THE GENERALIZED BURNSIDE RING AND THE
K-THEORY OF A RING WITH ROOTS OF UNITY
W. G. Dwyer*, E. M. Friedlander** and S. A. Mitchell
University of Notre Dame
Northwestern University
University of Washington
x1. Introduction
Determining the algebraic K-theory of rings of integers in number fields has*
* been the
goal of much research. In [10] D. Quillen showed that the Hurewicz map h : Q0(*
*S0) !
B GL (Z)+ (see 1.1 for the notation) induces an interesting map on homotopy gro*
*ups from
the stable homotopy groups of spheres to the algebraic K-theory of the ring Z o*
*f rational
integers. Quillen observed that if ` is an odd prime and if p 6= ` is another p*
*rime generating
the `-adic units, then the composite map
h + ae +
Q0(S0)!- B GL (Z) !- B GL (Fp)
induces a surjection
sss*(S0; Z=`) ! K*(Z; Z=`) ! K*(Fp; Z=`)
on mod ` homotopy groups. In geometric terms, one can write Q0(S0) ' Im J x Cok*
*erJ
where Im J is the factor of Q0(S0) associated to the J-homomorphism J : O ! Q0(*
*S0).
The space Im J is then equivalent after localization at ` to B GL (Fp)+ in such*
* a way that
the composite
h + ae +
Im J x CokerJ ' Q0(S0)!- B GL (Z) !- B GL (Fp)
can be identified after localization at ` with the projection r : Im J x CokerJ*
*!- Im J.
In [6] the third author complemented Quillen's result by proving that the Hu*
*rewicz
map h does not in fact detect anything in stable homotopy theory beyond what Qu*
*illen
_____________*
Partially supported by the N.S.F. **Partially supported by the N.S.F. and *
*NSA Grant # MDA904-
90-H-4006 .
Typeset by AM S-*
*TEX
1
2 DWYER, FRIEDLANDER AND MITCHELL
found. He did this by verifying that the localization of h at ` factors throug*
*h Im J, or,
more precisely, that the maps
h; h . s . r : Q0(S0) ! B GL (Z)+
are homotopic after localization at `, where s : Im J ! Q0(S0) is a suitable ri*
*ght inverse
after localization at ` for the factor projection r : Q0(S0)!- ImJ. It follows*
* that Quillen's
map h* : sss*(S0; Z=`) ! K*(Z; Z=`) annihilates the homotopy of CokerJ, and tha*
*t for any
ring R the structure of K*(R; Z=`) as a module over sss*(S0; Z=`) extends (uniq*
*uely) to a
structure of K*(R; Z=`) as a module over ss*(Im J; Z=`) = K*(Fp; Z=`).
Let ia denote a primitive `a-th root of unity and the group of `-primary ro*
*ots of unity
of Z[ia]: The inclusion GL 1(Z[ia]) leads to an enriched analogue
h : Q0(B+ ) ! B GL (Z[ia])+
of Quillen's Hurewicz map. Let ` and p be primes as before (` is odd and p gene*
*rates the
`-adic units) and let Fq denote Fp[ia]. In Z[ia] there is a unique prime above *
*p with residue
class field Fq. The main result of B. Harris and G. Segal [3] implies that the *
*composition
h + ae +
Q0(B+ )!- B GL (Z[ia]) !- B GL (Fq)
induces a surjection of mod ` homotopy groups. In fact, for a suitable space X *
*there is
after localization at ` a product decomposition Q0(B+ ) ' B GL (Fq)+ x X in suc*
*h a way
that the (localized) composition
B GL (Fq)+ x X ' Q0(B+ ) ! B GL (Z[ia])+ ! B GL (Fq)+
can be identified with the projection r : B GL (Fq)+ x X!- B GL (Fq)+ .
The purpose of this paper is to extend the results of the third author to th*
*is wider
context. Theorem 4.1 asserts that after localization at ` the maps
h; h . s . r : Q0(B+ ) ! B GL (Z[ia])+
are homotopic, where s : B GL (Fq)+!- Q0(B+ ) is a suitable right inverse for *
*the factor
projection r. It follows that for any algebra R over Z[ia] the structure of K*(*
*R; Z=`) as
a module over sss*(B+ ; Z=`) extends (uniquely) to a structure of K*(R; Z=`) as*
* a module
over K*(Fq; Z=`). We also make some further remarks (4.10) about the relationsh*
*ip of our
work to [6], and briefly consider the situation of infinite cyclotomic extensio*
*ns (4.12).
Let G be a finite `-group. To prove Theorem 4.1 we show that h, h.s.r have h*
*omotopic
compositions with any map BG ! Q0(B+ ). Our technique is to use the generalized
Burnside ring A(G; ) to study maps from BG to Q0(B+ ) and to use the representa*
*tion
ring RR G of G over R to study maps from BG to B GL (R)+ . The key algebraic re*
*sult is
Theorem 3.5, which amounts to an algebraic analogue of (4.1) and asserts that t*
*he map
from A(G; ) to RZ[ia]G naturally factors through a surjection A(G; )!- RFqG.
We conjecture that, after localization at `, the composition
h . s : B GL (Fq)+ ! B GL (Z[ia])+
extends to an infinite loop map. One can view Proposition 4.9 as evidence that *
*such an
extension exists. The reader is referred to [7] for a discussion of this conjec*
*ture and related
matters.
