The product formula in unitary deformation Ktheory
Tyler Lawson (tlawson@math.mit.edu) *
Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract. For finitely generated groups G and H, we prove that there is a weak
equivalence KG ^kuKH ' K(G x H) of kualgebra spectra, where K denotes the
"unitary deformation Ktheory" functor. Additionally, we give spectral sequences
for computing the homotopy groups of KG and HZ ^kuKG in terms of PU (n)
equivariant Ktheory and homology of spaces of Grepresentations.
1. Introduction
To a finitely generated group G one associates the category C of finite
dimensional unitary representations of G, with equivariant morphisms.
Elementary methods of representation theory allow this category to be
analyzed; explicitly, the category of unitary Grepresentations is natu
rally equivalent to a direct sum of copies of the (topological) category
of unitary vector spaces.
However, there is more structure to C. First, there is a bilinear tensor
product pairing. Second, C can be given the structure of an internal
category Ctop in Top . This means that there are spaces Ob (Ctop) and
Mor (Ctop), together with continuous domain, range, identity, and com
position maps, satisfying appropriate associativity and unity diagrams.
The topology on Ob (Ctop) reflects the possibility that homomorphisms
from G to U(n) can continuously vary from one isomorphism class
of representations to another. The identity gives a continuous functor
C ! Ctop which is bijective on objects.
Both of these categories have notions of direct sums and so are
suitable for application of an appropriate infinite loop space machine.
This yields a map of (ring) spectra as follows:
`
KC ' ku ! KCtop= KG . (1)
Irr(G)
Here K is the algebraic Ktheory functor, ku is the connective Ktheory
spectrum, and Irr(G) is the set of irreducible unitary representations of
G. Note that ss*(KC) ~=R[G] Z[fi] as a ring, where R[G] is the unitary
representation ring of G.
The spectrum KG is the unitary deformation Ktheory of G. It differs
from the C*algebra Ktheory of G  for example, in section 8, we find
________*
Partially supported by NSF award 0402950.
cO2005 Kluwer Academic Publishers. Printed in the Netherlands.
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that the unitary deformation Ktheory of the discrete Heisenberg group
has infinitely generated ss0.
When G is free on n generators, one can directly verify the formula
_ n !
`
KG ' ku _ ku .
A more functorial description in this case is that KG is the connective
cover of the function spectrum F (BG+ , ku). Even in simple cases KG
can be difficult to directly compute, such as when G is free abelian on
multiple generators.
In this paper, we will prove the following product formula for unitary
deformation Ktheory.
THEOREM 1. The tensor product map induces a map of kualgebra
spectra KG ^kuKH ! K(G x H), and this map is a weak equivalence.
The reader should compare the formula
R[G] R[H] ~=R[G x H]
for unitary representation rings.
The proof of Theorem 1 proceeds by making use of a natural fil
tration of KG by subspectra KG n. These subspectra correspond to
representations of G whose irreducible components have dimension less
than or equal to n. Specifically, we show in sections 5 and 6 that there
is a fibration sequence of spectra
KG n1 ! KG n ! kuPU (n)(Hom (G, U(n))=Sum (G, n)).
Here Sum (G, n) is the subspace of Hom (G, U(n)) consisting of those rep
resentations G ! U(n) which have a nontrivial invariant subspace. The
spectrum kuPU (n)(X) is a connective PU (n)equivariant Khomology
spectrum for X, discussed in section 3.
As side benefits of the existence of this filtration, Theorems 22 and
24 give spectral sequences for computing the homotopy groups of KG
and the homotopy groups of HZ ^ku KG respectively.
When G is free on k generators, Theorem 24 gives a spectral sequence
converging to Z in dimension 0, Zk in dimension 1, and 0 otherwise,
but the terms in the spectral sequence are highly nontrivial  they are
the homology groups of the spaces of ktuples of elements of U(n),
mod conjugation and relative to the subspace of ktuples which admit
a nontrivial invariant subspace. The method by which the terms in this
spectral sequence are eliminated is a bit mysterious.
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The motivation for studying these deformation Ktheory spectra
comes from algebraic Ktheory.
The underlying goal of many programs in algebraic Ktheory is to
understand the algebraic Kgroups of a field F as being built from the
Kgroups of the algebraic closure of the field, together with the action
of the absolute Galois group. Specifically, Carlsson's program (see [2])
is to construct a model for the algebraic Ktheory spectrum using_the
Galois group and the Ktheory spectrum of the algebraic closure F .
In some specific instances, the absolute Galois group of the field
F is explicitly the profinite completion G^of a discrete group G. (For
example, the absolute Galois group of the field k(z) of rational func
tions, where k is an algebraically closed of characteristic zero, is the
profinite completion of a free group.) In the case where F contains an
algebraically closed subfield, the profinite completion of the deforma
tion Ktheory spectrum KG is conjecturally homotopy equivalent to
the profinite completion of the algebraic Ktheory spectrum KF .
The layout of this paper is as follows. Section 2 gives the necessary
background on spaces acted on by a compact Lie group G to identify
equivariant smash products. The model theory of such functors was
considered when G is a finite group in [4], using simplicial spaces. Our
approach to the proofs of the results we need follows the approach
of [3]. Section 3 gives explicit constructions of an equivariant version
of connective Ktheory, and in section 4 the unitary deformation K
theory of G is defined. Sections 5 and 6 are devoted to constructing the
localization sequences for deformation Ktheory, and in particular ex
plicitly identifying the base as an equivariant smash product. In section
7 the algebra and module structures are made explicit by making use
of results of Mandell and Elmendorf. The main theorems are proved in
sections 8 and 9.
