An introduction to maps between surgery obstruction groups

An Introduction to Maps Between Surgery Obstruction Grouos by Ian Hambleton*,

Laurence Taylor**,

and Bruce Williams**

In order to apply surgery theory we need methods

for

determining whether a given surgery obstruction (f : N ÷ M,f)

~ Ln(~G,e)

is trivial.

Notice. that it is

more important to have invariants which detect L-groups than to be able to compute the L-group themselves. One approach is to use numerical invariants Arf invariant3, [M1], maps.

[Da],

multisignatures,

[H-Mad],

[P].

such as

or the new "semi-invariants"

Another approach is to use transfer

For example, (i)

Dress

[D] has shown that when

group,

Ln(~G)

G

is a finite

is detected under the transfer

by using all subgroups of

G

which are hyper-

elementary. (ii)

Wall G

[W9] has shown that when

is finite, then image

is detected by

Ln(~G2,~),

2-Sylow subgroup of

*Supported by NSERC **Supported by NSF

G.

M

is closed and

(~ : [M,G/TOP] + Ln(~G,e)) where

G2

is the

5O

(iii)

If

G

is a f i n i t e

detected

under

subquotients nionc,

The (when

goal

G

surgery

theory.

finite

as w e l l

assume

Suppose G.

T h e n we

that

H

JR1],§2),

"push

and the

...

is an i n d e x

... ÷ L n ( m H , m ) (see

such

transfer

from

procedure maps

codimension

subgroup

that

1

of a

of s u b g r o u p s

Hi

,e - 1.

is an i n d e x Thus

we m i g h t

2 subgroup.

exact

fl L n ( ~ G , ~

[T-W]).

transfer

a sequence

2 subgroup

forward"

quater-

a systematic

is any

exists

by u s i n g

(see

computing

H

i = 0,

is an i n d e x

get t h e

if

is

dihedral,

cyclic

arising

c He = G

foz

H

maps

that

Hi+ 1

are

and

for

L~(~G)

and p r o j e c t i o n

which

then there

H = H 0 c H 1 c H 2 c ...

then

is to give

transfer

G,

of

G

2-group)

Recall

2-group

2 subgroup

of

paper

is a f i n i t e "twisted"

transfer

seml-dihedral,

o f this

and the

2-group,

of an a r b i t r a r y sequence

) ÷ L n ( f !) ~

exact

group

...

sequence

!

... ÷ L n ( ~ G , m ) (see

JR1],

One of We t.

...

§7.6).

can v i e w

R = ~H. let

f~ n n ( ~ H , ~ ) + Ln(f! ) +

More

a = t 2 e H,

S = ~G

as a t w i s t e d

precisely, and we

suppose let

p

we

quadratic choose

: R ÷ R

be

extension

t e G - H. conjugation

Then, S = R [v'~] P

where

t

tx = p(x)t

is v i e w e d for all

= R

[t]/(t

2 -

as an i n d e t e r m i n a t e x E R.

a),

P

over

R

such

that

by

51

Let

y

d e n o t e the G a l o i s

automorphism

of

S

over

R

g i v e n by y

W e w a n t to e x t e n d the and

[M-H]

(x,y

: S ÷ S; x + yt ÷ x - yt

appendix

classical

2 where

results

f : R + S

c R).

in [L] chap.

is a q u a d r a t i c

7

extension

of fields.

R e c a l l that W a l l ring with u

[W2] d e f i n e d g r o u p s

anti-structure, a2(r)

is a unit,

For example,

i.e.

~

= u r u -I

Ln(ZZG,m)

Ln(R,~,u)

for any

is an a n t i - a u t o m o r p h i s m ,

for all

r e R,

= Ln(~G,a~,l) ,

and

where

~(Z

~(u) ngg)

= u -I :

Z n g ~ ( g l g -I.

S u p p o s e w e h a v e a m a p of rings w i t h a n t i - s t r u c t u r e f : (R,~0,u) where

S

is a t w i s t e d

Galois

automorphism

y-conjugate

where

quatratic

y.

This y i e l d s

~ ~

(~,u)

Y ~ ( s ) / a -!

~0

to get

(~,u)

s c S

= (~,~),

and

~ = ¢-a y ~ ( / ~ - l ) u .

a map

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

If w e t w i s t and we get

with

(S,y~,u).

for all

: (R,~n,~) where

Rp[~]

T h e n we also h a v e the f o l l o w i n g

: (R,~0,u) +

we can " t w i s t "

~(s)

extension

map

Yf Moreover,

~ (S,~,u)

a map

(ya,u)

÷ of

(S,~,~), ~.

we get that

(y~,u)

= (y~, - ~),

52 N

¥f

:

(R,a 0,

-

~)

+

(S,¥a,

-

~).

Then we get the following amazing isomorphisms. P! : Ln_l(f !) T Ln(f !)

F., : Ln-i (Yr.') ~÷ Ln(Yfx )

r"' : Ln( f! ) ± Ln+l( ~! )

r "' : Ln(Y f' ) _ + L n+l(Yf ' ")

!

The maps

F!

and

F"

are defined using an algebraic

version of integration along the fibre for line bundles. the case of group rings the isomorphism

F!

[WI] chap. 12.C, [C-SI] and explicit in [H].

In

is implicit in The general case

is due to Ranicki (after some prodding by us). (See [RI], §7.6 and the appendix by Ranicki in [H-T-W]). !

By combining

F!

for

f,

F"

for

~f,

and

scaling

isomorphisms : Ln(S,~,u) ~ Ln(S,y~ ~ - ~] = Ln(S,y~,,u)

(see 2.5.5)

Ranicki constructed the following comuutative braid of exact sequences

5S

(0.i)

Twisting Diagram for

f : (R,~o,U) + (S,m,u)

Yf! Ln(S,a,a) = Ln(S,y~,u)

f! L (S,~,u) n = Ln(S,Y~,-~)

Ln(R,a0,u)

/\

(R,ao,a) Ln-2 = Ln(R,~0,-u)

/\

Ln+l(f ! )

/--,, Ln+l(S,a,u)

~f!

/

Ln+l (Yf")

\

Ln_l(R,50,u)

=

=

Ln+l(S,Y~,-~)

Ln+l(R,50,-~)

Thus the problem of computing

/

/

Ln(f ! )

L n I(S,~, ~) = Ln_l(S,y~,u)

f!

and

Yf!

Ln_I(R,~o,U)

is intimately

related to the problem of computing the "twisted" maps and

f!

~f!.

Examples: I.

Suppose

semi-simple rings.

p L n = L n'

n

is even, plus

R

and

S

are

Recall that

simple rings (see [R2]).

Lp is trivial for semiodd Thus, all of the groups along the

bottom of (0.I) are trivial and the groups along the top form the following exact octagon (see also [War] and [Le]).

54

(0.2)

Semi-simple

8-Fold Way

f, ~ Ln(S,~,u)

Yf! q J - a / L n ( R , ~ 0 , u ) Ln(S,~,~)

Ln(R,aO,-~)

Ln(R,~0,u)

Ln(S,a,-~)

y!"-'-" f W'a~

(a)

f, Ln(S,~,-u) ~i

then

yf!o/~ Ln(R,~0,-u)

Suppose we have f : (F,id,l)

where

~f.1 q ~

F ÷ K

+

(K,id,l)

is a quadratic extension of fields.

Ln(F,id,-l)

~ Ln(K,id,-l)

m (0);

If

n = 0,

and we get the

following exact sequence (0.3)

0 + L0(K,y

) + L0(F ) ÷ L0(K ) + L0(F ) ÷ L0(K,y

) + 0

which extends the exact sequences of Lam [L], chap. 7 and Milnor-Husemoller (b)

[M-H], appendix 2.

Suppose we have f

where

D

: (K,id,l)

is a quaternionic

maximal subfield of involution of type

D. O.

L0(D;S p) = L0(D,~,-I) ,

÷

(D,~,l)

division algebra and

Since

~IK = id, ~

If we let and

K

must be an

L0(D;O) = L0(D,~,I) ,

L0(K,p) = L0(K,~00~I) ,

get the following exact sequence

is a

then we

55

(0.4)

0 ÷ L0(D;S p) ÷ L0(K,p)

÷ L0(D;O)

÷ L0(K ) ÷ L0(D;O ) +

L0(K,P) + L0(D;S p) + 0 (c) K

Suppose

D

be a quadratic

a division

extension

ring.

we get another

is a division

(trivial

d

such that

(anti)

D@FK

involution

F.