GENERALIZED BURNSIDE RING 3
1.1 Notation and conventions. The symbol "+" as a subscript denotes a disjoint
basepoint; as a superscript it denotes Quillen's plus construction. We write 1 *
*X+ for the
suspension spectrum of the pointed space X+ and QX+ (= Q(X+ )) for the correspo*
*nding
infinite loop space, where as usual Q = 1 1 . Similarly Q0X+ Q0(X+ ) denotes t*
*he
basepoint component of QX+ . We employ the notation [-; -] to denote pointed ho*
*motopy
classes of maps between pointed spaces or spectra.
x2. Preliminaries on representation
rings and the generalized Burnside ring.
Let G be a finite group. We will let AG denote the Burnside ring of G, i.e.*
*, the
Grothendieck ring of finite G-sets, with sum and product induced respectively b*
*y disjoint
union and cartesian product of G-sets. Suppose that H is another finite group.*
* The
Burnside module A(G; H) is defined as follows (cf. [5]). A (G; H)-set is a se*
*t P with a
left G-action and a free right H-action such that the two actions commute; equi*
*valently,
P is an H-free (G x H)-set, but it is convenient to put the two actions on oppo*
*site sides.
Then A(G; H) is the free abelian group on the isomorphism classes [P ] of finit*
*e (G; H)-sets,
subject to the relations [P q Q] = [P ] + [Q]. We now list some properties of *
*the group
A(G; H) and describe some ways in which it is related to representation rings.
(2.1). A (G; H)-set is indecomposable if and only if it has the form P = PK; *
*where
K G, : K ! H is a homomorphism and PK; = G xK H. There is an obvious
conjugation action of G x H on the set of all pairs (K; ), and PK; is isomorph*
*ic to PK0;0
if and only if (K; ) and (K0; 0) lie in the same orbit of this action. As an ab*
*elian group,
A(G; H) is freely generated by the elements [PK; ] as (K; ) runs through a set *
*of orbit
representatives.
(2.2). The group A(G; H) is a module over AG, with multiplication [P ][Q] = [P *
*x Q],
[P ] 2 AG, [Q] 2 A(G; H). Here G acts diagonally on P x Q and H acts only on th*
*e Q-
factor. There is a natural map of AG-modules oe : A(G; H) ! AG given by [P ] 7!*
* [P=H];
in addition, oe has a canonical right inverse j, given by j([S]) = [SxH] (here *
*H has a trivial
G-action). Composing oe with the usual augmentation ffl : AG! Z, we get an augm*
*entation
ffl0: A(G; H)! Z. The augmentation ffl is split by the ring homomorphism Z ! A(*
*G) which
sends an integer n 0 to the trivial G-set [n] with n elements.
(2.3). Let IG = Ker ffl and I(G; H) = Ker ffl0. Then if G is an `-group, the IG*
*-adic and
`-adic filtrations on A(G; H) are topologically equivalent on I(G; H).
(2.4). Suppose that H is abelian. Then A(G; H) is a commutative ring, with pro*
*duct
[P ][Q] = [P xH Q]. The maps j and oe of (2.2) make A(G; H) into an augmented a*
*lgebra
over AG.
Given a (G; H)-set P , define a map of spectra ffP : 1 BG+ ! 1 BH+ as the co*
*m-
posite 1
1 BG+ !o1 (EG xG (P=H)+ ) ! h 1 BH+
where o is the transfer and h classifies the principal H-bundle EGxG P over EGx*
*G (P=H).
In the special case P = PK; , ffP is the familiar composition of transfer o : 1*
* BG+ !
1 BK+ and B : 1 BK+ ! BH+ :
4 DWYER, FRIEDLANDER AND MITCHELL
(2.5). The correspondence P 7! ffP extends to a homomorphism
ff : A(G; H) ! [1 BG+ ; 1 BH+ ]
of AG-modules which is a ring homomorphism if H is abelian. Here [1 BG+ ; 1 BH+*
* ]
is a module over the stable cohomotopy ring ss0s(BG+ ), and so receives an AG-m*
*odule
structure from the ring homomorphism AG = A(G; *) ! ss0sBG+ . Note that 1 BH+ is
a commutative augmented ring spectrum if H is abelian.
(2.6).`Recall that QBH+ can be identified with the group completion of the mon*
*oid
n(n o H), and Q0BH+ can be identified with B(1 o H)+ . Now a (G; H)-set P cor*
*re-
sponds to a conjugacy class of homomorphisms G ! n oH, n = |P=H|, and so determ*
*ines
a unique homotopy class BG ! B(n o H). The map ff can then be reinterpreted in *
*an
obvious in terms of group completion or in terms of the plus construction.
(2.7). For any commutative ring R, let RR G denote the representation ring of G*
* over
R, i.e., the Grothendieck group of finitely generated R-projective RG-modules. *
*It is clear
that RR G is a ring under tensor product. Let R0RG denote the Grothendieck gro*
*up of
all finitely generated RG-modules. If R is a Dedekind domain, then the natural*
* map
RR G ! R0RG is an isomorphism. If G is an `-group and E is a field of character*
*istic `,
then the augmentation ideal of EG is nilpotent and every finitely generated mod*
*ule over
EG has a finite filtration with each filtration quotient isomorphic to the triv*
*ial G-module
E; consequently R0EG ~=Z, generated by the trivial G-module E.