A proof of the product formula for representations in GL (n), rather
than U(n), would also be desirable. This paper makes use of quite rigid
constructions that make apparent the identification of the base in the
localization sequence with a particular model for the equivariant smash
product. In the case of GL (n), the definitions of both the cofiber in the
localization sequence and the equivariant smash product need to be
replaced by notions that are more wellbehaved from the point of view
of homotopy theory.
2. Preliminaries on Gequivariant spaces
In this section, G is a compact Lie group, and actions of G on based
spaces are assumed to fix the basepoint; a free action will be one that
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is free away from the basepoint. We will now carry out constructions
of spaces in a na"ive equivariant context. When G is trivial, these are
the standard definitions for spaces.
For any natural number k, denote the based space {*, 1, . .,.k} by
k+ .
Let oGbe the category of right Gspaces which are a finite wedge of
the form _G+ with morphisms being Gequivariant. (Strictly speaking,
we take a small skeleton for this category.) The set oG(X, Y ) can
be given the mapping space topology, which gives this category an
enrichment in spaces. Explicitly,
Yk
oG(_kG+ , Z) ~= Z
as a space. We refer to the opposite category as G . If G is trivial we
drop it from the notation.
Definition 2.A G space is a basepoint preserving continuous func
tor oG! Top *.
Here Top * is the category of compactly generated weak Hausdorff
spaces with nondegenerate basepoint.
Any G space M has an underlying space M(G+ ^ ). This 
space inherits a continuous left Gaction because the left action of G
on the first factor of G+ ^ Y is right Gequivariant. In other words,
there is a continuous composite homomorphism
G ! oG(G+ ^ Y, G+ ^ Y ) ! F (M(G+ ^ Y ), M(G+ ^ Y )).
Remark 3. More generally, if H ! G is a map of groups, the formula
M 7! M( ^H G+ ) defines a restriction map from G spaces to H 
spaces with a left action of the Weyl group NH=H.
Let X, Z 2 oG, Y a based set. The continuous Gequivariant left ac
tion of G on G+ ^ Y , acting on the lefthand factor alone, gives rise to a
continuous right action of G on oG(G+ ^ Y, Z). There is a map OE : X !
oG(G+ ^ Y, X ^ Y ) given by x 7! OEx, where OEx(g ^ y) = xg ^ y. The
map OE is clearly Gequivariant. Composing this map with the functor M
gives a continuous Gequivariant map X ! F (M(G+ ^ Y ), M(X ^ Y )),
and the adjoint is a natural map X ^G M(G+ ^ Y ) ! M(X ^ Y ).
For any based right Gspace X and G space M, we can functorially
form a new space X G M as follows. First, X defines a functor
F G( , X) : G ! Top *.
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Definition 4.The space X G M is defined as follows:
0 1
a
X G M(Z) = @ M(Y ) ^F G(Y, X ^ Z)A = ~
Y 2 oG
Here the equivalence relation ~ is generated by relations (u, f*v) ~
(f*u, v) for f : Y ! Y 0, u 2 M(Y ), v 2 F G(Y 0, X ^ Z). More concisely,
X G M(Z) can be expressed as the coend
Z Y
M(Y ) ^F G(Y, X ^ Z).
Remark 5. If X. is a simplicial object in the category oG, there is a
natural homeomorphism X. G M ! M(X.).R(A short proof can be
given by expressing X. as a coend n Xn ^ n+and applying "Fubini's
theorem"  see [10], chapter IX.) The reason for allowing GCW com
plexes rather than simply restricting to these simplicial objects is that
some Ghomotopy types cannot be realized by simplicial objects. For
example, any such simplicial object of the form X+ is a principal G
bundle over X=G+ , and is classified by an element in H1discrete(X=G, G).
A general X+ is classified by an element in H1cont.(X=G, G).
We will refer to the space (X G M)(1+ ) as M(X); this agrees with
the notation already defined when X 2 oG. We will only apply this
construction to cofibrant objects in a certain model category of based
Gspaces; specifically, we will only apply this construction to based G
CW complexes with free action away from the basepoint. Such objects
are formed by iterated cell attachment of G+ ^ Dn+along G+ ^ Sn1+.
It will be useful to have homotopy theoretic control on X G M, for
the purposes of which we introduce a less rigid tensor product.
Definition 6.For M a G space, we can define a simplicial G space
LM. by setting LM(Z)p equal to
`
oG(Zp, Z) ^ oG(Zp1, Zp) ^. .^. oG(Z0, Z1) ^M(Z0).
Z0,...,Zp
The face maps are given by:
di(fp ^. .^.f0 ^m) = fp ^. .^.fiO fi1 ^. .^.f0 ^m if i < p
dp(fp ^. .^.f0 ^m) = fp ^. .^.f1 ^(Mf0)(m)
The degeneracy map si is insertion of an identity map after fi for
0 i p.
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The simplicial G space LM.has a natural augmentation LM.! M.
The augmented object LM. ! M has an extra degeneracy map s1 ,
defined by
s1 (fp ^. .^.f0 ^m) = id ^fp ^. .^.f0 ^m.
As a result, the map LM.(Z) ! M(Z) is a homotopy equivalence for
any Z 2 oG. (Note that LM. is the bar construction B( oG, oG, M).)
For X a right Gspace, consider the simplicial G space X G LM..
We have
` h i
X G LMp = X G oG(Zp, ) ^ oG(Zp1, Zp) ^. .^.M(Z0)
Z0,...,Zp
as a space; the tensor construction distributes over wedge products
and commutes with smashing with spaces. However, a straightforward
calculation yields the formula
h i
X G oG(Y, ) (Z) ~=F G(Y, X ^ Z),
the space of Gequivariant based functions from Y to X ^ Z.
PROPOSITION 7. For X a GCW complex with free action away
from the basepoint, the augmentation map X G LM. ! X G M
is a levelwise weak equivalence of spaces after realization.
Proof. It suffices to prove that LM.(X) ! M(X) is a weak equiv
alence for any GCW complex X. We will prove this by showing that
it is filtered by weak equivalences.
Suppose X is a GCW complex. For any n 2 N, define a restricted
mapping space F (Y, X)(n) to be the subspace of F (Y, X) consisting
of those functions whose image contains representatives of at most
n distinct Gorbits of X. The space M(X) is the limit of a natural
sequence of subspaces M(X)(n), which are defined by
_ !
a
M(X)(n)= M(Y ) ^F G(Y, X)(n) = ~,
Y
where the equivalence relation is the same as that defining M(X).
For any n > 0, this yields the following natural pushout square:
F (n+ , X)(n1)^ n RG M(G+ ^ n+ ) _____//F (n+ , X) ^ n RG M(G+ ^ n+ )
 
