The product formula in unitary deformation K-theory 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 ku-algebra spectra, where K denotes the "unitary deformation K-theory" functor. Additionally, we give spectral sequences for computing the homotopy groups of KG and HZ ^kuKG in terms of PU (n)- equivariant K-theory and homology of spaces of G-representations. 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 G-representations 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 K-theory functor, ku is the connective K-theory 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 K-theory of G. It differs from the C*-algebra K-theory of G - for example, in section 8, we find ________* Partially supported by NSF award 0402950. cO2005 Kluwer Academic Publishers. Printed in the Netherlands. ggammaarx.tex; 21/03/2005; 11:38; p.1 2 that the unitary deformation K-theory 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 K-theory. THEOREM 1. The tensor product map induces a map of ku-algebra 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 n-1 ! 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 K-homology 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 k-tuples of elements of U(n), mod conjugation and relative to the subspace of k-tuples which admit a nontrivial invariant subspace. The method by which the terms in this spectral sequence are eliminated is a bit mysterious. ggammaarx.tex; 21/03/2005; 11:38; p.2 3 The motivation for studying these deformation K-theory spectra comes from algebraic K-theory. The underlying goal of many programs in algebraic K-theory is to understand the algebraic K-groups of a field F as being built from the K-groups 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 K-theory spectrum using_the Galois group and the K-theory 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 K-theory spectrum KG is conjecturally homotopy equivalent to the profinite completion of the algebraic K-theory 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 K-theory, 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 K-theory, 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 well-behaved from the point of view of homotopy theory. 2. Preliminaries on G-equivariant -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 ggammaarx.tex; 21/03/2005; 11:38; p.3 4 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 G-spaces which are a finite wedge of the form _G+ with morphisms being G-equivariant. (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 base-point 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 G-action because the left action of G on the first factor of G+ ^ Y is right G-equivariant. 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 G-equivariant left ac- tion of G on G+ ^ Y , acting on the left-hand 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 G-equivariant. Composing this map with the functor M gives a continuous G-equivariant 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 G-space 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 *. ggammaarx.tex; 21/03/2005; 11:38; p.4 5 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 G-CW com- plexes rather than simply restricting to these simplicial objects is that some G-homotopy 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 G-spaces; 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+ ^ Sn-1+. 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(Zp-1, Zp) ^. .^. oG(Z0, Z1) ^M(Z0). Z0,...,Zp The face maps are given by: di(fp ^. .^.f0 ^m) = fp ^. .^.fiO fi-1 ^. .^.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. ggammaarx.tex; 21/03/2005; 11:38; p.5 6 The simplicial G -space LM.has a natural augmentation LM.! M. The augmented object LM. ! M has an extra degeneracy map s-1 , defined by s-1 (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 G-space, consider the simplicial G -space X G LM.. We have ` h i X G LMp = X G oG(Zp, -) ^ oG(Zp-1, 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 G-equivariant based functions from Y to X ^ Z. PROPOSITION 7. For X a G-CW 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 G-CW complex X. We will prove this by showing that it is filtered by weak equivalences. Suppose X is a G-CW 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 G-orbits 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)(n-1)^ n RG M(G+ ^ n+ ) _____//F (n+ , X) ^ n RG M(G+ ^ n+ ) | | | | fflffl| fflffl| M(X)(n-1) ___________________________//_M(X)(n) ggammaarx.tex; 21/03/2005; 11:38; p.6 7 R Here n G is the wreath product Gn o n. X is a G-CW complex, so the horizontal arrows are cofibrations. This identifies the cofiber of the map M(X)(n-1) ! 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 G-CW 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)|(n-1) ! |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)|(n-1) _____//|LM.(X)|(n)_____//C(n, X) ^ n G |LM.(G+ ^ n+ )| | | | | | | fflffl| fflffl| Rfflffl| M(X)(n-1) ________//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 right-handRvertical 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 G-CW 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 (Zp-1, Zp) ^ . .