Let

is still ~0

on

D,

example

extension)

Suppose we have

÷ (R x R, a 0 x ~0,u x u)

is the diagonal

and (0.2) breaks

÷ (D®FK,~0@Id,1)

quadratic

(d : (R,~0,u) where

F

Then for any

(D,~0,1)

(d)

of

ring with center

map.

into the short

Then

Ln(S,~,+u)

exact

sequences

is trivial of the form

d v

1 ÷ Ln(R,~0,u )

2. 12C,

; Ln(R,e0,u)

Codimension

[C-S1],

[C-$2],

x Ln(R,a0,u)

1 Sursery

(see

÷ Ln(R,a0,u)

[B-L],

[H], and [RI] Chap.

[Me],

+ 1

[Wl] Chap.

7)

Suppose we have f : (~H,am,l) where

H

is an index 2 subgroup

is a closed manifold a connected map

+ (~G,am,I)

with

submanifo!d

~ : G + G/H = {+i}.

of

G.

Also,

(~iX,el X) = (G,e).

such that

~I(~(Y + X))

Then by combining

suppose Let

Xn

yn-i

be

induces

the

results

of Wall

56 [wi],

chap.

12C,

Cappell-Shaneson

we get the following

commutative

[C-SI],

and Hambleton

[H],

diagram with exact rows

Ln(?ZH,~)

[ fI

a, ... ÷ S(X)

'+ Ln(?ZG,~)

, [X,G/TOP]

!a t

[Y,G/TOP]

p.*

Ln_I(ZZG,~)

' Ln_I(Z~G,~

Ln_l( ?Z'H,~oj, 1 )

where

t E G - H

Assume an element f

and

L

~i Y ~ Wl X in

S(X),

is homotopic

n

,1)

then

> Ln_l(f !)

n ~ 5. 8,(f)

fl

If

~ Ln_2(?Z.H,~m,~)

[C-SI],

f : M + X

is trivial

such that

are simple homotopy

Cappell-Shaneson

Ln(ZZH,a~,-I)

= Ls n

and

to a map

f[l(x - Y) ÷ X - Y

observed

Ln(f ! )

[C-$2]

represents

if and only if

f~l(y)

+ y

and

equivalences.

and Hambleton

[H] have

that since

image(a,

Yf!at(x) element

: [M,G/TOP]

÷ Ln(~G,~))

can b e v i e w e d as t h e p r i m a r y x ~ Ln(~G,m)

arising

c ker(Yf!a t)

obstruction

from surgery

t o an

on closed

manifolds.

In this

p a p e r we c o m p u t e t h e t w i s t i n g f : (?Z.H,a~,I)

÷

(ZZG,a~,I),

diagram

(O.1) where

57

G

is a finite 2-group,

and

Ln = Lp . n

Our motivation is

that we have used these results to compute L~(~G,m)

(see [H],

~

: [M,G/TOP] +

IT-W], and [H-T-W]).

Roughly speaking, we proceed as follows (i)

We show that the Twisting Diagram (0.1) for decomposes

into a sum of diagrams

irreducible @-representations (ii)

of

f

indexed by the G.

We use quadratic Morita theory to construct

an

isomorphism between each component diagram and the twisting diagrams associated to integral versions of Examples

1. (a),(b),(c),(d)

i.e.

maximal orders in division rings. (iii)

By using classical results on quadratic forms over division rings and localization sequences we are able to finish the calculation.

In Part I we carry out this program for the groups along the top and bottom of the twisting diagram (0.1).

In Part II

we compute the actual diagrams. This paper is a preliminary version of [H-T-W] where we give details, in addition to

compute the twisting diagrams L~,

for other L-groups

and give geometric applications.

We thank Tony Bak, Bill Dwyer, Ib Madsen, Bob Oliver,

Chuck Giffen, Karl Kronstein,

Andrew Ranicki,

Carl Riehm, and Richard

Swan for conversations which helped to clarify our thinking.

58

PART !:

Computation of the LP-grouDs

Let

G

be a finite 2-group and

H an index 2 subgroup.

In the o r i e n t e d case, the groups

[B-K],

have b e e n

computed by B a k - K o l s t e r

[KI],

Carlson-Milgram

These results are nicely s u m m a r i z e d

[C-M].

[K2],

LP(Z~G)

Pardon

[P], and

by T h e o r e m A in [H-M] where they give a d e c o m p o s i t i o n of LP(zzG)

indexed by the i r r e d u c i b l e ~ - r e p r e s e n t a t i o n of

G.

Besides e x t e n d i n g their computations to the u n o r i e n t e d L-groups, LnP(Z~G,m)

and the c o d i m e n s i o n i surgery groups

we also have to overcome the following problem.

LP(~G,~,I), All of the

above computations were b a s e d upon c h o o s i n g a m a x i m a l i n v o l u t i o n invariant order,

MG

w h i c h contains

Z~G.

Unfortunately,

it

is not always true that

M G n ~H

invariant order in

(Bruce Magurn has o b s e r v e d that this

~H.

can h a p p e n even w h e n

G

is a m a x i m a l i n v o l u t i o n

is the dihedral group of order 8).

Thus it is not clear that the above computations

and decom-

positions are functorial and we have had to m o d i f y t h e i r m e t h o d somewhat.

We have a t t e m p t e d to keep Part I fairly

self-contained,

but we w o u l d like to emphasize that Part I

is b a s e d upon the w o r k of the above authors and Wall's damental sequence of papers

[WI] - [W8].

we try to clarify certain questions

In Section

fun-

(2.5)

i n v o l v i n g quadratic

Morita theory.

§I.

(i.i)

Basic D e f i n i t i o n s

and O v e r v i e w

I n t e r m e d i a t e L-sroup The use of arithmetic

squares

forces us to use L-groups

5g

other than

Lp . n

relationships

Thus we shall start by r e c a l l i n g

b e t w e e n the various

A ring with antistructure R,

an a n t i - a u t o m o r p h i s m

that

a2(a)

--uau -I

L-groups.

(R,a,u)

~ : R ÷ R

for all

For any right R-module

M,

the

and a unit

a E R, D~(M)

is an a s s o c i a t i v e u E R

and such that

= HomR(M,R)

ring

such

~(u) -- u-1

is the right

R-module where (f • r)(m)

-- a(r)

• f(m)

where

Since inner automorphisms

act trivially

Ki(R)

and

induces

involutions

Ki(R))

which we also denote by

If then

Y

on

is an ~-invariant

L~(R,~,u)

f ~ DaM, m c M,

and

on K-theory,

r E R. D~

Ki(R) -- coker(Ki(Tz,)

÷

~.

subgroup

of

Ki(R),

denotes the standard L-group

i=0

or i,

defined in [Ca],

[R3] §9, and [RI] p. 688.

If

R = ~G

and

~

the following geometric

: Z n g + Z ng~(g)g -I, g L-~roups,

Ls(?ZG,~)

{w ab } cK 1 -- L n (Z~G,a~,I)

(see

[Wl])

L~(~G,~)

= Ln (~G,a~,l)

(see

[Sh])

LP(~G'~)n

= L~0(~G'~

(see [Ma],

If

Y2 c YI c Ki(R),

subgroups, sequence

'I)

i = 0 or i

then we get the following (see

[R3] 9.1)

then we get

[P-R])

are both ~-invariant Rothenberg

exact

Y2 Y1 ... ÷ L n (R,~,u) ÷ L n (R,~,u) ÷ H~(YI/Y 2) ÷ ...

(i.i.I)

where

H~(YI/Y 2)

is the Tate cohomology group

associated to the action of

~/2

on

YI/Y2

Hn(~/2,YI/Y 2) via

m.

Also,

KI(R) (1.1.2)

Ln

If

Y = K0'

Oc~0(R)

(R,~,u) = L n

(R,~,u)

then

(1.1.3)

L~(~ 1 × R~,~ 1 x ~2,Ul × u 2) : L~(al,~l,Ul>

× L~(R2,~2,u2), '

and (1.1.4)

L~(RI,al,Ul) : L~(R2,a2,U 2) and

(R2,a2,u2)

whenever

(Rl,~l,Ul)

are quadratic Morita equivalent

(see Section 2 for definition)

Since

KI(RI × R2) # Ki(RI) × Ki(R2)

and since

Ki

not a Morita invarlant,

(1.3) and (1.4) are false for most

Y : e.g.

This problem is overcome by intro-

Y = 0 c K0

ducing the following variant L-groups. If

X

is an ~-invariant subgroup of

get L-groups, for any If

(1.1.5)

L~(R,a,u).

(See [W3] for

i ~ 0). ImageK0(ZZ) c X c K0(R),

then

ff~(R,~,u) = L~(R,~,u),

Ki(R) , i = 1

then we and [B-W]

is

61

where

X = image

If

of

X

in

K0 (R)"

imageKl(?Z ) c X ~ KI(R) ,

then we get an exact

sequence (1.1.6) where

... ÷ LXn(R,~,u) ÷ LnX(R ~,u) + Hn(imageK0(2Z)) + X = image

of

X

in

.