 
fflffl fflffl
M(X)(n1) ___________________________//_M(X)(n)
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R
Here n G is the wreath product Gn o n. X is a GCW complex,
so the horizontal arrows are cofibrations. This identifies the cofiber of
the map M(X)(n1) ! M(X)(n)with the space
C(n, X) ^R M(G+ ^ n+ ).
n G
Here C(n, X) is the quotient of the space Xn by the subspace consisting
of elements (x1, . .,.xn) where xi = * for some i or xi = gxj for some
g 2 G,Ri 6= j. Since X was a free GCW complex, C(n, X) admits a free
( n G)CW complex structure.
We can applying this same construction to LM.(X); the space
LM.(X)(n) is the realization of the simplicial space LM.(X)(n). The
cofiber of the corresponding map LM.(X)(n1) ! LM.(X)(n)is the
geometric realization
fifi fi
fifiC(n, X) ^ LM (G ^ n )fifi~C(n, X) ^ LM (G ^ n ).
fi n RG . + + =fifi n RG . + +
There is a map of cofibration sequences as follows:
R
LM.(X)(n1) _____//LM.(X)(n)_____//C(n, X) ^ n G LM.(G+ ^ n+ )
  
  
fflffl fflffl Rfflffl
M(X)(n1) ________//M(X)(n)________//C(n, X) ^ n G M(G+ ^ n+ )
To show inductively that LM.(X)(n)! M(X)(n)is a weak equiv
alence, it therefore suffices to show that the righthandRvertical map is
a weak equivalence for allRn. C(n, X) is a cofibrant ( n G)space, so
smashing with it over n G preserves weak equivalences. The result
follows because LM.(G+ ^ n+ ) ! M(G+ ^ n+ ) is a weak equivalence
for all n.
COROLLARY 8. If a map X ! Y of free GCW complexes is k
connected, so is the map M(X) ! M(Y ).
Proof. It suffices to show that the map LM.(X) ! LM.(Y ) is k
connected. However, this is a map of simplicial spaces which levelwise
is of the form
W G o o
Z0,...,ZpF (Zp, X) ^ G (Zp1, Zp) ^ . .^. G (Z0, Z1) ^ M(Z0)


W fflffl
Z0,...,ZpF G(Zp, Y ) ^ oG(Zp1, Zp) ^ . .^. oG(Z0, Z1) ^ M(Z0).
This map is kconnected because the map F G(Zp, X) ! F G(Zp, Y )
is. Since LM.(X) ! LM.(Y ) is kconnected levelwise, so is the map of
geometric realizations.
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For any G space M, we have an associated (na"ive pre)spectrum
{M(G+ ^ Sn )}, which is the same as the spectrum of the underlying
space M(G+ ^ ). A map of G spaces M ! M0 is called a stable
equivalence if the associated map of spectra is a weak equivalence.
PROPOSITION 9. For any G space M and free based GCW complex
X, the map X ^G M(G+ ^ ) ! X G M is a stable equivalence.
Proof. It suffices to show that X ^G M(G+ ^ Sn ) ! M(X ^ Sn )
is highly connected for large n. Using the levelwise weak equivalence
X G LM. ! X G M, it suffices to show that this statement is true
for spaces of the form oG(Y, ) for Y 2 oG.
In this case, we have the following diagram.
W n W n
X ^G Y(G+ ^ S ) ______//Y X ^G (G+ ^ S )
 
 
Q fflffl Q fflffl
X ^G Y(G+ ^ Sn ) _________//_Y(X ^ Sn )
 
 
 