(2.8). For a commutative ring R there is an obvious ring homomorphism AG ! RR G
induced by mapping a G-set P to the permutation module RP . More generally, sup*
*pose
given a homomorphism j : H ! R*. Then there is a natural homomorphism
'(j) : A(G; H) ! RR G
of AG-modules, given by P 7! RP RH R. Here R is an RH-module via j. If H is
abelian, '(j) is a ring homomorphism. We remark that the image of '(j) lies in*
* the
subroup of RR G generated by the R-free RG-modules. Observe that if P = PK; , *
*then
'(j)(PK; ) is the induced representation RGRK R, where the action of K on R is *
*given by
j . . Equivalently, '(j)(PK; ) is a "monomial representation", that is, the cor*
*responding
homomorphism G ! GL nR maps into the monomial matrices n o R* (up to conjugacy),
where n = |G=K|. If is a subgroup of R* and j : ! R* is the inclusion we will*
* write
'R instead of '(j).
(2.9). There is a natural ring homomorphism
fi : RR G ! [BG+ ; B GL (R)+ x K0R]
obtained essentially by associating to an R-projective G-module M the map deter*
*mined by
the G-structure map of M, G ! Aut(M)`and viewing B GL (R)+ x K0R as the homotop*
*y-
theoretic group completion of M B Aut(M). If H is abelian, then j : H ! R* in*
*duces a
GENERALIZED BURNSIDE RING 5
functor P 7! RP RH R of bipermutative categories (free right H-sets) ! (f.g. *
*projective
R-modules) and hence a map of ring spectra 1 BH+ ! KR. The adjoint map
(j) : BH+ ! BGLR+ x K0R
restricted to BH is just the obvious composite BH Bj!B GL 1(R) ! B GL (R)+ x {1*
*}. It
induces a ring homomorphism
(j)* : [BG+ ; QBH+ ] ! [BG+ ; B GL (R)+ x K0R].
We will write 0(j) for the restricted map Q0BH+ ! B GL (R)+ . If is a subgroup*
* of
R* and j : ! R* is the inclusion, we will write R instead of 0(j).
(2.10). Let A^(G; H) and R^RG denote the IG-adic completions of respectively A(*
*G; H)
and RR G. Then there are maps ^ffand ^fiwhich fit into a commutative diagram
^ff
A(G; H) ----! ^A(G; H)----! [BG+ ; QBH+ ]
? ? ?
'(j)?y ^'(j)?y (j)*?y
^fi
RR G ----! R^RG ----! [BG+ ; B GL (R)+ x K0R]
in which the composite of the two upper arrows is ff (2.5) and the composite of*
* the two
lower arrows is fi (2.9).
The first assertion of the following theorem is a result of G. Lewis, J.P. M*
*ay, and J.
McClure [4] derived from the affirmation of the Segal Conjecture; the second as*
*sertion is
a result of D. Rector [8].
2.11 Theorem. ([4], [8]) Let G; H be finite groups. Then the map ^ffof (2.10) *
*is an
isomorphism. Moreover, if R is a finite field, then map ^fiof (2.10) is also an*
* isomorphism.
x3. Monomial representations of finite `-groups
Throughout this section G denotes a finite `-group. If R is a ring, let (R) *
*denote the
`-torsion subgroup of R*, that is, the group of all `n-th roots of unity, n 0,*
* and let a(R)
be defined by the equation |(R)| = `a(R), 0 a(R) 1. The condition |(R)| > 2 is
equivalent to a(r) 1 if ` is odd or to a(R) 2 if ` = 2.
Our goal in this section is to investigate the map '(j) : A(G; ) ! RR G (2.8*
*) in the
special situation in which is a subgroup of (R) and j : ! R* is the inclusion*
*; recall
(2.8) that in this case we denote '(j) by 'R :
We begin with the following (known) description of the structure of a simple*
* G-module
over a field of characteristic different from `. Our discussion follows the an*
*alysis of P.
Roquette [11].
3.1 Proposition. Let F be a field of characteristic different from ` with |(F )*
*| > 2, and
let V be a simple right F G module. Then End FG V is a cyclotomic extension *
*field F 0
of F , and there exists a subgroup K G and homomorphism : K ! (F 0) such that
F 0= F [(K)] and V ~=F 0FK F G as an (F 0; F G)-bimodule.
6 DWYER, FRIEDLANDER AND MITCHELL
Proof. Denote by D the division algebra EndFG V . We must show that D is a cycl*
*otomic
field extension of F . Clearly, we may assume that G acts faithfully on V . I*
*f V admits
a non-trivial decomposition V = W1 . . .Wk as D-modules such that G permutes t*
*he
Wi, then V is obtained by induction from a proper subgroup of G. Arguing induct*
*ively
on the order of G, we may assume that V admits no such decomposition. By [11], *
*G must
then be a group of "normal rank one" and hence either G is cyclic or ` = 2 and *
*G is
dihedral, semidihedral or quaternionic. If G is cyclic, then one readily verifi*
*es that D is a
cyclotomic extension F 0of F and that the action of G on V is given by a homomo*
*rphism
: G ! AutFG V ~=(F 0)* such that F 0= F [(G)]. We proceed to show that other n*
*ormal
rank one groups do not occur.