KI(R).

Again we get Rothenberg sequences as in (I.i.i), and Ki(R ) O=Ki_I(R) Ln (R,~,u) = L n (R,a,u).

(1.1.7)

Furthermore,

(1.1.8)

~

Y2

YI×Y2 Ln (R I x R2,a I x ~2,Ul x u 2) m L I(R!,~I,U I) x L n (R2,~2,u 2) and (1.1.9)

L~(RI'aI'Ul) ~ L¢(Y)(R2'm2'U2)n and

(R2,~2,u 2)

whenever

(R l,al,ul)

are quadratic Morita equivalent.

(@ : Ki(R I) ~ Ki(R 2)

is the isomorphism induced by

the Morita equivalence) (I.i. I0)

Convention:

L l(R,a,u) ~ L

OcK0

Henceforth

will denote

(R,~,u).

Our first goal is to compute finite 2-group and

Ln(R,m,u)

(~,u)

L~(~G,~,u)

for

any anti-structure.

First consider the following long exact sequence

G

any

62

(i.l.ll)

....

~,~(~,~,~,)

(1.2)

~ ~(~2~,~,,u)

Computation of

(1.2.1)

Theorem:

structure

(a,u)

~ ~ ( ~ . ~ ~. ~2~,:,,:,)....

LnP(~2G, m,U)

For any finite 2-group and any antion

~ 2 G,

we get ~/2

p

Ln(ZZ2G,e,u)

~+ LnP(ZZ/2,id,l)

=

0

if

n e 0(2)

if

n e 1(2)

Theorem 1.2.1 follows from the following two results. (1.2.2)

Reduction Theorem:

If

R

is a complete local ring

then for any 2-sided ideal I, (i)

K0(R) ~ K0(R/I) ,

(ii)

Proof: (1.2.3)

LP(R'~'U)n +- L~(R/I,~,u),

(assuming

~(I) = I)

See [W5], [B]. Lemma:

ker (~/p)G Proof:

and

+ ~/p

If

G

is a finite p-group, then

is nilpotent.

See [SE], p. 57.

Notice (1.2.3) implies that kernel

(~pG

+ (~/p)G

÷ ~/p)

is complete.

(1.3)

Computation..of

LP(ZZGn . + ~2G~a~u).

It is well known that QG = ~A¢,

(see [Sl] or [Y])

where the product is taken over the set of isomorphism classes of irreducible Q-representatlons.

Each

A¢

is a simple ring,

83 and

A¢ = a¢~G

for some central idempotent

be expresses as a sum of nontrivial Since or

~

where

~(¢)

a¢ ~ ~ [ ½ ] G

~[½]-maximal

is another

a(m¢)

is either

irreducible

(.see [Y], p. 4) and

order in

A@

which can not

central idempotents.

is an inner automorphism

a (¢)

In fact a

2

a¢

aS

~-representation.

A¢ = a¢(~[½]G)

(.see [Re], p. 379).

is

Resi~Ic --

tion gives a decomposition of rings with antl-structure. (1.3.1)

(~[%]G,a,u)

¢=~(¢)~(A¢,~¢,u¢) The A¢ x Ae(¢)

=

x ¢=~C¢)~ (~¢ xA (¢),~¢x~C¢),u ~ x u (¢)) . are called type GL factors and make no

contribution to any Wall group.

We prove the following result in Section 2. (.1.3.2)

Decomoosition Theorem:

and any anti-structure

on

~G ,

For any finite 2-group

G

we get the following

canonical isomorphlsms. L~(~G

÷ ~2G,m,u)

T Ln(~[½]G

÷

~2G,m,u) ^

-9"

¢

Ln(A ~ + A¢(2),m~,u¢)

¢=~(¢) (Recall that

Ln

denotes

Consider the following a primitive

K1 OcK0) Ln t L n ~[½]-algebras,

where

~j

is

2J-th root of 1 and - denates complex conjugation.

64

(1.3.3) i)

rN

2)

RN = ~ [ } ] [ ~ N + 2

+ ~N+2 ]

3)

FN = ~[½][~N+2

- ~N+2 ]

4)

HN = ( ~ )

~[½][~N+I ]

=

( ~ )

® RN_2,

= {~

+

~i

Remark:

Each of these

of rank

2N .

where

+ ~j

+ ~k

I ij = -ji = k, i 2= j2 = -I}

~ [ ~i] - a l g e b r

a

s

is a free

~[½]-module

Now, consider the following anti-structures.

(1.3.4) l)

(!d,l), (-,I), (T,I), (F,I)

On

rN,

Id

is the identity;

T(~N+I) = - ~N+I field of

T; ~

-

or, equivalently, has fixed field

2)

On

R N, (id,l),

3)

On

F N, (Id,l), (-,I)

4)

On

HN,

(~i,I),

is complex conjugation; FN_ 1

is the fixed

FN_ 1 .

(T,I)

(4,1)

where

~l(i) = ~(i) = i ~l(j) = ~(j) = j

^

and

alIRN_2 -- !d, ~IRN_ 2 = T.

In Section 2 we also prove the following result.

65

(1.3.5) and

Identification Theorem:

(~,u)

If

G

is any anti-structure on

irreducible ~-representation

~

is a finite 2-group,

ZZG ;

with

then for any

m(¢) = ¢,

^

we get that

^

Ln(A ~ + A¢(2),~¢,u ¢) ~ Ln(A ¢ + A¢(2),B¢,v ¢)

where

A¢ = a¢2Z[I]G

and

(A¢,B¢,+_v#)

with anti-structure in list (1.3.4).

is one of the rings Recall that

Ln(A¢,B~,v ¢) ~ Ln+2(A¢,~,-v¢).

In Section 3 we compute Ln(A ÷ A(2),~,v) rings with anti-structure in List (1.3.4)

for all of the

and tabulate the

results in Table I. Theoretically we could then calculate L P ( ~ G , a , u ) but we n restrict the antl-structure slightly at the start of Appendix I in order to easily identify (A¢,S¢,v¢) on List (1.3.4) Appendix I, part I).

(see

In part 2 we settle the remaining questions

involved in using l.l.ll.

§2.

Proofs of the Decomposition Theorem (1.3.2) and of the Identification Theorem (1.3.5)

(2.1)

Excision in Arithmetlc Squares Suppose

Then

S-IA

= lim A/sA s~S

S

is a multiplicative subset of a ring

is the localization of

R

away from

is the S-adic completion of

A-

1

A,

~

l

S-IA ÷ S-I~ is the arithmetic square associated to

(A,S).

and

S,

A.

66

(2.1.2)

K-theory Excision Theorem:

For any integer

i,

Ki(A + A) ~ Ki(S-IA + S-I~)

(2.1.3)

Corollary:

integer

i, Ki(~G

For any finite 2-group

G

and any

÷ ~2 G) ~ K i ( ~ [ ½ ] G + ~ 2 G) ~

Ei(A ¢ ~ A¢(2))

¢

irred. Q-rep. (same notation as in (1.3)) (2.1.4)

L-theory Excision Theorgm:

(See [RI].

Also [B],

[B-W], [C-M], [p], and [W7].) We assume that structure

(a,u)

A

in (2.1.1) is equipped with an anti-

such that

SIS

is the identity.

Localization

and completion then induce anti-structures on the other rings in (2.1.1).

Let

X c Ki(S-IA)

and

Y c K.(A)

be ~-invariant

1

subgroups. and let

Let

C = kernel of Ki(A) ÷ Ki(S-IA)/X @ Ki(~)/y '

I = image of X @ Y ÷ Ki(S-IA) @ Ki(~ ) + Ki(S-I~).

Then, LC+Y(A + A) ÷ LX+I(s-IA ÷ S-I~) n

(2.1.5) structure

n

Corollary: (a,u)

CI(G) Ln (~G

on

"

For any finite 2-group ~G~

+ ~2G,~,u)

letting

X

and

Ki+l 1 ~ Ln (~[~]G

¢

a(¢)=¢

G

and any anti-

Y

be trivial we get

÷ ~2G,~,u)

LKi+I(A + ^ n ¢ A¢(2)'~¢'u¢)

67

where

CI(G) =

ker K I ( ~ G )

rest of the notation

÷ Ki(m[½]G)

@ Ki($2G )

and the

is the same as in (1.3).

c.(G) If

i = l,

then the

which were computed by Wall Notice that

(2.2)

then

LC0 (G) n (~G

^ + ~ 2 G)

(2.1.5) would imply the D e c o m p o s i t i o n

(1.3.2).

