r y c o b o r d i s m s ;
then, r
= g(u).
By definition, g(u) is a p r i m i t i v e e l e m e n t u n d e r the m a p p i n g
induced by the H - s t r u c t u r e on C P ~ . Any o p e r a t i o n of AU
the m u l t i p l i c a t i v e o p e r a t i o n t a k e s p r i m i t i v e e l e m e n t s into p r i m i t i v e ones.
e l e m e n t s is o n e - d i m e n s i o n a l and, s i n c e q~(g(u)) = u + o(u), then ~(g(u)) =
U* (CP =) Q O-+ U* (CP ~)
Proof.
U* (CP ~) (~ Q,
c o m m u t e s with the diagonal, i.e.,
But, the ~ u - m o d u l e of p r i m i t i v e
g(u).
Let q~(u) be a f o r m a l p o w e r s e r i e s i n f ( u ) . As is known (see [6]), f r o m the s e r i e s / (u) ~ fLv[[u]] @ Q
one r e c o n s t i t u t e s , u n i v o c a l l y , the m u l t i p l i c a t i v e o p e r a t i o n ~o E AU | Q. We d e n o t e b y g~(u) the s e r i e s o b tained f r o m g(u) by the a c t i o n of ~ on its c o e f f i c i e n t s . We h a v e :
g (u) = ~ (g (u)) = g= (/(u)),
consequently,
g (]-1 (u)) = g~ (u),
i.e.,
/-1 (u) = g-1 (g~ (u)).
(1.4)
Let x EW(fiU | Q, F) be a v e c t o r such that wn(x) = ~a[cpn-1]. We then obtain f r o m (1.1), (1.4), and (1.2.3)
t h a t f - l ( u ) = E(x), i.e., the A r t i n - H a s s e exponent established the c o n n e c t i o n b e t w e e n the a c t i o n of the m u l t i plieative o p e r a t i o n on the coefficient r i n g and its a c t i o n on the e o b o r d i s m s of i n f i n i t e - d i m e n s i o n a l p r o j e c tive s p a c e .
C O R O L L A R Y 1.5. L e t w 1, w 2, . . . , E flU- T h e r e e x i s t s m u l t i p l i c a t i v e o p e r a t i o n cp E AU, with q~ [cpn-1] =
w n, if and only if t h e r e exists a c o l l e c t i o n xl, x 2, . . . E flu such that
Wr, ~
dir~ u, [ t . , l y
j Xd
,
X 1 ~
J.
P r o o f . This follows d i r e c t l y f r o m the p r e v i o u s d i s c u s s i o n and f r o m {1.2).
This l a t e s t c o r o l l a r y m a k e s it p o s s i b l e to obtain d i v e r s e i n f o r m a t i o n about the m u l t i p l i c a t i v e o p e r a tions of AU. F o r e x a m p l e :
C O R O L L A R Y 1.6.
F o r any m u i t i p i i c a t i v e o p e r a t i o n ~0 EAU we h a v e :
(1) Td(~p[cPph-l]) -= 1 (mod p),
256
(2) L(~ [cPph-1]) -= 1 (mod p) for odd p,
(3) L(~a [cp2k-1]) _= 0 (rood 2),
(4) X(cp[cpn-1]) =- 0 (mod n).
H e r e , Td is the Todd genus, L is the H i r z e b r u c h L - g e n u s , and X is the Euler c h a r a c t e r i s t i c .
Proof. ALl these a s s e r t i o n s a r e of the same type, so we shall prove only the first. Let x E W(~ U, F)
be a v e c t o r such that Wn(X) = ~ [ c p n - i ] . It exists, by virtue of {1.5). Let Yi = Td xi. Then, by virtue of
~'
d " n/d
(1.2) we have: Tdw~ (x) ~ .~1,~ yd
. In p a r t i c u l a r
T d w h (x)
~'~
.
n-1
= ~=~p~yPpi
. But, Yl = wl(x) = ~(1) = 1,
whence follows the required a s s e r t i o n .
COROLLARY 1.7. Operation ~ap 6 AU @ Zp, c o r r e s p o n d i n g to v e c t o r Vp(1), acts as follows:
q% [CPph-l] = [Cppn-1],
%, [CP'] = 0 when n ~ ph __ 1.
Proof. This follows d i r e c t l y from a s s e r t i o n (6) of T h e o r e m (2.1.9) and f r o m (1.1) (see [6]).
LITERATURE
1.
2.
3.
4.
5.
6.
7.
CITED
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Math., 176., No. 3, 126-140 (1936).
Honda, " F o r m a l groups and zeta functions," in the collection of t r a n s l a t i o n s : Matematika, 133, No. 6,
3-21 (1969).
M. L a z a r d , ' S u r les groupes de L i e f o r m e l s a u n p a r a m e t r e , " Bull. Soc. Math. France, 8.3, 251-274
(1955).
D. Mumford, L e c t u r e s on Curves on an Algebraic Surface, Princeton Univ. P r e s s (1966).
P. C a r t i e r , "Sur les groupes f o r m e l s a b e l i e n n e , " C o m p t e s rendus, 265, No. 1, 49-129 (1967).
S . P . Novikov, "Methods of algebraic topology f r o m the point of view of c o b o r d i s m theory," Izv. Akad.
Nauk SSSR, Ser. Matem., 31, No. 4, 855-951 (1967).
D. Quillen, "On the f o r m a l group laws," Bull. A m e r . Math. Soc., 75, No. 6, 1293-1298 (1969).
257