Assume then that ` = 2, a(F ) 2, and that G is normal rank one but not cycl*
*ic;
in this case G fits into an extension Z=2n ! G ! Z=2, n 2. Let y generate Z=2n
and let x 2 G project to the generator of Z=2. Then either xyx-1 = y-1 (dihedra*
*l and
quaternionic cases) or xyx-1 = y2n-1-1 (semidihedral case). In all cases xT x-1*
* = T -1,
where T = y2n-2 has order four. Consider V as a module over = Z=4. Since <*
*T > is
normal and V is primitive, the restriction of V to is isotypical, and nontr*
*ivial since V
is faithful. Since a(F ) 2, we conclude that T = i for all , where i 2 (F ) i*
*s a fixed
element of order 4. But then T (x) = xT -1 = -i . x, which is a contradiction.
Proposition 3.1 enables us to prove the following surjectivity result, which*
* asserts that
sometimes the representation ring of G is generated by monomial representations.
3.2 Proposition. Let F be a field of characteristic different from ` with |(F )*
*| > 2. Let
denote (F ). Then
'F : A(G; ) ! RF G
is surjective with right inverse oe : RF G ! A(G; ).
Proof. Since F G is semi-simple, RF G is freely generated as an abelian group b*
*y the set
of isomorphism classes of simple F G-modules. Thus, the surjectivity of 'F as*
* well as
the existence of the left inverse oe follows immediately once we verify that ev*
*ery irreducible
F G-module lies in the image of 'F . As discussed in (2.8), any representation *
*of G obtained
by induction from a one-dimensional representation : K ! of some subgroup K o*
*f G
is of the form 'F (PK; ). Thus, it suffices to prove that every irreducible rep*
*resentation of
F G is induced from some one-dimensional representation of a subgroup K of G.
By (3.1), each irreducible F G-module is of the form V = L FH F G, where H *
*G is a
subgroup acting on the cyclotomic field extension L=F via a homomorphism : H !*
* (L)
with L = F [(H)]: Let K denote -1 ((F )) H. Since F is preserved by the action*
* of
(K), we conclude the existence of a natural homomorphism of F G-modules
F FK F G ! L FH F G:
This map is a surjective homomorphism of F -vector spaces of the same dimension*
*, and
thus an isomorphism. (The equality of these dimensions comes down to the observ*
*ation
that since |(F )| > 2 and L = F [(H)] is obtained by adjoining to F additional *
*`-primary
roots of unity, the degree of L over F is equal to the order of the quotient gr*
*oup (L)=(F ),
which in turn is equal to the index of K in H.)
GENERALIZED BURNSIDE RING 7
Remarks. (a) Proposition 3.2 is obviously false if a(F ) = 0, although if ` is *
*odd it is
possible to recover the conclusion by making an additional assumption about F [*
*6, 1.15].
(b) If ` = 2 and F = Fq with q = 3 mod 4, Proposition 3.2 is false for G = Z=8.*
* (c) The
assertion of Proposition 3.2 for fields of characteristic 0 is a special case o*
*f a theorem of
J. Tornehave [12] (which also treats the case a(F ) = 1 for ` = 2 when char(F )*
* = 0).
3.3 Representation ring calculations. We will now use Morita theory to carry out
certain standard representation ring calculations in a very explicit way. As a*
* matter of
terminology, if F is a number field with ring of integers R, we will call R[1=`*
*] the ring of
`-integers of F . The situation we are interested in is as follows. We let F *
*be a number
field with |(F ) > 2|, and V1; : :;:VN a complete nonredundant list of isomorph*
*ism classes
of simple right F G-modules. For 1 i N we denote by Fi the endomorphism field*
* (3.1)
of Vi and by i: Ki! (Fi) a homomorphism provided by 3.1, so that Vi~= FiFKi F G
as an (Fi; F G)-bimodule. We let S be the ring of `-integers of F , Si the ring*
* of `-integers
of Fi, and ViS Vi the (Si; SG)-bimodule SiSKi SG. (Observe that the tensor prod*
*uct
defining ViS makes sense because i(Ki) (Si) = (Fi)). Finally, E is a finite re*
*sidue
class field of S such that (S)!- (E) is an isomorphism, Ei is the field E S Si*
*, and ViE
the (Ei; EG)-bimodule EiSiViS. (The fact that Ei is a field follows from the hy*
*pothesis
on a(F ), the assumption (S) ~=(E) and the fact (3.1) that Fi is obtained from *
*F by
adjoining `-primary roots of unity: together these imply by elementary number *
*theory
that the prime of S determining E is inert in the extension Fi=F .)
There are maps fFi: RF G ! K0Fi given by fFi[M] = [ViFG M], maps fSi: RSG !
K0Si given by fSi[M] = [ViSSG M], and maps fEi : RE G ! K0Ei given by fEi[M] =
[ViE EG M]. (Here K0(-) is defined to be the Grothendieck group of finitely gen*
*erated
projective left modules.)
3.4 Proposition. In the above situation, there is a commutative diagram
RE G ---- RSG ----! RF G
? ? ?
xfEi?y xfSi?y ?yxfFi
QN QN Q N
ZN ~= i=1 K0Ei ---- i=1 K0Si ----! i=1K0Fi~= ZN
in which the horizontal arrows are induced by the evident ring homomorphisms. T*
*heQver-
ticalQarrows in this diagram are isomorphisms. Moreover, if both of the groups *
* Ni=1K0Ei
and Ni=1K0Fi are identified with ZN by using the standard bases provided by *
*rank one
free modules, then the two maps RSG!- ZN given by this diagram are the same.