Representation

Definition: noncyclic, (2.2.1)

are the Ln-groups

[W8].

if we can show that

LPn(ZZG ÷ ~ 2 G ) , Theorem

!

regroups

Lnl

Theory

for Finite 2-~roups

A finite 2-group normal

w

is special

if it has no

abelian subgroups.

Proposition:

A group

n

is special

if and only if

it is one of the following groups (5)

(ii)

cyclic,

C N =

dihedral,

D N = , N > 3 2 N-I

(iii)

semi-dihedral,

x

2 N-2 1 - >, N >

SD N = 3. Each special group representation

w

has a unique

¢(~).

For any irreducible p : G ÷ GL(Vp),

faithful, irreducible,

we let

@-representation

of a group

G

~-

=

68 Dp = End@G(Vp). Schur's lemma implies that

(2.2.2)

Theorem:

N

(i)

G,

H

¢

on

with normal

is special

up to

Proof:

there exists a subgroup

If we pull

(iii)

is a division ring.

such that H/N

(ii)

P

For any irreducible @-representation

a finite 2-group subgroup

D

@(H/N)

G,

De m D$,

we get where

back to

H

and then induce

¢. @ = @(H/N).

(See [F])

(2.2.3)

Table D@(~)

IT

CN

FN_ I @ Q

DN

RN_ 3 ® Q

SD N

FN_ 3 ® Q

QN

HN_ 1 ® Q

Thus the rings from list (1.3.3) are in the division rings

D@(w)

~[½]-maximal

orders

Notice that the centers of

these division rings are precisely the fields which are subfields of

~(~j)

for some

Q(~i )' ~(~i + ~i )' primitive (2.2.4) for some

and

j,

namely fields of the form

~(~i - ~i )"

(Recall

~j

is a

2J-th root of 1.) Weber's Theorem: j.

Let

0

Suppose

K

is a subfield of

@(~j)

be the ring of algebraic integers in

69

K,

and let (i)

R = 0[I]. K/Q

Then

is unraraified over all odd primes.

it is totally prime (ii)

d

ramified,

and the unique

Over

2,

dyadic

is principal.

The ideal class

group

F(K) ~_ K0(0)

= K0(R)

has

odd order (iii)

The narrow

ideal

class

group

(x))

F (K) ~ (group of ideals) --/principal ideals ~ s u c h that x > 0 \ f o r all real places also has odd order Proof: theory

For (ii), implies

that

have odd order; is unramified

see Theorem if

then

K = @(~i + [i )' K

(2.2.5)

K = ~(~i )

Corollary:

implies

For any

K0(H N)

have odd order.

Proof:

Notice

where

if

i.

Since

HN

H N ® ~, (36.3)

for

or

does not E/K

which

E®K@(~ i)

of

Q(~i ).

K0(~),

FN ,

K0(FN),

then

in a subfield

is a maximal

field

Thus

K = ~(~i + ~i )"

N, K0(FN)~

rings

then

extension

(iii)

F*(K)

entension

But,

R = FN, RN,

0 = ring of algebraic

for some algebra

that

quadratic

Class

and

has a quadratic

at all finite primes.

would be an unramified, (il) for

10.4 in [Was].

of

and

R = 0[ 1 ] ~(~i)

order in the division

in [Re] implies

that

K0(HN) : r~(~(~ N + IN)).

70

(2.3)

(Linear)

- Morita

Theory:

(see

[Bass

I] and

[Re]

rings

A

for

details).

Definition: B

A Morita

is a 4 - t u p l e

BMA

and

bimodule

equivalence

(M,N,u,r)

ANB;

u

isomorphisms T(n

between

where

M

: M@AN ÷ B such

® m)

and

and

T

two N

and

are b i m o d u l e s

: N®BM + A

are

that

• n' = n

• u ( m @ n'),

• m'

• r(n

and u ( m ~ n)

= m

@ m')

for all all

F o r any r i n g

A,

we

let

finitely

generated

projective

(2.3.1)

Theorem:

Assume

between

A

and

B.

PA

we get

the

e N,

and

e M.

category

of

R-modules.

(M,N,u,T)

Then,

m, m'

denote

right

n, n'

is a M o r i t a

an e q u i v a l e n c e

equivalence of c a t e g o r i e s

@A N PA ~" ~ PB

®BM and an i s o m o r p h i s m

K.(A) ~

~.(B).

l

Furthermore, of

1

center(A)

= B - A - bimodule

endomorphisms

M = center(B).

Examples

(2.3.2)

Derived

Suppose

of

Mn

M

for some

Morita

c 0b

equivalence

(F A)

n > 0,

and

i.e.

A

M

is a d i r e c t

summand

is a p r o ~ e n e r a t o r .

Then

71 A

and

B = EndA(M)

HomA(M,A),u,~)

where

the evaluation If

of the simple

G,

then we let

(V ¢)

A¢c

R

if there

M

such that

If

A

@G.

R-algebra,

fractions

Then, whenever

K.

R

is also a R-maximal

is Azumaya

prime

ideals Suppose

of a finite Azumaya

Definition: of Morita

representation

of

where

Furthermore,

A

A

and a progenerator

A

is central

[K-O].)

i.e.

domain with field of

is an Azumaya A®RK.

(See

R-algebra,

Conversely,

K-algebra

A

with

Ap ~ Mn(K p)

if

A

center

is R,

for all finite

[Rog].)

¢ : G ÷ GL(V¢) group of order

and the division

as R-algebras.

then

order in

(See

A¢

then a R-algebra

B

if and only if

R.

Re-algebra

i of Prop.

becomes

in

ring,

is a Dedekind

order in a simple

A

Thus

A@RB ~ EndR(M)

Assume

then

is

denote the simple module

is a R-algebra

is an Azumaya

a R-maximal

and

Z ;Ki(D¢)-

center A = R.

A

T

equivalent.

is a commutative

is Azumaya PR

V¢

are Morita

Ki(~G)

of

(M,N =

• m' = m • n(m')

is a @-irreducible

component

D# = EndA

If

via

map.

group

(.2.3.3)

equivalent

u(m @ n)

¢ : G + GLn(Q)

a finite

ring

are Morita

is a irreducible m.

Then

~-representation

A~ = a¢ • ~ [ ~ ] G

Re = center(A¢).

(See

is an

[F], Corollaire

8.1.) For any commutative equivalence

an abelian

classes

ring

R,

Br(R)

is the set

of Azumaya R-algebras.

group under tensor product

over

It R.

72 Suppose

R

is a Dedekind

a finite

extension

of

(.2.3.4)

Theorem:

Let

orders

Q

or

~p

equivalent

for some prime

Aj c Aj

in simple K-algebras

are Morita

domain with quotient

for

j = 1,2

Aj, j = 1,2.

if and only if

K

p.

be R-maximal

Then

AI

field

and

AI A2

and

A2

are Morita

equivalent. Proof:

See [Re], Theorem

(2.3.5)

Corollary:

(2.3.6)

Theorem:

Then,

(21.6).

The map Suppose

Br(R)

+ Br(K)

is a momomorphism.

that G is a finite

for any i, Ki( ~ G ÷ ~ 2 G ) ÷ ~ Ki(A ¢ + ~¢(2)

¢

where

¢

runs over the irreducible

G and A¢

First

(2.3.4)

after consulting

and Table

apply

corollary

paragraph

Theorem

K0(~G)

is a finite group.

(Swan):

(2.3.6), becomes

Rothenberg

representations

of

The result

two of (2.3.2);

follows (2.2.2)

from (iil);

2.2.3.

(2.4.1)

Then

)

(1.3.3).

(2.1.3).

Proof of the Decomposition

K0(~G)

rational

is one of the rings on list

Proof:

(2.4)

2-group.

If

(2.2.5),

G

argument

(1.3.2)

is a finite

and (1.2.2

an isomorphism

sequence

Theorem

group,

then

(i)) imply that

when we localize

then implies

that

at 2.

C0(~G) A

÷

73

c0(~G) Ln Thus

(1.3.2)

(2.5)

(~G

~ ~ 2 G) ÷

is a special

Quadratic

Definition:

(A,m,u)

(M,N,~,T)

where we make

N

~2G)

(Compare with

A quadratic. Morita

equivalence

+

equivalence

and

B-A

(B,~,v)

bimodule

[F-Mc]

and [F-W])

between

two rings

is given by a Morita

plus a B-A-bimodule

into a

.

case of (2.1.5).

Morita Theory

with antl-structure

L~(~G

isomorphism

using

a

h

and

S.