^. G (Z0, Z1) ^ M(Z0) | | W fflffl| Z0,...,ZpF G(Zp, Y ) ^ oG(Zp-1, Zp) ^ . .^. oG(Z0, Z1) ^ M(Z0). This map is k-connected because the map F G(Zp, X) ! F G(Zp, Y ) is. Since LM.(X) ! LM.(Y ) is k-connected levelwise, so is the map of geometric realizations. ggammaarx.tex; 21/03/2005; 11:38; p.7 8 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 G-CW 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 K-homology In this section, we construct for each n a PU (n)-space whose underly- ing spectrum is homotopy equivalent to ku, the connective K-theory spectrum. Let W be a fixed n-dimensional 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 ggammaarx.tex; 21/03/2005; 11:38; p.8 9 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 non-basepoint [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 well-defined: 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 K-theory 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.) ggammaarx.tex; 21/03/2005; 11:38; p.9 10 4. Unitary deformation K-theory In this section we will have a fixed finitely generated discrete group G. Carlsson, in [2], defined a notion of the "deformation K-theory" of G as a contravariant functor from groups to spectra, and in the introduction of this article an analogous notion of "unitary deformation K-theory" 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 K-theory 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 K-theory 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 K-homology 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 G-plane V of dimension k is a pair (V, ae) where V is a k-plane in U and ae : G ! U(V ) is an action of G on V . We now describe a (non-equivariant) -space KG . Define n fifi o KG (X) = (Vx, aex)x2X fiVx a G-plane, Vx ? Vx0 ifx 6= x0, V* =.0 This is a special -space. The underlying H-space is a KG (1+ ) ' V (n) xU(n) Hom ( , U(n)), n where V (n) is the Stiefel manifold of n-frames 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 G-plane, 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 G-plane. Degeneracy maps are given by insertion of 0-dimensional G-planes. ggammaarx.tex; 21/03/2005; 11:38; p.10 11 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 p-simplex 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 Ei-1 ! 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 ku-modules, so the above is induced by an exact couple of ku-module 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 G-representations whose irreducibles have dimension less than or equal to i, and the quotient category should be the category of sums of irreducible representations ggammaarx.tex; 21/03/2005; 11:38; p.11 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 i-dimensional 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 : s-1 (a) ! ~-1 (ga) is a ggammaarx.tex; 21/03/2005; 11:38; p.12 13 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 Ei-1. 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 Ei-1 is the honest fiber over any point is clear, but we need to show that Ei-1 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 K-theory 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,j-1 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,j-1! Bi,j-1is a quasifibration. Let Yj be the space of triples (A, ae, W ), where W is an ij-dimensional 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,j-1; in other words, ae contains less than j distinct i-dimensional 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 ae-invariant ij-dimensional subspace of U containing all the i-dimensional 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 i-dimensional summands. There is a map Xj ! Bj-1 given by sending (A, ae, W ) to (A, ae), and a similar map X0j! Ej-1. These maps all assemble into the diagram below. Ei,j-1oo___X0j _____//Yj0 |pi| || |p| fflffl| fflffl| fflffl| Bi,j-1 oo___Xj _____//Yj ggammaarx.tex; 21/03/2005; 11:38; p.13 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,j-1is a quotient map; two points become iden- tified by forgetting the "framing" subspace W , the non-i-dimensional 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 i-dimensional 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 non-i-dimensional 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 right-hand 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 Ei-1: An equivalence class of points (A, ae, W ) 2 Yj0consists of a choice of ij-dimensional 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 ij-dimensional planes in U, and V is the bundle over the Grassmannian consisting of ij-dimensional planes in U and elements of Ei-1 acting on their orthogonal comple- ments. Given any point (A, ae, W ) of Xj, the fiber in X0jis Ei-1 acting on the orthogonal complement of W . Suppose that (A, ae) in Bi,j-1is 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,j-1consists of all possible actions of Ei-1 on the orthogonal complement of W 0. The map from the fiber ggammaarx.tex; 21/03/2005; 11:38; p.14 15 over (A, ae, W ) to the fiber over (A, ae) is the inclusion of Ei-1 acting on W ? to Ei-1 acting on (W 0)? . This inclusion is a homotopy equivalence. Therefore, Ei,j! Bi,jis a quasifibration with fiber Ei-1. Taking colimits in j, Ei ! Bi is a quasifibration with fiber Ei-1. COROLLARY 13. The maps KG i-1 ! 