X ^G oG(Y, G+ ^ Sn ) _______//_ oG(Y, X ^ Sn )
The top vertical arrows are isomorphisms on homotopy groups up to
roughly dimension 2n since G+ ^ Sn is (n  1)connected. The upper
most horizontal arrow is an isomorphism. Therefore, the bottom map
is an equivalence on homotopy groups up to roughly dimension 2n, as
desired.
3. Connective equivariant Khomology
In this section, we construct for each n a PU (n)space whose underly
ing spectrum is homotopy equivalent to ku, the connective Ktheory
spectrum.
Let W be a fixed ndimensional inner product space. For any d 2 N,
we have the Stiefel manifold V (nd) of isometric embeddings of W Cd
into C1 , where Cd has the standard inner product. The space V (nd)
has a free right action by I U(d)`by precomposition.`Denote the quo
tient space by H(d), and write H = dH(d). (Note H ' dBU (d).)
H has a partially defined direct sum operation: if {Vi} is a family of
elements of H such that Vi ? Vj for i 6= j, there is a sum element Vi
in H.
There is also an action of U(n) I on V (nd) that commutes with
the action of I U(d), hence passes to an action on the quotient H(d).
Since ~I I = I ~I, the scalars in U(n) act trivially on H(d), so the
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action factors through PU (n). We therefore get a right action of PU (n)
on H. The direct sum operation is PU (n)equivariant.
For any Z 2 oG, define
n fifi o
kuPU (n)(Z) = f 2 F PU(n)(Z, H) fif(z) ? f(z0) if[z] 6= [z0].
A point of kuPU (n)(Z) consists of a vector space isomorphic to W dimf(z)
associated to each nonbasepoint [z] of Z=G such that the vector spaces
associated to [z] and [z0] are orthogonal if [z] 6= [z0].
Given a map ff 2 oG(Z, Z0) and f 2 kuPU (n)(Z), we get an element
kuPU (n)(ff)(f) 2 kuPU (n)(Z0) as follows:
M
kuPU (n)(ff)(f)(z0) = f(z)
ff(z)=z0
This is welldefined: if the preimage of z is the family zi, then the zi all
lie in distinct orbits. The map kuPU (n)(ff)(f) is also clearly PU (n)
equivariant, and takes distinct orbits to orthogonal elements of H.
The spectrum attached to the underlying space of kuPU (n)is weakly
equivalent to the connective Ktheory spectrum ku  see [17].
We also define a second G space ku=fi as follows. For any z 2 oG,
ku=fi(Z) = "N[Z=G].
More explicitly, ku=fi(Z) is the quotient of the free abelianPmonoid on
Z=GPby the submonoid N[*]. For ff 2 oG(Z, Z0), ku=fi(ff)( nz[z]) =
nz[ff(z)]. (The reason for the notation is that the underlying spec
trum is the cofiber of the Bott map  this will be made more explicit
in section 7.)
For X a free right PU (n)space, X PU (n)ku=fi is the infinite sym
metric product Sym 1 (X=PU (n)).
There is a natural map ffl : kuPU (n)! ku=fi of G spaces: if f 2
P i dimf(z)j
kuPU (n)(Z), define ffl(f) = [z] _______n[z]. The map ffl represents the
augmentation ku ! HZ on the underlying spectra, the first stage of
the Postnikov tower for ku.
The G space kuPU (n)determines a homology theory for PU (n)
spaces. Specifically, we can define
i j
kuPU*(n)(X) = ss* X PU (n)kuPU (n) .
Here ss* denotes the stable homotopy groups of the spectrum. In fact,
since the underlyingispectrum ofjkuPU (n)is special, we can compute
kuPU*(n)(X) = ss* kuPU (n)(X) for X connected. (See [16], 1.4.)
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4. Unitary deformation Ktheory
In this section we will have a fixed finitely generated discrete group G.
Carlsson, in [2], defined a notion of the "deformation Ktheory" of G as
a contravariant functor from groups to spectra, and in the introduction
of this article an analogous notion of "unitary deformation Ktheory"
KG was sketched. The following are weakly equivalent definitions of the
corresponding notion of KG :
`
 The group completion of nEU (n) xU(n)Hom (G, U(n)).
 The Ktheory of a category of unitary representations of G. This
category is an internal category in Top : i.e., the objects and mor
phism sets are both given topologies.
 The simplicial object which is the Ktheory of the singular complex
of the category above. (This is essentially the definition given in
[2].)
We will now describe another model for the unitary deformation K
theory of G, equivalent to the first definition given above. The construc
tion is based on the construction of connective topological Khomology
of Segal in [17]. (Also see [18].)
Let U = C1 be the infinite inner product space with orthonormal
basis {ei}. The group U = colim U(n) acts on U. A Gplane V of
dimension k is a pair (V, ae) where V is a kplane in U and ae : G ! U(V )
is an action of G on V .
We now describe a (nonequivariant) space KG . Define
n fifi o
KG (X) = (Vx, aex)x2X fiVx a Gplane, Vx ? Vx0 ifx 6= x0, V* =.0
This is a special space. The underlying Hspace is
a
KG (1+ ) ' V (n) xU(n) Hom ( , U(n)),
n
where V (n) is the Stiefel manifold of nframes in U. We will now de
scribe the simplicial space X. = KG (S1). Since KG is special, X. '
1 KG .
For p > 0, Xp is the space
n fifi o
(Vi, aei)pi=1fi(Vi, aei) a Gplane, Vi ? Vj ifi.6= j
(X0 is a point.) Face maps are given by taking sums of orthogonal G
planes or removing the first or last Gplane. Degeneracy maps are given
by insertion of 0dimensional Gplanes.
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The geometric realization of this simplicial space can be explicitly
identified. Let Y be the space of pairs (A, ae), where A 2 U and ae
is a homomorphism G ! U commuting with A. Call two such el
ements (A, ae) and (A0, ae0) equivalent if A = A0 and ae, ae0 agree on
all eigenspaces of A corresponding to eigenvalues ~ 6= 1. Write the
standard psimplex p as the set of all 0 t1 . . . tp 1. Then
there is a homeomorphism X. ! (Y= ~) given by sending a point
((Vi, aei)pi=1, 0 t1 . . . tp 1) of Xp x p to the pair (A, ae),
where A acts on Vi with eigenvalue e2ssitiand by 1 on the orthogonal
complement of Vi, while ae acts on Vi by aei and acts by 1 on the
orthogonal complement of Vi. This map is a homeomorphism by the
spectral theorem. (The essential details of this argument are from [8]
and [12].)
We will refer to this space X. ~=(Y= ~) as E. It is space 1 of the
spectrum associated to KG, in the sense that 1 KG ' E.
This method is applicable to various other categories of representa
tions of G that we will now examine in detail.
For any i 0, there is a sub space KG i of KG such that KG i(X)
consists of those elements of (Vx, aex)x2X of KG (X) such that aex breaks
up into a direct sum of irreducible representations of dimension less
than or equal to i. Each KG i is a special space.
We have infinite loop spaces Ei = KG i(S1). For any i 2 N, Ei is
the subspace of E which is the image of the space of pairs (A, ae) such
that ae is a direct sum of irreducible representations of G of dimension