: M ÷ N We

also require that mT(h(mlU)

® m 2) = T(h(m 2) @ mlv) for all

(2.5.1) and

Theorem:

(B,B,v)

A quadratic

induces

Morita

equivalence

ml,m 2 ~ M

between

(A,e,u)

isomorphisms

: Ki(A) Z Ki(B),

(equivariant with respect to ~, and B,)

H~(Ki(A))

@ H~(Ki(B)),

L~(A,a,u)

+ L¢(X)(B,B,v) n

and where

X

is any

~,-invariant subgroup

(2.5.2)

Derive~

(A,~,u)

is a ring with anti-structure

(,linear)

Morita

center B.

e

Mor~ta Equivalence

equivalence

Assume

where we use (i)

Quadratic

h

: M ÷ N

to make

B

admits

h

becomes

B

to make

between

N

a unique

A

(M,N,u,T)

and

is a right

B.

Let

A-module

into a right

isomorphism

a left S-module.

B

Ki(A).

Suppose is a R = center A =

isomorphism,

A-module.

anti-automorphism

a B-A-bimodule N

and

Theorem:

of

Then, such that

when we use

(BIR = siR);

and

74

(ii)

there exists (M,N,~,~,h) between

(2.5.3)

is a quadratic

(A,a,u)

Corollary:

structure where

a unique unit

Suppose

A

(D,B,v) Proof:

Then

A-modules,

(.2.5.4) ~G

(A,a,u)

V

w

is a ring with anti-

D = the division

and

algebra

(~,u)

G. ~

to

are both simple right

a right A-module

Suppose

Let

(B,v).

V a = HomA(V,A)

there exists

Corollary:

K.

is quadratic Morita equivalent

Ln(~G,~,u)

and

(A,~,u)

and let

for some finite group

(2.5.5)

Morita equivalence

(B,B,v).

for some anti-structure Since

such that

is a simple algebra over a field

V = simple right A-module EndA(V).

and

v E B

isomorphism

h

is an anti-structure

: V ÷ V ~.

on

Then, ~¢

Ln(D¢,~,v~).

Definition:

If

(R,a,u)

is a ring with anti-structure

is a unit in

R,

then the scaling of

(a,u)

by

w

is the new anti-structure (a,u) w =

where

B(r) = wa(r)w -I For any R-module

for all M,

(~,v)

r c R,

there exists

DaM + DBM;

f ÷ (fw

and

v = w~(w-l)u.

an isomorphism

: x ÷ wf(x)).

Thus we get an isomorphism ow

Alternatively,

: Ln(R,m,u) ÷ Ln(R,(m,u)W )

~w

can be gotten by applying

(2.5.2) with

75 MR = R

(2.5.6)

and

h : R + HomR(R,R)

Definition:

= R

Suppose

sends

is a commutative

involution

~0"

equivalence

classes of rings with anti-structure

A

Then

R

the map that

Br(R~0~

r

air = s 0

rw -I.

ring with

is the set of quadratic

is a Azumaya R-algebra and

to

Morita

(A,e,u),

Br(R,~ 0)

where

is an abelian

group under tensor product.

Warning:

We shall see in (2.5.9)

that the quadratic

analogue

of (.2.3.4) is not true in general. Let

Br0(R,~ 0)

be the kernel of the forgetful map

Br(R,~ 0) ÷ Br(R).

Ass~ne that K.

Let

R

I = the group of R-fractional

g : K* ÷ I

be the map that sends

By sending elements the map

Warnins:

is a Dedekind domain with quotient

~0

: K ÷ K

The map

and fractional

ideals

x E K*

to the u n d e r l y i n g module

[~]

K,

and let

to the ideal

ideals to their

we get an action of

I + Ko(R)

in

field

~/2

images

on

K*

which sends a fractional is not equivariant.

(x). under

and

I.

ideal

Indeed,

s0 [~0 (~-I)] m [~]

(2.5.7)

Theorem:

= HomR(~,R)

There exists

made

into a right R-module

via

~0"

an i s o m o r p h i s m

: Br0(R,a 0) + H0(ZZ/2,K* ÷ I) ~f

{(x,--)

E K* ~ I i~0(x)

{(Y~0(y-I),

-- x - l , ( x ) a

(Y)8~0(S) I (Y,8)

=

~0(~)}

~ K* @ I}

76

The map represents

Y

is defined as follows.

an element

A = EndR(i,1). Let

in

Br0(R,a0).

V = M®RK

and

Suppose

Choose

A = A@RK.

can choose a right A-module i s o m o r p h i s m yields an a n t i - s t r u c t u r e K : EndA(V)

and

adjoint of

h

B = ~0

Let

(v,a)

where

so that

as in (2.5.3) we : V + V~

EndA(V).

ad(h)

a

M

Notice that

: V x V + K

Then

which

be the

Y(A,~,u)

is

is the f r a c t i o n a l ideal

h(M x M).

The map

~

has the following interpretation.

is choosen so that derived from

M

a c R.

Assume

h

Then the (linear) M o r i t a e q u i v a l e n c e

and the pairing

e q u i v a l e n c e of categories nonsingular

on

: V + V ~ = HomK(V,K).

r e p r e s e n t e d by g e n e r a t e d by

(B,v)

h

(A,~,u)

h

: M x M ÷ a

determines

Sesq(A,~,u) ÷ Sesq(R,a0,v).

an

But,

forms are sent to a-valued m o d u l a r forms.

The following result was suggested to us by Karoubl.

(2.5.8)

Proposition:

Any ring w i t h a n t i - s t r u c t u r e

is quadratic M o r l t a equivalent to

8

Proof: and

Let

h : M + N apply

= ~u_la(c)

be the standard basis for

be the dual basis for

be given by

(2.5.2)

where

I )r

{el,e 2}

{e~,e~}

(M2(A),8,1)

(A,~,u)

N = M*.

h(e I) = ue~

and

M = A @ A

Then we let

h(e 2) = e~

and we

to the derived M o r l t a equivalence.

This implies that

Br0(R,m0)

in the sense of F r o h l l c h - W a l l

is isomorphic to

IF-W].

Thus

(2.5.7)

B0(R,~/2) is at least

77

implicit

in [F-W].

(2.5.9)

Samole

(i)

Calculations:

R = K If

charK + 2,

which

then

are represented by

Br0(K,~ 0) = (I) (ii)

Br0(K,id)

when

R = finite extension

(K,a0,1)

s 0 # id of

s 0 # id,

we let

Br0(R,a 0) = (1), Br0(R,a 0)

(ill)

a ring of algebraic in

0,

charK = 2.

~ {+i}

R

is inert over R

is

Br(R,a 0) ÷ Br(K,~ 0)

and (iv) we let and

Z

r,

The isomorphism

and

r.

is not an injection.

R = 0(Z )

where

0

is

is a set of prime ideals (1.3.3).

Br0(R, id) = 2 R* @ F/F 2

F = coker(K* + 7),

Then,

and

ramified over

e.g. the center of the rings in List

(ill) where

when

integers

or

(K,a0,-l).

be the fixed subring of a 0 .

has order 2 when

Notice that

In cases

r

and

P

Br0(R, id) ~ Bro(K,id)

If

has two elements

2 R* = [x E R* I x 2 = 1}.

comes from the following braid

78 which is induced by the following exact sequence of

~

~/2-

modules i ÷ R* + K* ÷ [* + F + i

Remark:

The localization

sequence

(see JR1], §4.2) implies

that

L~(R,.id,1)-_ Similarly,

if

coker(LP(K,id,l)÷

(A,~,u)

a ~ F/F 2 c Br0(R,id ) coker(L~(A,id,l)

L0P(K,id,1)

+

represents and

@ P

and where

@

P

a

A = A@RK ,

Case !:

K

LP(Ap +

L~(R,id,1)

is nontrivial and

~1 e R,

= L~(A,~,U). then

: 2 × order L~(A,a,u).

is unramified over the fixed field of

Furthermore, Br0(R,~0).

0.

Then

Otherwise,

s0

and

Br0(R,~ 0)

Bro(R,~ 0) + Br0(K,e 0) @ wBr0(Rp,~ 0) O if

(A,~,u)

represents

Then

LP(R,~0,1) # L~(A,~,u). Case 2:

L~(A,id,l)

Assume s 0 ~ id.

order 2, but the map

element in

L~(A,~,u)

Ap,id,1) + LR(Rp + Kp,id,1).