Proof. The action of G on the various modules involved gives a commutative diag*
*ram of
rings and homomorphisms
EG ---- SG ----! F G
?? ? ?
y ?y ?y
Q N E Q N S Q N
i=1End EiVi ---- i=1End SiVi ----! i=1End FiVi
8 DWYER, FRIEDLANDER AND MITCHELL
The Artin-Wedderburn theorem implies that in this diagram the right vertical ar*
*row is an
isomorphism. Let di = dimFiVi. By construction ViS is a free module of rank di *
*over Si.
SinceQ|G| is invertible in S the ring SG is a maximal S-order in F G and for ge*
*neral reasons
N S ~ Q N Q N ~ Q N
i=1End SiVi = i=1Mdi(Si) is a maximal S-order in i=1End FiVi = i=1Mdi(*
*Fi)
(see [6, 1.7] and [9]); it follows that in the above diagram of rings the middl*
*e vertical arrow
is an isomorphism. The left vertical arrow is then also an isomorphism, since *
*it is the
tensor product of the middle arrow over S with E.
The commutativity of the diagram of (3.4) is evident, and the isomorphism st*
*atements
of (3.4) now follow from Morita theory; more precisely, what follows in the cas*
*e of the
map (xfFi) for instance is the fact that for each i the functor given by tensor*
*ing over
End FiVi with Vi gives an equivalence from the category of left End FiVi module*
*s to the
category of Fi modules. The fact that the two indicated maps RSG ! ZN agree amo*
*unts
to the observation that both maps assign to a class [M] then vector (r0; : :;:r*
*N ), where
ri= rankSi(ViSSG M).
The following theorem is a discrete analogue of Theorem 4.1. Theorem 3.5 ext*
*ends a
result of the third author [6, 1.12] from the Burnside ring A(G) to the general*
*ized Burnside
ring A(G; ). Recall that ia denotes a primitive `a-th root of unity.
3.5 Theorem. Let R = Z[ia], where a 1 if ` is odd and a 2 if ` = 2. Let E be
a residue field of R such that the reduction map R ! E induces an isomorphism f*
*rom
= (R) to (E), and let oe : RE G ! A(G; ) be a right inverse to 'E (cf. 3.2). T*
*hen
'R . oe . 'E = 'R :
In other words, there is a unique map h = 'R . oe making the following triangle*
*[224z
commute 'R
A(G;?) ! RR G
?
'E ?y % h :
RE G
3.6 Remark. The maps 'E and 'R in (3.5) are maps of modules over AG; since 'E i*
*s a
surjection (3.2) the map h, if it exists, is also a map of modules over AG.
Proof of 3.5. Since 'E is surjective by Proposition 3.2, it suffices to prove t*
*hat ker'E =
ker'R . Let S = R[1=`] and let F be the quotient field of R. By (3.4) there *
*is an
isomorphism RE G ~=RF G which fits in the following commutative diagram:
A(G;?) ! RSG?
?? ?
y . ?y
RE G ~= RF G
The localization theorem yields an exact sequence
M o j
R0R=PG!- R0RG(~=RR G)!- R0SG(~=RSG) ! 0 :
P|`
GENERALIZED BURNSIDE RING 9
(see 2.7) where RR=P G ~=Z, generated by the trivial G-module [R=P] of dimensio*
*n one.
The transfer map o sends [R=P] to [R] - [P], where R and P are again regarded as
trivial G-modules. Now the map RR G ! K0R obtained by forgetting the G-action *
*is
canonically split by the construction which takes a projective R-module and giv*
*es it the
trivial G-action. Hence K0R is canonically embedded as a direct summand in RR G*
*, and
the preceeding remarks show that kerj is contained in the class group eK0R, and*
* is in fact
equal to the subgroup of eK0R generated by the elements [R] - [P] as P runs thr*
*ough the
non-principal primes of R over `. Since 'R maps A(G; ) into the R-free RG-modu*
*les,
im 'R \ kerj = 0. Consequently, in order to show that ker'E = ker'R it is enoug*
*h to
show that ker'S = ker'F .
We will use the notation and indexing of (3.3) and (3.4). To show that ker'S*
* = ker'F ,
it is enough to show that for any (G; )-set P and integer i between 1 and N the*
* element
fSi('S(P )) 2 K0Si is the class of a free Si-module. Observe (2.8, 3.3) that by*
* definition
fSi('S(P ))= ViSSG 'S(P )
= SiSKi SG SG 'S(P )
= SiSKi 'S(P )
= SiSKi SP S S
where Ki acts on Si via i : Ki ! (Si). Choose such a P , fix the integer i; the*
* above
formula shows that fSi('S(P )) depends only on the structure of P as a (Ki; )-s*
*et. We
may decompose P as a disjoint union of indecomposable (Ki; ) sets (see 2.1), ea*
*ch of the
form PH; where H Ki is a subgroup and : H ! a homomorphism. The summand
of fSi('S(P )) corresponding to such an PH; is the module
SiSKi SKiSH S = SiSH S :
where in the right-hand tensor product H acts on S by and on Si by the composi*
*te of
i with the inclusion H ! Ki. It is clear that SiSH S is a quotient module of Si*
*. Since
this module is also a projective module over Si, it must be either zero or isom*
*orphic to Si;
in particular, it must be free.