less than or equal to i.
This gives a sequence of inclusions
* = E0 E1 E2 . . .
of infinite loop spaces. Each of these inclusions is part of a quasifibration
sequence Ei1 ! Ei ! Bi where the base spaces will be explicitly
identified. This gives rise to the following "exact couple" of spectra
* _____________//_E1_____________//E2_____________//_E3`.`.?.?``B``B
?? __ BB ___ BB xxx
O?? ___ OBB ___ O BB xxx
? ""___ B ""__ B xx
B1 B2 B3
Additionally, the inclusions of infinite loop spaces Ei are induced
by maps of kumodules, so the above is induced by an exact couple of
kumodule spectra.
The intuition for the description of Bi is that the category of G
representations whose irreducible summands have dimension less than
i forms a Serre subcategory of the category of Grepresentations whose
irreducibles have dimension less than or equal to i, and the quotient
category should be the category of sums of irreducible representations
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12
of dimension exactly i. The topology on the categories involved compli
cates the question of when a localization sequence of spectra exists in
this situation, since the most obvious attempts to generalize of Quillen's
Theorem B would not be applicable. We will construct the localization
sequence explicitly.
There is a quotient space Fi of KG i by the equivalence relation
(Vx, aex)x2X ~ (Vx0, ae0x)x2X if for all x 2 X:
 The subspace Wx of Vx generated by irreducible subrepresentations
of ae of dimension i coincides with the corresponding subspace for
ae0.
 ae and ae0 agree on Wx.
Again, Fi is a special space.
Define Bi to be the space Fi(S1). Bi is the quotient of Ei by the
following equivalence relation. We say (A, ae) ~ (A0, ae0) if:
 The subspace W of U generated by irreducible subrepresentations
of ae of dimension i is the same as the subspace of U generated by
irreducible subrepresentations of ae0 of dimension i.
 ae and ae0 have the same action on W .
 A and A0 have the same action on W .
Note that each equivalence class contains a unique pair (A, ae) such
that ae acts trivially on the eigenspace of A for 1 and on the com
plementary subspace ae is a direct sum of irreducible idimensional
representations.
5. Proof of the existence of the localization sequence
The proof that pi : Ei ! Bi is a quasifibration (and hence induces a
long exact sequence on homotopy groups) proceeds inductively using
the following result of Hardie [7]:
THEOREM 10. Suppose that we have a diagram
Q oo_h_f*(E)_ _____//E
~ s p
fflffl fflffl fflffl
Q0 oo_g___A ___f___//B
where f is a cofibration, p is a fibration, f*(E) is the pullback
fibration, and ~ is a quasifibration. If h : s1 (a) ! ~1 (ga) is a
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weak`equivalence`for all a 2 A, then the induced map of pushouts
Q f*(E)E ! Q0 A B is a quasifibration.
PROPOSITION 11. The map pi : Ei ! Bi is a quasifibration with
fiber Ei1.
Remark 12. This is what we might expect, as the map Ei ! Bi is
precisely the map which forgets the irreducible subrepresentations of
dimension less than i. The fact that Ei1 is the honest fiber over any
point is clear, but we need to show that Ei1 is also the homotopy
fiber.
Proof. We will proceed by making use of a rank filtration. These
rank filtrations were introduced in [12] and [14]. In particular, Mitchell
explicitly describes this rank filtration for the connective Ktheory
spectrum.
For any j, let Bi,jbe the subspace of Bi generated by those pairs
(A, ae) such that ae contains at most a sum of j irreducible representa
tions. There is a sequence of inclusions Bi,j1 Bi,j. Write Ei,jfor the
subset of Ei lying over Bi,j.
The map Ei,0! Bi,0is a quasifibration, since Bi,0is a point. Now
suppose inductively that Ei,j1! Bi,j1is a quasifibration.
Let Yj be the space of triples (A, ae, W ), where W is an ijdimensional
subspace of U, A is an element of U(W ), and ae is a representation of
G on W commuting with A and containing irreducible summands of
dimension i or less. Let Xj be the subset of Yj of triples (A, ae, W ) such
that (A, ae) represents a pair in Bi,j1; in other words, ae contains less
than j distinct idimensional irreducible summands on the orthogonal
complement of the eigenspace for 1 of A.
Next, we define a space Yj0of triples (A, ae, W ), where (A, ae) 2 Ei,j
and W is an A and aeinvariant ijdimensional subspace of U containing
all the idimensional irreducible summands of ae. There is a map Yj0!
Yj given by forgetting the actions of A and ae off W . Let X0jbe the fiber
product of Xj and Yj0over Yj; it consists of triples (A, ae, W ) where ae
contains less than j distinct idimensional summands.
There is a map Xj ! Bj1 given by sending (A, ae, W ) to (A, ae), and
a similar map X0j! Ej1. These maps all assemble into the diagram
below.
Ei,j1oo___X0j _____//Yj0
pi  p
fflffl fflffl fflffl
Bi,j1 oo___Xj _____//Yj
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14
There is an evident map from the pushout of the bottom row to
Bi,j, and similarly a map from the pushout of the top row to Ei,j.
The map Xj ! Bi,j1is a quotient map; two points become iden
tified by forgetting the "framing" subspace W , the nonidimensional
summands of ae, and the summands of ae on the eigenspace for 1 of
A. For points of Yj not in Xj, the framing subspace W is determined
by the image (A, ae) in Bj since ae must have j distinct idimensional
irreducible summands covering all of W , and A can have no eigenspace
for the eigenvalue 1. Therefore, the map from Yj to the pushout of
the bottom row is precisely the quotient map gotten by forgetting the
framing W and any nonidimensional summands or summands lying
on the eigenspace for 1 of A. However, this identifies the pushout with
Bi,j. In exactly the same way, the pushout of the top row is Ei,j. The
induced map of pushouts is the projection map Ei,j! Bi,j.
The map Xj ! Yj is a cofibration because it is the colimit of geomet
ric realizations of a closed inclusion of real points of algebraic varieties.
(The subspace W is allowed to vary over the infinite Grassmannian. If
we restrict its image to any finite subspace we get an inclusion of real
algebraic varieties.)
The righthand square is a pullback by construction, and the map
pi is assumed to be a quasifibration.
The map p is a fiber bundle with fiber Ei1: An equivalence class
of points (A, ae, W ) 2 Yj0consists of a choice of ijdimensional subspace
W of U, a choice of element in (A~, ~ae, W ) in Yj to determine the action
of A and ae on W , and a choice of (A0, ae0) acting on the orthogonal
complement of W such that ae0 is made up of summands of dimension
less than i. In other words, there is a pullback square:
Yj0_______//V
 