Z = the set of all prime ideals in

trivial.

then

L~(Ap + A0,id,l)) , where

order L~(R,id,l) (iv)

Kp,id,1))

an element

But it is not true in general that For example if

L0P(Rp +

has is

the nontrivial

79

Br0(R,a 0) T $ Br0(Rp,~0), where we can sum all finite primes ramified over the fixed field of

p

in

Z

which are

~0 "

These results are proven by using the isomorphism H0(~/2; K* ÷ I) Z fi0(~/2;

where Remark:

If

book [C].

g(K)

# p~

R* x P pES

P

is the idele class group of

R = 0,

v arch

v

K.

then case (iv) is related to Connor's

In fact, HO(~/2; K* ÷ I) ~ Gen(K/K ~0 )

(see chap. I in [C]), and if L~(A,a,u) z Hx(~), symmetric,

(2.6)

where

~(A,a,u) = [(x,=)],

Hx(=)

then

is the Witt group of

x-

=-modular forms studled in chap. IV of [C].

Proof of the Id@ntif%cation Theorem (1.3:5) Theorem (1.3.5) will follow from (2.5.1)

relative version)

(or rather its

if we can prove that (A¢,a¢,u¢)

Morita equivalent to (A¢,8¢,±I)

where (A¢,8¢,I)

is quadratic is one of the

rings with anti-structure in List (1.3.4) From the proof of Theorem (2.3.6),

we know that A¢ is

linearly Morita equivalent to F N, F N, R N, or H N for some N. Let R denote the center of A¢.

Then (A¢,a¢,u¢) a Br(R,a 0) for

some m0" The proof divides into three cases. I)

De is commutative, m0 = Id. Then (A¢,a¢,u¢) e Bro(R,Id).

(2.2.4) (ll), Br0(R,Id) = ~ / 2 ~

From (2.5.9) (iii) and

and from (2.5.9) (i) we see

80 that

(R,Id,l) 2)

and (R,Id,-l)

are the two elements.

De is non-commutative, The calculation

are the two distinct Br(R).

From

in I) shows that

elements

(HN,~I,I)

in Br(R,Id)

which map to [H N] e

so done.

~0 ~ Id. First notice that R with each n o n - t r l v i a l

occurs on List

(1.3.4).

From

we see Br0(R,~ 0) = (i). Remark:

(2.2.4)

Using

A

Henceforth,

on

A.

then,

is to compute

Ln(A ÷ A2,~,I)

(1.3.3)

Recall that

Ln

we shall suppress writing the i in

The results

for any s0,

Sequence

is any of the rings from List

any (anti)-involution

case 2,

is one to one.

Localization

The goal of this section

(iv)

we are finished.

Notice that if R = FN, FN, or RN,

§3.

involution

(i) and (2.5.9)

(2.3.6)

Br(R,~ 0) + Br(K,~ 0)

where

and (HN,al,-I)

(2.3.6) we know that A¢ is linearly Morita equiv-

alent to HN, 3)

a0 = Id.

are summarized

in

Table

and denotes

B

is OcK 0 Ln

(8,1).

I.

(3.1) General Backsround ' Suppose

K

is an algebraic

integers

0.

algebra;

A = R-maximal

involution

of

Let

A.

R = 0[½];

number

D : central,

order in

We assume

Consider the following

field with ring of algebraic

D;

K0(A)

and

simple, B

any (anti)-

has odd order.

arithmetic

square

K-division

81

A÷D + ¢ ~, ~. ~

.

Then, (3.1.1) (by L-theory

Ln(A ÷ D,B) ~ Ln(A ÷ D,8)

Excision

Theorem

(2.1.4))

^

= @ Ln(A

P

÷ D ,8) P

(by 4.1.2 and 4.1.5

where we sum over all maximal such that

(3.1.2) maximal

R

If

If

in

R

p

is a

and such that

B(p) = P

we assume

maps to the trivial K

p

k.

81R = id,

where

Suppose

Morita Theorem:

such that

for some

ideals

8(p) = p.

Local Quadratic ideal in

in [RI])

that

element

in

is the algebraic

81R $ id,

(A,8,1) Br(K,id)

closure

we assume that

over the fixed field for

81K.

~ Br(R,id)

p

of

- {+I},

K.

is unramified

I

Then, Ln(Ap -~ D p ,B) ÷~ Ln(R p ÷ K P ,B)

Proof: (3.1.3)

Apply

(2.5.1)

Divissage

R

such that

p

is unramified

i

Rp,

and (2.5.9)

Theorem:

B(p) = p.

we get

If

(ii).

Suppose

p

is a maximal

81R # id, we also assume

over the fixed field for

81K

.

ideal in that

Then,

since

82

Ln(Rp ÷ K p' 8) ÷~ LP(k^,B), n where

k

Proof:

is the residue field

P

R/O.

See 4.2.1 in [RI].

If

(A,~)

satisfies the assumptions

in (3.1.2) and (3.1.3),

then we get the following localization diagram with exact rows and columns

(3.1.4) •

•

•

LL+I(A÷A 2, ~ )-~Li(D, B )÷ni( A2' ~ )@@LP( kp, B )-~Li(A÷A2, B )....

Li(a,B)

, Li(D,B)

~ @LP(kp,B)-~Li_l(A,B)

0 -map L i IA2, B)

Since

ideals in

i

^

~ ~ A 2,

@ L~(kp,B) R

^

we get that D2

A2 ~ D2 "

(where

0

contains Recall

we are summing over the set of maximal

such that

N.D.

If

is a simple ring.

~(p) = p.

Notice that this is the

same as summing over the ~-invarlant, 0

1

Li_I(A2,S)--Li_I(A2,B)

a unique dyadic prime, then that in

+...

N.D. maximal ideals in

stands for nondyadic).

We shall compute

L,(A + A2,~)

Notice that the domain and range of of L-groups of semi-simple rings.

by computing the map $

@.

is expressed in terms

83

(3.1.5) Semi-simole Theorem: then

LPi+I(A,S)~ = 0

Proof:

If

A

is a semi-simple

for any involution

ring,

S.

See [R2]. A

(3.1.6)

Reduction Theorem:

For any abelian group

(3.2)

Type O-Commutative

We assume

8 = id

Then for any field

Li(Rp, S) = Li(kp,8) ,

G,

where

kp = Rp/p.

2 G = {g ~ Glg 2 = i}.

Case:

(FN,id) , (RN,id) , (FN,id),

which we suppress writing.

K

with

charK ~ 2,

L~(K) = W(K),

the classical Witt ring of symmetric bilinear pairings over K in

(see ILl, [M-H], W(K)

[O'M], and [W4], p. 135).

comes from the tensor product of pairings.

I(K) = kernel r : W(K) ÷ ~ / 2

The group

L~(K) ~ (I)

singular pairing hyperbolic

b

where

is the rank map.

i.e.

b

sequence plus

k

If

for

n ~ 3,2,1,0(4)

is a finite field with

Kp/Qp

with

is

(3.1.5) then imply that

Examples If

non-

3.5).

disc: l(k) T k*/k .2 (ii)

Let

because any skew-symmetric

Ln(K) ~ O,0,2K*,I(K)

(3.2.1)

r

has a symplectic basis,

(see [M-H],

The Rothenberg

(i)

Multiplication

chark ~ 2,

has order 2.

[Kp,~p] = £,

then

then

84

disc : Z(R)/12(Rp) ~ R~/ff~2; Hasse-Witt:

I2(Kp) ~ 2Br(K2 ) ~ {+l}.

then the map

^ ^ L0(K p) ÷ L0(.Rp ÷ Kp) ~ L (kp),

li(K ) P

li-l(k ) P

onto

Ii(Kp) ÷ k*p/k*2p invariant.

^, ^,2 v

÷ Kp/Kp

÷ ~/2

For any

p,

Thus

+

1

Itr

I(Kp)/I2(Kp)

P

sends

(.see [M-H], IV, 1.4).

/tdi'c

0

N.D.,

We also get the following exact sequence

p/ P

where

is

can he identified wlth the Hasse-Witt

1 + ~* 8,2

where

p

If

+ L0(kp)/I(kp) ,

is the integral closure of

0*p : U(Kp) x ~Zp

~

P

in

P

(see [$2], XIV, §4),

p(.Kp) = roots of unity, Thus ~L2

if

p

is

N.D.

^, ^,2 0p/0p = ~/2 ~+I (lii) (iv)

I(@) -__ (0) If

K/@

and

with

if

p

sig : I(]R) ÷ 2 ~ . [K,Q] = r I + 2r 2

the number of embeddings disc

is dyadic

of

K

where into

rI

is

~,

then

: I(K)/12(K) ~ K*/K .2,

Hasse-Witt:

I2(K)/13(K) Z 2Br(K),

and

slg : I3(K) $ @ i3(Rv) ± (8~) rl , where v varies over the real embeddings of K.

v

85

Suppose algebraic

K is a field on 2.2.3

integers

in

K,

and

, 0

is the ring of

R : 0[½].