x4. The main theorem
Let L denote the functor [1] on the category of spaces which assigns to each*
* space X its
localization LX at `. If X is a connected loop space or a simply connected spac*
*e, then the
homotopy group ssiLX is isomorphic to Z(`) ssiX, where Z(`)is the ring obtained*
* from Z
by inverting all primes except `. There is a natural map X ! LX which in all of*
* the cases
of interest to us nduces an isomorphism on homology or cohomology with coeffici*
*ents in
Z(`). We will sometimes use the same notation for a map f : X ! Y and for the i*
*nduced
map Lf : LX ! LY . The notation LB GL (R)+ stands for L(B GL (R)+ ).
For the definition of the maps E and R which appear in the following stateme*
*nt, see
(2.9).
10 DWYER, FRIEDLANDER AND MITCHELL
4.1 Theorem. Suppose that R is the ring Z[ia], where a 1 if ` is odd and a 2 *
*if
` = 2. Let = (R) and let E denote a residue class field of R with ~=(E). Th*
*en
there is a map s : LB GL (E)+ ! LQ0(B+ ) such that in the diagram
s E + s R +
LB GL (E)!- LQ0(B+ ) --! LB GL (E) !- LQ0(B+ ) --! LB GL (R)
the following two conditions hold
(1) the composite E . s is homotopic to the identity map of LB GL (E)+ , and
(2) the composite R . s . E is homotopic to the map R .
Remark. The existence of infinitely many quotient fields E of R satisfying the *
*condition
of (4.1) is guaranteed by the Tchebotarev density theorem.
Remark. Theorem 4.1 implies that up to homotopy the following diagrams commute:
LB GL (E)+? id! LB GL (E)+ LQ0(B+?) R! LB GL (R)+
? ?
s?y % E E ?y % R.s
LQ0(B+ ) LB GL (E)+
The composite map R .s is determined uniquely up to homotopy (4.8) by the requi*
*rement
that the diagram on the right commute. The map s itself does not appear to hav*
*e any
uniqueness property.
Remark. Since LQ0(B+ ) and LB GL (R)+ are infinite loop spaces, Theorem 4.1 imp*
*lies
that up to homotopy there are product decompositions
LQ0(B+ ) ~=LB GL (E)+ x X1
LB GL (R)+ ~=LB GL (E)+ x X2
with respect to which R amounts to the projection LB GL (E)+ x X1!- LB GL (E)+
followed by factor inclusion LB GL (E)+!- LB GL (E)+ x X2.
Remark. The results of [2] show that if E is a field of characteristic differen*
*t from ` which
is a union of finite fields and |a(E)| > 2, then the homotopy type of LB GL (E)*
*+ depends
only on the isomorphism type of the group (E).
For the convenience of the reader we state the following standard propositio*
*ns. The
first one is due originally in a more general form to D. Sullivan, and can be p*
*roved using
his technique of compact representable functors.
4.2 Proposition. Let X be a CW-complex and X0 X1 X2 . .X.a filtration of
X by subcomplexes such that [iXi = X. Let Y and Z be loop spaces. Assume that t*
*he
groups H"j(Xi; Z(`)) are finite for all i, j, and that the homotopy groups of Y*
* and Z are
finitely generated modules over Z(`). Then, given maps f : X ! Z and g : Y ! Z *
*with
fn = f|Xn , there exists h : X ! Y with g . h ~ f iff for each n there exists h*
*n : Xn!- Y
with g . hn ~ fn.
GENERALIZED BURNSIDE RING 11
4.3 Remark. Proposition 4.2 implies that if X and Z are spaces of the indicated*
* type,
then two maps f; f0 : X!- Z are homotopic iff for each n their restrictions to*
* Xn are
homotopic (this follows from the observation that f and f0 are homotopic iff (f*
*; f0) : X !
Z x Z lifts up to homotopy over the diagonal map Z ! Z x Z). We will apply (4.2*
*) in
situations in which the spaces Y and Z are derived from LQ0(B+ ) ( a finite gro*
*up), from
LB GL (E)+ (E a finite field) or from LB GL (R)+ (R the ring of integers in an *
*algebraic
number field). In the first case the relevant homotopy groups are finite by sta*
*ble homotopy
theory, in the second case they are finite by the calculation of Quillen, and i*
*n the third
they are finitely generated, again by a theorem of Quillen.
4.4 Proposition. Let G be a finite group, H an `-Sylow subgroup of G and i : H*
* ! G
the inclusion. Then the map LQ(Bi+ ) : LQ(BH+ ) ! LQ(BG+ ) has a right inverse.
4.5 Remark. The right inverse referred to in in 4.4 is derived from the transfe*
*r (2.5).
Proposition 4.4 implies that if G is a finite group with `-Sylow subgroup H, an*
*d g : Y ! Z
is a map of `-local infinite loop spaces, then a map f : BG ! Z lifts up to hom*
*otopy to
a map BG ! Y iff the restriction of f to BH lifts up to homotopy to a map BH ! *
*Y .
Similarly, a map BG ! Y is null homotopic iff its restriction to BH is.