 
fflffl fflffl
Yj _____//Gr(ij)
Here Gr (ij) is the Grassmannian of ijdimensional planes in U, and
V is the bundle over the Grassmannian consisting of ijdimensional
planes in U and elements of Ei1 acting on their orthogonal comple
ments.
Given any point (A, ae, W ) of Xj, the fiber in X0jis Ei1 acting on the
orthogonal complement of W . Suppose that (A, ae) in Bi,j1is in canon
ical form: ae acts by a sum of irreducible dimension i representations on
some subspace W 0 W and trivial representations on the orthogonal
complement, and A has eigenvalue 1 on the orthogonal complement of
W 0. Then the fiber over (A, ae) in Ei,j1consists of all possible actions
of Ei1 on the orthogonal complement of W 0. The map from the fiber
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15
over (A, ae, W ) to the fiber over (A, ae) is the inclusion of Ei1 acting on
W ? to Ei1 acting on (W 0)? . This inclusion is a homotopy equivalence.
Therefore, Ei,j! Bi,jis a quasifibration with fiber Ei1. Taking
colimits in j, Ei ! Bi is a quasifibration with fiber Ei1.
COROLLARY 13. The maps KG i1 ! KG i! Fi realize to a fibration
sequence in the homotopy category of spectra.
Proof. This follows because all three of these spaces are special.
6. Identification of the space Fn
Using the results of section 2, we will now identify the spaces Fn as
equivariant smash products.
Let Sum (G, n) be the subspace of Hom (G, U(n)) of reducible G
representations of dimension n. Define Rn = Hom (G, U(n))=Sum (G, n).
There is a free action of PU (n) on Rn by conjugation. According to
a result of Park and Suh ([13], Theorem 3.7), the algebraic variety
Hom (G, U(n)) admits the structure of a U(n)CW complex. Since all
isotropy groups of Rn contain the diagonal subgroup, this structure is
actually the structure of a PU (n)CW complex, and Rn has an induced
CW structure.
PROPOSITION 14. There is an isomorphism of spaces
Rn PU (n)kuPU (n)! Fn.
Proof. By the universal property of the coend
Z Y
Rn PU (n)kuPU (n)(Z) = kuPU (n)(Y ) ^F G(Y, Rn ^ Z),
we can construct the map by exhibiting maps
kuPU (n)(Y ) ^F G(Y, Rn ^ Z) ! Fn(Z),
natural in Z, that satisfy appropriate compatibility relations in Y .
Recall that`a point of kuPU (n)(Y ) consists of an equivariant map
f : Y ! H = V (nd)=I U(d) such that f(y) ? f(y0) if y 6= y0.
Suppose f ^ g 2 kuPU (n)(Y ) ^ F G(Y, Rn ^ Z). For every y 2 Y the
element g(y) = r(y) ^ z(y) determines an irreducible action r(y) of G on
Cn. The element f(y) 2 V (nd)=I U(d) is the image of some element
fg(y)2 V (nd), which determines an isometric embedding W Cd ! U.
Combining these two gives an action of G on an ndplane of U, together
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16
with a marking z(y) of the plane by an element of Z. The action of
I U(d) commutes with the Gaction on W Cd, so the choice of
lift fg(y)does not change the resulting Gplane. For g 2 G, r(gy) =
g . r(y) = gr(y)g1 , and f(gy) = (g I)f(y)(g1 I), so the resulting
plane only depends on the orbit Gy. The resulting Gplane breaks up
into irreducibles of dimension precisely n. Assembling these Gplanes
over the distinct orbits gives a collection of orthogonal hyperplanes
with Gactions, marked by points of Z, which break up into a direct
sum of ndimensional irreducible representations. As r(y) approaches
the basepoint of Rn, the representation becomes reducible, so the map
determines a welldefined element of Fn(Z). The compatibility of this
map with maps in Y is due to the fact that it preserves direct sums.
This map is bijective; associated to any point of Fn(Z) there is a
unique equivalence class of points which map to it. We leave it to the
reader to verify that the inverse map is continuous.
7. E1 algebra and module structures
In this section we will make explicit the following. The tensor product
of representations leads to the following multiplicative structures:
1. ku is an E1 ring spectrum.
2. KG is an E1 algebra over ku.
3. The sequence of maps KG 1 ! KG 2 ! . .!.KG is a sequence of
E1 kumodule maps.
4. There are compatible E1 kulinear pairings KG n ^ KG m ! KG nm
for all n, m.
5. The map KG n ! Rn PU (n)kuPU (n)is a map of E1 kumodules.
All of the above structures are natural in G. From this point onward,
we are only interested in derived categories of module spectra, and so
all smash products are meant in the derived sense.
To begin, we will first recall the definition of a multicategory. A
multicategory is an "operad with several objects", as follows. See [6].
Definition 15.A multicategory M consists of the following:
1. A class of objects Ob (M).
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17
2. A set Mk(a1, . .,.ak; b) for each a1, . .,.ak, b 2 Ob (M), k 0 of
"kmorphisms" from (a1, . .,.ak) to b.
3. A right action of the symmetric group k on the class of all k
morphisms such that oe* maps the set Mk(a1, . .,.ak; b)) to the set
Mk(aoe(1), . .,.aoe(k); b).
4. An "identity" map 1a 2 M1(a; a) for all a 2 Ob (M).
5. A "composition" map
Mn(b1, . .,.bn; c) x Mk1(a11, . .,.a1k1; b1) x . . .
! Mk1+...+kn(a11, . .,.ankn; c)
which is associative, unital, and respects the symmetric group ac
tion.
We will not make precise these last definitions; they are essentially
the same as the definitions for an operad. A map between multicate
gories that preserves the appropriate structure will be referred to as a
multifunctor.
Example 16. Any symmetric monoidal category (C, ) is a multicat
egory, with
Ck(a1, . .,.ak; b) = C(a1 . . .ak, b).
For example, the categories of spaces or symmetric spectra under ^
are multicategories.
There is a (lax) symmetric monoidal functors U from spaces to
symmetric spectra [11]. This naturally leads to a multifunctor from the
multicategory of spaces to the multicategory of symmetric spectra.
Remark 17. The smash product of spaces of simplicial sets is
defined using left Kan extension. As a result, we can equivalently define
a multicategory structure on spaces without reference to the smash
product by declaring the set of kmorphisms from (M1, . .,.Mk) to N
to be the set of collections of maps
M1(Y1) ^. .^.Mk(Yk) ! N(Y1 ^. .^.Yk)
natural in Y1, . .,.Yk.
We will now define two multicategories enriched over topological
spaces. The first multicategory F is a parameter multicategory for
an E1 filtered algebra. The second multicategory M2 is a parame
ter multicategory for maps of E1 modules. Let E(n) be the space of
linear isometric embeddings of U n in U. Together the E(n) form an
E1 operad.
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18
Definition 18.The multicategory F has objects R (the ring), A (the
algebra), and An for n 1 (the algebra filtrations). Define a func
tion filton the objects by R 7! 1, An 7! n, A 7! 1. The spaces of
multicategory maps as defined as follows:
ae Q
Fk(B1, . .,.Bk; C) = ;E(n) ififQfilt(Bi)f>ifilt(C)lt(B
i) filt(C)
Composition is given by operad composition for E.
Definition 19.The multicategory M2 has objects R (the ring), M1,
and M2 (the modules). The mapping space M2k(B1, . .,.Bk; C) is equal
to E(n) in the following cases:
 C = Bi = R
 C = M1 and there is a unique i such that Bi = M1; all other Bi
are equal to R
 C = M2 and there is a unique i such that Bi = M1 or M2; all
other Bi are equal to R
Otherwise the mapping space is ;. Composition is given by operad
composition for E.
PROPOSITION 20. There are multifunctors T : F ! spaces and
Sn : M2 ! spaces, continuous with respect to the enrichment in
spaces, such that:
 T (R) = ku, T (A) = KG , T (An) = KG n
 The images under T of the identity maps in E(1) are the standard
maps ku ! KG 1 ! KG 2 ! . .!.KG
 S(R) = ku, S(M1) = KG n, S(M2) = Fn
 The image under S of the identity map in M2(M1, M2) = E(1)
is the map KG n ! Fn
Proof. Write KG Unfor the space KG n indexed on the universe U.
For groups G1, . .,.Gk, there is a welldefined exterior tensor product of
representations:
k
K(G1)Un1(Z1) ^. .^.K(Gk)Unk(Zk) ! K(G1x. .x.Gk)Un1...nk(Z1 ^. .^.Zk)
Postcomposition with linear isometric embeddings U k ! U then gives
a map of spaces
E(k)+ ^ K(G1)n1 ^. .^.K(Gk)nk ! K(G1 x . .x.Gk).
ggammaarx.tex; 21/03/2005; 11:38; p.18
19
If all Gi are equal to G or the trivial group, we can pull back along the
diagonal map to get a map
E(k)+ ^ B1 ^. .^.Bk ! C,
where the Bi and C are either KG or ku. This map has a continuous ad
joint which defines the multifunctor T . This map preserves composition
and units.
Similarly, the exterior tensor product of Grepresentations with triv
ial representations preserves the dimension of irreducible subrepresen
tations. In the same manner, we get maps
E(k)+ ^ B1 ^. .^.Bk ! Fn,
whenever all Bi are equal to ku except for at most one, and the adjoints
of these maps define the multifunctor S.
These multifunctors are multifunctors of categories enriched in topo
logical spaces. We can now apply Theorem 1.4 of [6] to find weakly
equivalent models which have the structure of strict ring, module, and
algebra spectra. (The singular complex functor must first be applied to
move from the category of spaces to the category of simplicial sets,
and then the spaces must be realized as symmetric spectra.) We will
abuse notation and not change the names. Combining previous results
with this, we have the following.
COROLLARY 21. There exists a ring symmetric spectrum ku and
contravariant functors K()n and K() from finitely generated discrete
groups to connective kumodule symmetric spectra with the following
properties.
 There are kumodule maps KG 1 ! KG 2 ! . .!.KG , and KG
is weakly equivalent to the homotopy colimit.
 There are strictly commutative and associative kumodule pair
ings KG n ^ku KG m ! KG nm which commute with the above maps.
(We allow the case when n or m are equal to 1, using the con
vention KG 1 = KG .)
 For any n, the homotopy cofiber of the map KG n1 ! KG n is
weakly equivalent as a kumodule to kuPU (n)^PU (n)Rn.
We also note that the cofiber sequence 2ku ! ku ! HZ can
be smashed over ku with kuPU (n). The quotient HZ ^ku kuPU (n)is
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20
weakly equivalent as a PU (n)spectrum to the spectrum denoted ku=fi
in section 3.
8. The exact couple for KG
There is the following chain of equivalences of spectra:
_ !
HZ ^ kuPU (n) ^ Rn ' ku=fi ^ Rn ' HZ ^(Rn=PU (n))
ku PU(n) PU(n)
Define QIrr(G, n) = Rn=PU (n). QIrr(G, n) is the quotient space of
isomorphism classes of representations G of dimension n modulo decom
posable representations. (The notation is to avoid confusion with the
standard notation for the subspace of isomorphism classes of irreducible
representations.)
Corollary 21 identifies the following cofiber sequences. The homo
topy colimit of the top row is KG .
* ______//_KG1____________//KG2 ________________//_KG3. . .
  