Since

0

has a

^

unique prime over 2, K 2 sequence

(3.1.4)

are trivial.

is a field and the Localization

implies that

L3(R + R2 )

We also get the following

and

L2(R ÷ R2 )

commutative

diagram

with exact rows and columns. (3.2.2) 0 +

0 +

~2 I2(K)--~ I2(K2 ) @ @ I(k ) + + PND w 0 + LI(R + fi2) + L0(K) +~ L0(K 21 @

@ LP(k 01 ÷ L0(R + fi2) ÷ 0 P N.D. + fl

0

0

The snake lemma then yields the following exact sequence (3.2.3) 0 + ker~ 2 + LI(R + R2 ) ~ ker$ I ~ coker~2 ÷ L0(R ~ ~2) ÷ coker~l ÷ 0

Computation of If

I

:

@I:

group of 0-fractional

ideals in

K

and

r :

ideal class group, then we get the following exact sequence i + 0* ÷ K* ÷ I ÷ F ÷ I

Since

r

has odd order by Weber's Theorem

(2.2.4),

86

we get the following short exact sequence i 1 + 0 " / 0 ~2 ÷ K ~ / K ~2 ÷ t / I 2 ÷ 1

Since any 0-fractional ideal can be expressed uniquely as a product of prime ideals, we can identify

I

with the

free abelian group generated by the maximal ideals in

0.

Thus, ~/Z

2

Z

e ~/2. p~2G,~,u) L (X) is found on column UX on r

the row "odd" if r is odd or on the row "even" if r is even. If X has type 0 or S p'

EX on the row ~ = 3, 2, I, or 0: ~ E r E r+2

Part 2:

(mod 4)

Compute

Lr(X) is found in column

we use subtable O: (mod 4)

if Type X is O;

if Type X is S . P

LP( 7ZG, e,u)

We have reduced this problem to understanding a pair of exact sequences

0 ÷ L;r+l ( 7ZG ÷ ~ 2 G , ~ , u )

L;r ( 7ZG ->~2C,~,u)

- ~2r+~

L;r( 7ZG,c~,u) ----~L2r(2Z2G,(~,u) 0

(for r = 0, I). Some terminology will be useful. A representation (or its character X) is called cyclic if it can be obtained by pulling back the faithful irreducible rational representation of CN along some epimorphism y: G + CN, N >_ 0.

A representation (or its

character) is called dihedral if it can be obtained by pulling back the

114

faithful

irreducible rational representation

of D N along some epimorphism

Y: G ÷ D N, N > 3. The epimorphism y determines X but not vice versa. ~( determines

The kernel of

the kernel of ~,~ but X only determines y up to an automorphism

of the quoient. p

Next we determine ~2r"

Since L r (2Z2G,~,u)

-~ 2Z /2 2Z ,

~2r is either

trivial or one to one. Theorem AI.2.1: ~2r is determined by: ~0 is one to one

iff

there is a type 0 cyclic representation;

~2 is one to one

iff

there is a type Sp cyclic representation.

It is easy to describe t h e right-hand Theorem AI.2.2:

The left-hand

extension.

~2r is a split epimorphism.

extension

is more difficult

to describe,

since even if

K2r is onto, two different things can happen. Associated maps;

to a dihedral representation

YI: G ÷ ±i = DN/CN_ I

representation g,~,b)

iff

is twisted

and

y: G -~ D N

there are two other

Y2: ker Yl ÷ CN-I -~ ±I.

A dihedral

( with respect to the geometric antistructure

¥i is 0 invariant and y2(y-10(y))

= -I

for any (and hence

every) y g G-ker YI" A cyclic representation

is twisted

iff

the composite

G Y-~ C N -~ ±I

sends b to -i. We have Theorem AI.2.3: If K0 is onto, then it is split unless there is a type UI twisted cyclic or a type O twisted dihedral representation.

Then ~0 is

not split. If K2 is onto, then it is split unless there is a type UI twisted cyclic or a type S

P

twisted dihedral representation.

Then ~2 is not split.

If K2r is not split, then any x e L;r(2Z G,~,u) has infinite order.

with K2r(X) = -I

115

To apply the above results it is desirable to be able to find cyclic and dihedral representations.

A return to the theory behind the results in

Part I yields the critera below. A character X is cyclic iff E X = r N or R 0 and X(e) = 2 N or i. character X is dihedral is twisted iff .

iff EX = ~

(b 2N) = -X(e);

and X(e) = 2 N+2.

A

A cyclic character X

a dihedral character X is twisted iff

^N+I

X((g-~O(g)) z

) = -X(e)

for at least one g e G.

Another way to give such representations directly.

is to give the epimorphism y

In this case there is a quicker way to find the type than by

using step 2 of Part i. In the cyclic case, extend ¥: G + CN ~(-g) = -y(g).

to 7: ±G + C N

by defining

Recall that ~ induces a map from G to ±G. X has type UI

iff

X has type 0 or S

~(g-l) = ~(~(g))

for all g e G

~(g) = y(~(g))

iff

for all g c G

P the type is 0 if ~(u) = I; Any twisted dihedral representation define Tg

~(g)X(gO(g)u):

=

T

g

has type 0 or S . P

is either 0

0 if there exists a g ~ G with T

the type is Sp if ~(u) = -I.

= X(e):

X(e), or -X(e). we have type S

g a g 6 G with T

g

For any g e G, We have type if there exists

P

= -X(e).

Any cyclic character with x(e) = 1 is called linear: character with x(e) = 2 is called Ruadratic.

any cyclic

A linear cyclic character

either the trivial character or a cyclic character with y: G + C I. quadratic characters the linear characters

are the cyclic characters with y: G + C 2. are in one to one correspondence

is

The

Notice that

with HI(G; ~ / 2 ~ ) .

Ex@mples: i)

(~G,~,e)

~(g) = g-I

:

any non-linear

U;

all the linear ones are type O.

cyclic representation

Therefore ~2 is trivial:

is type

P0 is one to

one. There are no twisted cyclic or twisted dihedral representations,

so

LP(i !] . . . .

of exact sequences

in the decomposition

Since GL types make no contribution the remaining

">LPr(EgG + ~ 2 G , ~ , u )

+

in

119

In case III we either use Table 2 or Table 3. subtable to use; An i n t e g e r ~

which row of that subtable;

and which columns to use.

mod 4 (or mod 2 on the U + U subtable)

of three groups on each row: contribution

We must decide which

determines

a sequence

this sequence will be isomorphic

to the

of this factor of the map to sequence AII.I.I.

If the type is III2 we use Table 2. If the subtype is mixed it is Type X 0 ÷ Type X. subtype

(A. II.I.2)

subtable

row

0

0 + 0

EX0 c E X

~ ~ r (mod 4)

Sp

0 ÷ 0

EX0 c E X

~ ~ r+2 (mod 4)

U

U ÷ U

UXD + U X

~

0 + U

0 ~ UX

EX0 c EX

~ 5 r (mod 4)

Sp÷ U

0 + UX

EX0 = E X

~ ~ r+2 (mod 4)

U ÷ 0

UX + 0

--

~E

UX ~ 0

-

~

U + S

~ r (mod 21

r (rood 4) E r+2 (mod 4)

P

Remarks:

The -- in the row column means that the subtable

in question

has only I r o w . In the 0 + U

(or S + U ) case we may need to go back to P steps 2 and 3 in Appendix I to compute E X. Note that we do not need EX if we are using subtable

O ÷ UII,

EXo suffices.

If the type is 1113 we use Table 3.

If the subtype is mixed,

Type X -> Type X0. subtype

(AII.I.3)

subtable

row

0

0 -> 0

EX c EX0

~ - r (rood 4)

Sp

O ÷ 0

E X c EX0

~-

r+2 (mod 4)

U

U ÷ U

UX ÷ UX 0

~-

r (rood 2)

0 ÷ U

0 -> UX

E X c EX0

~-

r (rood 4)

S -~ U P U ÷ 0

O + UX

E X c EX0

~--- r+2 (rood 4)

UX ÷ O

--

~-

UX + 0

--

~ - - r+2 (mod 4)

U ÷ S P

r (rood 4)

it is

120

Remark: The only visable difference between AII.I.2 and AII.I.3 is that in the row column we have interchanged the role of X 0 and X.

A closer

study shows that on the O ÷ UI N subtable we need EX to use Table 2 but that on Table 3, this subtable has only one row.

Part 2:

Relative transfer maps

This time our goal is to describe !