4.6 Proposition. If Y is a space which is local at ` and f : X1 ! X2 is a map *
*which
induces an isomorphism H*(X1; Z(`)) ! H*(X2; Z(`)), then f induces a bijection *
*[X2; Y ]!-
[X1; Y ].
4.7 Proposition. Suppose that G is a finite `-group and that Y is a connected l*
*oop space
with finite homotopy groups. Then the natural map [BG; Y ]!- [BG; LY ] is a bi*
*jection.
Remark. This follows from the fact that Y can be expressed up to homotopy as a *
*product
Y 0x LY , where Y 0is the localization of Y with respect to the ring Z[1=`]; s*
*ince the
homotopy groups of Y 0are uniquely `-divisible, the pointed set [BG; Y 0] is tr*
*ivial. We will
apply this proposition with Y = Q0B+ ( a finite group) or Y = B GL (E)+ (E a fi*
*nite
field).
Proof of 4.1. To find a map s : LB GL (E)+ ! LQ0(B+ ) satisfying 4.1(1) it is e*
*nough
to find
(1) (by 4.6) a map B GL (E) = [nB GL n(E) ! LQ0(B+ ) which is a lift of the*
* map
B GL (E)!- LB GL (E)+ , or
(2) (by 4.2) for each n 1 a map B GL n(E) ! LQ0(B+ ) which is a lift of th*
*e map
BGLn(E)!- LB GL (E)+ , or
(3) (by 4.4) for each n 1 a map BGn!- Q0(B+ ) which is a lift of the map
BGn!- B GL (E)+ , where Gn denotes some chosen `-Sylow subgroup of GL *
*n(E).
To accomplish this last observe that the composite
BGn!- B GL n(E)!- B GL (E)+
corresponds under the map fi of (2.10) to (in - [n]), where in 2 RE Gn is given*
* by the
inclusion Gn GL n(E) and [n] is the dimension n trivial representation. By (3*
*.2) the
element in - [n] lifts to some element jn of A(Gn; ). The map ff(jn) then by (2*
*.10) gives
the required map BGn!- Q0(B+ ).
12 DWYER, FRIEDLANDER AND MITCHELL
A chain of reasoning parallel to the above (see 4.3) shows that 4.1(2) can b*
*e proved by
verifying that for each n 1 the two maps
R ; R . s . E : LQ0(B+ )!- LB GL (R)+
become homotopic after precomposition with the map
BKn ! B(n o ) ! Q0(B+ ) ! LQ0(B+ )
where Kn n o is an `-Sylow subgroup. We will denote this map BKn!- LQ0(B+ )
by fn. Consider the map gn = s . E . fn : BKn ! LQ0(B+ ), and let "fn; "gn: BKn*
*!-
Q0(B+ ) be the lifts of fn and gn provided by (4.7). By (2.11) there are elemen*
*ts an; bn 2
A^(Kn; ) such that "fn= ^ff(an) and "gn= ^fi(bn). Since (LE ) . fn ~ (LE ) . gn*
*, it follows
from (4.7) that E . "fn~ E . "gnand hence (2.11) that ^'E(an) = ^'E(bn). The IG*
*-adic
completion of (3.5) (see 3.6) now implies that ^'R(an) = ^'R(bn), which gives b*
*y (2.10) that
R . "fn~ R . "gnand by naturality of localization that (LR ) . fn ~ (LR ) . gn.
4.8 Dyer-Lashof operations. Let R = Z[ia]. In [3], the map B GL (R)+ ! B GL (E)+
induced by the quotient map R ! E is shown to have a right inverse after locali*
*zation at `,
provided that (R) ~=(E) and a 2 if ` = 2. Theorem 4.1 gives us a canonical hom*
*otopy
class of such a right inverse if a 1, since (in view of the fact that the loca*
*lization at ` of
E has a right inverse up to homotopy) the map = R . s is clearly determined (u*
*p to
homotopy) by the condition that . E = R . This observation leads to a simple p*
*roof of
the following.
4.9 Proposition. Let R = Z[ia], where a 1 and a 2 if ` = 2. Let E be a resid*
*ue
field of R with (R) ~=(E), and let = R . s : LB GL (E)+!- LB GL (R)+ be the m*
*ap
provided by (4.1). Then commutes up to homotopy with the Dyer-Lashof action ma*
*ps
of LB GL (E)+ and LB GL (R)+ derived from the infinite loop structures of these*
* spaces.
Proof. We indicate the proof briefly. Let X = LQo(B(R)+ ), Y = LB GL (E)+ , a*
*nd
Z = LB GL (R)+ . For any space W let DnW = En xn W n(n 1). Construct the
diagram
DnE Dn
DnX ----! DnY ----! DnZ
?? ? ?
y ?y ?y
E
X ----! Y ----! Z
in which the vertical maps are Dyer-Lashof action maps arising from the appropr*
*iate
infinite loop space structures. The left hand square commutes up to homotopy b*
*ecause
the map E is an infinite loop map. The large rectangle obtained by omitting the*
* central
column commutes up to homotopy because R = . E and the map R is an infinite
loop map. Commutativity of the right hand square now follows from the fact that*
* DnE
has a right inverse up to homotopy, namely, Dns.
4.10 Fewer roots of unity. Theorem 4.1 easily leads to a proof at odd primes of*
* the
original [6] factorization and splitting results of the third author, a proof w*
*hich in its
totality is slightly different in detail from (but identical in concept to) the*
* proof in [6].