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 nd-plane of U, together ggammaarx.tex; 21/03/2005; 11:38; p.15 16 with a marking z(y) of the plane by an element of Z. The action of I U(d) commutes with the G-action on W Cd, so the choice of lift fg(y)does not change the resulting G-plane. For g 2 G, r(gy) = g . r(y) = gr(y)g-1 , and f(gy) = (g I)f(y)(g-1 I), so the resulting plane only depends on the orbit Gy. The resulting G-plane breaks up into irreducibles of dimension precisely n. Assembling these G-planes over the distinct orbits gives a collection of orthogonal hyperplanes with G-actions, marked by points of Z, which break up into a direct sum of n-dimensional irreducible representations. As r(y) approaches the basepoint of Rn, the representation becomes reducible, so the map determines a well-defined 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 -ku-module maps. 4. There are compatible E1 -ku-linear 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 -ku-modules. 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). ggammaarx.tex; 21/03/2005; 11:38; p.16 17 2. A set Mk(a1, . .,.ak; b) for each a1, . .,.ak, b 2 Ob (M), k 0 of "k-morphisms" 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 k-morphisms 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. ggammaarx.tex; 21/03/2005; 11:38; p.17 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 well-defined exterior tensor product of representations: k K(G1)Un1(Z1) ^. .^.K(Gk)Unk(Zk) ! K(G1x. .x.Gk)Un1...nk(Z1 ^. .^.Zk) Post-composition 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 G-representations 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 ku-module symmetric spectra with the following properties. - There are ku-module maps KG 1 ! KG 2 ! . .!.KG , and KG is weakly equivalent to the homotopy colimit. - There are strictly commutative and associative ku-module 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 n-1 ! KG n is weakly equivalent as a ku-module 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 ggammaarx.tex; 21/03/2005; 11:38; p.19 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 right-half-plane spectral se- quence of the form Ep,q1= kuPUq(n)-p+1(Rp-1) ) ssp+q(KG ). Remark 23. The grading convention is such that dr maps Ep,qrto Ep-r,q+r-1r. 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 right-half-plane spectral se- quence of the form Ep,q1= Hq-p+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 K-theory 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 K-theory spectra. The proof requires the following lemmas. LEMMA 27. A map M0 ! M of connective ku-module 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 ku-module spectrum M00is weakly contractible if and only if HZ ^ku M00' *. However, smashing M00with the cofiber sequence 2ku ! ku ! HZ of ku-module 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 G-representation, and W is an irreducible H-representation. 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 H-linear 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 H-linear, we find that _ must be H-linear, 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 H-isotypic component of U is G-invariant, so there is an irreducible unitary representation W of H such that U ~=W das an H-representation. Consider the G-vector space Hom H(W, U). There is an irreducible G-representation V (which we do not assume to have an inner product) and a nonzero G-map V ! Hom H (W, U). The adjoint of this map is a nonzero G x H-map V W ! U. Both sides are irreducible, and hence this map is an isomorphism of G x H-representations. The irreducible G-summands of U all come from a unique isomor- phism class of irreducible G-representations. Since the above map is nonzero, this G-representation 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 ku-algebras 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 ku-modules 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 ku-module 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: Mn-1 ____________//_Mn________________//Mn=Mn-1 fn-1|| fn|| gn|| fflffl| fflffl| fflffl| K(G x H)n-1 _____//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=Mn-1 and kuPU (n)^PU (n)Rn(G xH) are connective ku-module spectra. Applying Lemma 27, it suffices to prove that the map HZ ^ku gn is a weak equivalence. Because the map Mn-1 ! 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 p-1 )^ (KH q=KH q-1). p.q=n ku The spectra KG p=KG p-1 , 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=Mn-1 ' 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. 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