(AII.2.1)

... + L ~ ( ~ G ÷ ~ 2 G , ~ , u ) --!~+ L ~ ( = H ÷ ~ 2 H , a , u ) ÷ L~,! 7Z/:~ i 0

0

7Z/~_ ~-'

7Z/z 7z/2

0

7/_12

7~.2

::: 0

7L

17/

7~/0.

{/2

,~ZIZ

! =..~

0 ---', UZ ~ 1

-!

0

II ,

U_~..,-+ o

2

" - ~, i

--.z

i-e~

0

--a

!

'~-z

;

0

o

17]

i 0 ", . . . .

0

0

:7ll2

-/~/2 7/2[ , 0

0

~-z

1~4-' ::72

(~

3~'3':

'~"~'

*

i-e4"' i o

I '

l o

O ----~U:E i

R,._,c& i '

:

o

H..,cH______m.0

B., c ~

0

i

i0 i

!g ,

Z~ghiTLla

i,, 0 I

U--~ U

S~jJ --t

z._,--, uL -TZ

O

w~

!zJ~

, RojJ

, -FT~.M

I

7Z/2

7~/~.

',7/12 'q --.t , 2.l ,7//2

0

I

:0

0

• 1 I

,' ~ , ~ , .

,

I 0

iL 0

b

:, 7f"" 'l 9 u-a

um --, o :~

7z2

~-3

! 0

0 g :a

7/--/2

0

7zi2

;

0

i 7Z/2

7Z~'~ (:9

0

7i/2.

! f.a ',Zl2 &~at&

z&

2

+I

125

REFERENCES

[B]

A. Bak, K-theory of forms,

Ann. of Math. Study, 98(1981).

[B-K]

A. Bak and M. Kolster, The computation of odd dimensional projective surgery groups of all finite groups, Topology, 21(1982), 35-64.

[B-W]

A. Bak and B. Williams, in preparation.

[Bassi]

H. Bass,

[Bass2]

H. Bass, Lectures on topics in algebraic K-theory, Tata Institute (1967).

[B-L]

W. Browder and G. R. Livesay, Fixed point free involutions on homotopy spheres, T6koku J. Math., 25(1973), 69-88.

[Ca]

S. Cappell, A splitting theorem for manifolds and surgery groups, Bull. Amer. Math. Soc., 77(1971), 281-286.

[C-S1]

S. Cappell and J. Shaneson, Pseudo-free group actions I., Proc. 1978 Arhus Conference on Algebraic ToooloF~y, Springer Lecture Notes763(1979), 395-447.

[C-S2]

S. Cappell and J. Shaneson, A counterexample on the oozing problem for closed manifolds, ibid(1979), 627-634.

[C-M]

G. Carlsson and J. Milgram, The structure of odd L-groups, Proc. 1978 Waterloo Conference on Al~ebrai9 ' Topology, Springer Lecture Notes 741(1979), 1-72.

[C-F]

J. W. S. Cassels and A. FrShlich, Alsebraic Number Theory, Washington: Thompson (1967).

[C]

P. Connor, Notes on the Witt classification of Hermitian innerproduct spaces over a rin~ of al~ebraic integers, Texas (1979).

IDa]

J. Davis, The surgery semicharactertic,

[D]

A. Dress, Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math., 102(1975), 291-325.

IF]

J. M. Fontaine, Sur,la d6composition des alg~hres de groupes, Ann. Sci. Ecoie Norm. Sup., ~(1971], 121-180.

[F-Mc]

A. Fr~hlich and A. M. McEvett, Forms over rings with involution, J. Algebra 12_=(1969), 7~-i04.

IF-W]

A. Fr~hlich and C. T. C. Wall, Equivariant Brauer groups in algebraic number theory, Bull. Soc. Math. France, 25(1971), 91-96.

[H]

I. Hambleton, Projective surgery obstructions on closed manifolds, to appear.

Surgery and higher K-theory,

Al~ebraic K-theory, Benjamin

(1969).

to appear.

126

[H-Mad]

I. Hambleton and I. Madsen, Semi-free actions on ~ n with one fixed point, Current trends in Algebraic Topology - Proc. 1981 Conference at University of Western Ontario.

[H-M]

I. Hambleton and J. Milgram, The surgery obstruction groups for finite 2-groups, Invent. Math., 63(1980), 33-52.

[H-T-W]

I. Hambleton, preparation.

[K]

M. Kneser, Lectures on Galois cohomolo~y groups, Tata Institute (1969).

[K-O]

M-A. Knus and A. 0janguren, Th@orie de la descente et A l ~ b r e s d'Azumaya, Springer Lecture Notes 38___99(1974).

[KI]

M. Kolster, Computations of Witt groups of finite groups, Math. Ann., 24__!i(1979), 129-158.

[K2]

M. Kolster, Even dimensional of finite groups, to appear.

[L]

T. Y. Lam, The algebraic theory of quadratic forms, Benjamin (1980), 2nd printing with revisions.

[Le]

D. W. Lewis, Exact sequences of graded modules a graded Witt ring, to appear.

[M1]

I. Madsen, Reidemeister torsion~ surgery invariants and spherical space forms, Proc. London Math. Soc., 46(1983), 193-240.

[Ma]

S. Maumary, Proper surgery groups and Wall-Novikov groups, Proc. 1972 Battelle Seattle Conference on Algebraic K-theory~ Vol. III, Springer Lecture Notes 343(1973), 526-539.

[Me]

S. L6pez de Medrano, Springer (1971).

[M-H]

J. Milnor and D. Husemoller, Springer (1972).

[O'M]

O. T. O'Meara, Introduction Springer (1962).

[P]

W. Pardon, The exact sequence for a localization for Witt groups I!: Numerical invariants of odd-dimensional surgery obstructions, Pac. J. Math., 102(1982), 123-170.

[P-R]

E. K. Pederson and A. A. Ranicki, Projective theory, Topology, 19(1980), 239-254.

[RI]

A, A. Ranicki, Exact sequences of surgery, Princeton (1981).

[R2]

A. A. Ranicki, On the algebraic L-theory rings, J. Algebra, 50(1978), 242-243.

[R3]

A. A. Ranicki, The algebraic theory of surgery I. Foundations, Proc. Lond. Math. Soc., (3)40(1980), 87-192.

L. Taylor,

and B. Williams,

projective

Involutions

in of classical

surgery groups

over

on manifolds,

Symmetric bilinear to quadratic

forms,

forms,

surgery

in the alsebraic

theory

of semisimple

127

[Re]

I. Reiner,

Maximal orders,

[Rog]

K. W. Roggenkamp, A remark on separable Canad. Math. Bull., 12(1969), 453-455.

[Sl]

J. P. Serre, Linear representations Springer (1977).

of finite groups,

[$2]

J.P.

(1979).

[Sh]

J. Shaneson, Wall's surgery obstruction groups G x ~, Ann. of Math., 90(1969), 296-334.

for

[SE]

R. Swan, K-theory of finite groups Lecture Notes 149(1970).

Springer

[T-W]

L. R. Taylor and B. Williams, to appear.

[Wl]

C. T. C. Wall, Press (1970).

[W2]

C. T. C. Wall, On the axiomatic foundation of the theory of Hermitian forms, Proc. Camb. Phil. Soc., 67(1970), 243-250.

[W3]

C. T. C. Wall, Foundations of algebraic L-theory, Proc. 1972 Battelle Seattle Conference on Algebraic K-theory~ Vol. III, Springer Lecture Notes 343(1973), 266-300.

[W4]

C. T. C. Wall, On the classification forms II. Semisimple rings, Invent. ll9-141.

[WS]

C. T. C. Wall, III. Complete 19(1973), 59-71.

[W6 ]

C. T. C. Wall, 241-260.

[W7]

C. T. C. Wall, V. 261-288.

[W8]

C. T. C. Wall, VI. 1-80.

[W9]

C. T. C. Wall, Formulae for surgery obstructions, Topology, 15(1976), 189-210.

[War]

M. L. Warschauer, The Witt group of degree k maps and asymmetric inner product spaces, Springer Lecture Notes, 914(1982).

[Was ]

L. C. Washington, Springer Graduate

[Y]

T. Yamada, The Schur subsrou p of the Brauer group, Springer Lecture Notes, 397(].974).

Serre,

Local fields,

Academic

Press

Springer

(1975).

orders,

and orders,

Surgery on closed manifolds,

Surgery on compact manifolds,

Notre Dame University Notre Dame, Indiana 46556

of Hermitian Math., 18(1972),

semilocal

IV. Adele rings, Global rings, Group rings,

Academic

rings,

ibid.,

ibid., 23(1974), ibid., 23(1974), Ann. of Math.,

103(1976),

Introduction to cyclotomic fields, texts in mathematics, 83(1982).

McMaster University Hamilton, Ontario LSS 4KI