  
fflffl fflffl fflffl
ku ^ R1 kuPU (2)^PU(2)R2 kuPU (3)^PU(3)R3
The following spectral sequence results.
THEOREM 22. There exists a convergent righthalfplane spectral se
quence of the form
Ep,q1= kuPUq(n)p+1(Rp1) ) ssp+q(KG ).
Remark 23. The grading convention is such that dr maps Ep,qrto
Epr,q+r1r.
Smashing the previous diagram over ku with HZ yields the following.
The homotopy colimit of the top row is HZ ^ku KG .
* ______//_HZ ^ku KG 1_______//_HZ ^ku KG 2______//HZ ^ku KG 3. . .
  
  
fflffl fflffl fflffl
HZ ^ QIrr(G, 1) HZ ^ QIrr(G, 2) HZ ^ QIrr(G, 3)
Again, the diagram results in a spectral sequence.
THEOREM 24. There exists a convergent righthalfplane spectral se
quence of the form
Ep,q1= Hqp+1 (QIrr (G, p  1)) ) ssp+q(HZ ^ KG ).
ku
ggammaarx.tex; 21/03/2005; 11:38; p.20
21
Example 25. When G is finite or nilpotent, the cofiber sequences are
all split. When G is finite, this is clear. When G is nilpotent, results
of [9] show that the space of irreducible representations of dimen
sion n is closed in Hom (G, U(n)), which provides the desired splitting
kuPU (n)^PU (n)Rn ! KG n.
As a result, we have a weak equivalence
_ !
`
KG ' kuPU (n) ^ Rn .
PU(n)
In this case, the spectral sequence of Theorem 24 degenerates at the
E1 page. For example, consider the integer Heisenberg group:
2 3
1 Z Z
4 0 1 Z 5
0 0 1
The E1 = E1 page of the spectral sequence of Theorem 24 is as follows:
 ...

 0 0 0 0 0 0

 0 0 0 0 0 0

 Z 0 0 0 0 0

 Z2 Z 0 0 0 0

 Z Z2 Z 0 0 0 . . .
_________________________________
 Z Z2 Z 0 0

 Z Z2 Z 0

 Z Z2 Z

 ...

Example 26. When G is free on k generators, we can compute the
deformation Ktheory spectrum explicitly. In this case, KG ' ku _
(_k ku).
When G is free on two generators, explicit computations with the
spectral sequence of Theorem 24 give the following picture of the E1
page.
ggammaarx.tex; 21/03/2005; 11:38; p.21
22
 ...

 0 0 ? ?

 0 Z ? ?

 0 Z2 ? ?

 Z Z ? ?

 Z2 0 ? ?

 Z 0 ? ? . . .
______________________
 0 0 ?

 0 0

 ...