Rather than formulate the most general statement we will just give one example.
GENERALIZED BURNSIDE RING 13
4.11 Theorem. [6] Assume that ` is odd and that p is a prime which generates th*
*e `-adic
units. Then there is a map : LB GL (Fp)+!- LB GL (Z)+ such that the composite
Fp
LQ0(B{1}+ ) ~=LQ0S0 --! LB GL (Fp)+!- LB GL (Z)+
is homotopic to the map Z : LQ0S0!- LB GL (Z)+ .
Proof. Let R = Z[i1] and let = (R). The hypothesis on p implies that there is*
* a
unique prime of R above p, and we will let E be the corresponding residue class*
* field of R,
so that (E) ~=. Since Z[i1] is a free module of rank ` - 1 over Z and ` - 1 is *
*relatively
prime to `, the transfer construction in algebraic K-theory can be used to prod*
*uce a
map t : LB GL (R)+!- LB GL (Z)+ which is a left inverse up to homotopy to the*
* map
iZ : LB GL (Z)+!- LB GL (R)+ induced by the ring map Z ! R. Let iFp : LB GL (F*
*p)+!-
LB GL (E)+ be the map induced by Fp!- E, and let Z : LB GL (Fp)+!- LB GL (Z)+*
* be
the composite
iFp R t
LB GL (Fp)+ --! LB GL (E)+ --! LB GL (R)+!- LB GL (Z)+
where R is the map R . s provided by (4.1). A simple direct calculation which c*
*ombines
(4.1) and naturality shows that Z = has the desired property.
4.12 Even more roots of unity. It is natural to ask for an analogue of (4.1) in*
* the
situation in which all `-primary roots of unity are present. We can offer the *
*following
somewhat unsatsfactory result along these lines.
4.13 Theorem. Let R be the ring Z[ii|i 1], let p be a prime which generates the
`-adic units, and let E be the quotient field R=p ~=Fp[ii|i 1]. Then there exi*
*sts a map
R : LB GL (E)+!- LB GL (R)+ , such that, for any finite subgroup of (R) the *
*two
maps
R ; R . E : LQ0(B+ )!- LB GL (R)+
are homotopic.
Proof. Let Rn = Z[in], En = Rn=p, n = (Rn), and let n : LB GL (En)+ !-
LB GL (Rn)+ denote the map provided up to homotopy by (4.1). There are diagrams
En n
LQ0(Bn+) ----! LB GL (En)+ ----! LB GL (Rn)+
?? ? ?
y ?y ?y
En+1 n+1
LQ0(Bn+1+) ----! LB GL (En+1)+ ----! LB GL (Rn+1)+
in which the left hand square evidently commutes up to homotopy, as does the la*
*rge
rectangle obtained by removing the center column. It thus follows from remarks *
*above (4.8)
that the right hand square commutes up to homotopy. After chosing if necessary *
*models
for the spaces LB GL (En)+ so that the natural maps LB GL (En)+ !- LB GL (Em *
*)+
14 DWYER, FRIEDLANDER AND MITCHELL
(m > n) are represented by cofibrations, it is now easy to construct by inducti*
*on on n a
commutative ladder (of genuine maps)
LB GL (E1)+ ----! LB GL (E2)+ ----! . . .----! LB GL (En)+ ----! . . .
? ? ?
1 ?y 2 ?y n ?y
LB GL (R1)+ ----! LB GL (R2)+ ----! . . .----! LB GL (Rn)+ ----! . . .
such that n is homotopic to n. The mapping telescope of the upper rail of this *
*diagram
is equivalent to LB GL (E)+ and the mapping telescope of the lower rail to LB G*
*L (R)+ .
It is easy to see that the induced map between these telescopes is the desired *
*map R .
The reader may wonder about the restriction to finite subgroups (R) in the*
* above
theorem. To remove this restriction it would be natural to try to add a third (*
*upper) rail
to the above ladder, a rail of the form
LQ0(B1+)!- LQ0(B1+)!- . .L.Q0(Bn+)!- . .:.
This new rail would be connected to the now center one by the maps En and to the
lower one by the maps Rn . We are unable to construct such an extended ladder f*
*or the
following reason. Let An, Bn and Cn denote the n'th spaces in respectively the*
* upper,
middle, and lower rails. The most direct approach to making`the desired constr*
*uction
gives inductively, at the n'th`stage, a map gn : An+1 AnBn!- Cn+1 and it is *
*required
to extend gn over a map An+1 AnBn!- Bn+1 to a map Bn+1!- Cn+1. The homotopy
class of maps Bn+1!- Cn+1 provided by (4.1) extends the restriction of gn to A*
*n+1 and
also extends the restriction`of gn to Bn, but this is not enough. The difficul*
*ty is that
the restriction map [An+1 AnBn; Cn+1]!- [An+1; Cn+1] x [Bn; Cn+1] is not nec*
*essarily
injective.
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* Lecture Notes in
Math. 304, Springer, Berlin, 1972.
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[7]S.A. Mitchell, On the Lichtenbaum-Quillen conjectures from a stable homotopy*
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GENERALIZED BURNSIDE RING 15
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University of Notre Dame, Notre Dame, Indiana 46556
Northwestern University, Evanston, Illinois 60208
University of Washington, Seattle, Washington 98195
Processed March 17, 19*
*92