The differential d1 : E1,21! E0,21is an isomorphism. The terms Ep,q1
are zero on the set {p > 0, p+q < 2}, and also on the set {q > p2+p+2}.
The terms where q = p2 + p + 2 are all isomorphic to Z.
This spectral sequence converges to Z in dimension 0, Z2 in dimen
sion 1, and 0 in all other dimensions. The classes in E0,01and E0,11are
precisely those classes which survive to the E1 term.
9. Proof of the product formula
In this section we will prove Theorem 1, the product formula for defor
mation Ktheory spectra.
The proof requires the following lemmas.
LEMMA 27. A map M0 ! M of connective kumodule spectra is a
weak equivalence if and only if the map HZ ^ku M0 ! HZ ^ku M is a
weak equivalence.
Proof. By taking cofibers, it suffices to prove the equivalent state
ment that a connective kumodule spectrum M00is weakly contractible
if and only if HZ ^ku M00' *.
However, smashing M00with the cofiber sequence 2ku ! ku ! HZ
of kumodule spectra shows that HZ ^ku M00' * if and only if the Bott
map fi : 2M00! M00is a weak equivalence. This would imply that
the homotopy groups of M00are periodic; since M00is connective, the
result follows.
LEMMA 28. Irreducible unitary representations of G x H are precisely
of the form V W for V, W irreducible unitary representations of G
and H respectively.
ggammaarx.tex; 21/03/2005; 11:38; p.22
23
Proof. The tensor product of two irreducible representations is ir
reducible: Suppose V is an irreducible Grepresentation, and W is an
irreducible Hrepresentation. Then
Hom (V W, V W ) ~=Hom (V, V ) C Hom (W, W ).
Since V is irreducible, Hom G (V, V ) ~=C, given by scalars, and similarly
for W . Let {ei} be a basis for the vector space Hom (W, W ). Suppose
OE is a G x Hlinear endomorphism of V W . Then
X
OE = OEi ei 2 Hom G (V W, V W ) ) OEi 2 C,
so OE = 1 _ for some _ 2 Hom (W, W ). Since OE is also Hlinear, we find
that _ must be Hlinear, so _ is scalar. Therefore, V W is irreducible.
Any irreducible unitary representation of G x H is a tensor product:
Suppose U is an irreducible representation. Since the actions of G and
H commute, any Hisotypic component of U is Ginvariant, so there is
an irreducible unitary representation W of H such that U ~=W das an
Hrepresentation.
Consider the Gvector space Hom H(W, U). There is an irreducible
Grepresentation V (which we do not assume to have an inner product)
and a nonzero Gmap V ! Hom H (W, U). The adjoint of this map is a
nonzero G x Hmap V W ! U. Both sides are irreducible, and hence
this map is an isomorphism of G x Hrepresentations.
The irreducible Gsummands of U all come from a unique isomor
phism class of irreducible Grepresentations. Since the above map is
nonzero, this Grepresentation must be isomorphic to V , and hence
V admits an inner product. (Note that an irreducible representation
admits at most one invariant inner product up to scaling.)
Remark 29. Lemma 28 is precisely the portion of the proof which
fails when we consider representations of G x H in other groups such
as orthogonal groups and symmetric groups.
Proof. [of Theorem 1] The proof consists of constructing a filtration
of the spectrum KG ^ kuKH that agrees with the filtration on K(G xH).
We apply the results of Corollary 21 to get a map of kualgebras as
follows.
KG ^ KH ! K(G x H) ^ K(G x H) ! K(G x H)
ku ku
Similarly, whenever p.q n there is a corresponding map of kumodules
KG p ^ KH q! K(G x H)n.
ku
ggammaarx.tex; 21/03/2005; 11:38; p.23
24
This diagram is natural in p, q, and n. If we define new kumodule
spectra Mn = hocolim p.q nKG p^ kuKH q, then there are induced ku
module maps
fn : Mn ! K(G x H)n.
Since the maps (hocolim KG p) ! KG and (hocolim KH q) ! KH are
weak equivalences, there is a weak equivalence
hocolim Mn ' hocolimp,qKGp^ KH q ' KG ^ KH .
ku ku
Therefore, it suffices to show Mn ! K(G x H)n is a weak equivalence
for all n. We have an induced map of cofiber sequences:
Mn1 ____________//_Mn________________//Mn=Mn1
fn1 fn gn
fflffl fflffl fflffl
K(G x H)n1 _____//K(G x H)n _____//kuPU (n)^PU (n)Rn(G x H)
To prove the theorem inductively it suffices to show that the map
gn is a weak equivalence.
The spectra Mn=Mn1 and kuPU (n)^PU (n)Rn(G xH) are connective
kumodule spectra. Applying Lemma 27, it suffices to prove that the
map HZ ^ku gn is a weak equivalence.
Because the map Mn1 ! Mn is a map from the homotopy colimit
of a subdiagram into the full diagram, we can explicitly compute the ho
motopy cofiber of this map. The homotopy cofiber is weakly equivalent
to the wedge
`
(KG p=KG p1 )^ (KH q=KH q1).
p.q=n ku
The spectra KG p=KG p1 , and the corresponding spectra for H, are
those which were identified as equivariant smash product spectra in
Corollary 13 and Proposition 14. Smashing over ku with HZ gives us
the following identity.
` i j i j
HZ ^ Mn=Mn1 ' HZ ^QIrr (G, p) ^ HZ ^QIrr (H, q)
ku p.q=n HZ
_ !
`
' HZ ^ QIrr(G, p) ^QIrr (H, q)
p.q=n
The map HZ ^ku gn can be identified with the map
i j
HZ ^ _p.q=nQIrr (G, p) ^QIrr (H, q)! HZ ^QIrr (G x H, n)
which is induced by the tensor product of representations. The tensor
product map : _p.q=nQIrr (G, p) ^ QIrr(H, q) ! QIrr(G x H, n) is a
ggammaarx.tex; 21/03/2005; 11:38; p.24
25
continuous map between compact Hausdorff spaces. It is bijective by
Lemma 28. Therefore, it is a homeomorphism.
Acknowledgements
The author would like to thank Gunnar Carlsson, Chris Douglas, Bjorn
Dundas, Haynes Miller, and Daniel Ramras for many helpful conversa
tions, and to Mike Hill for his comments.
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