THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY THEORY H. FAUSK, L.G. LEWIS, JR, AND J.P. MAY Abstract.Let G be a compact Lie group. We describe the Picard group Pic(HoGS ) of invertible objects in the stable homotopy category of G-sp* *ectra in terms of a suitable class of homotopy representations of G. Combining* * this with results of tom Dieck and Petrie, which we reprove, we deduce an exa* *ct sequence that gives an essentially algebraic description of Pic(HoGS ) i* *n terms of the Picard group of the Burnside ring of G. The deduction is based on* * an embedding of the Picard group of the endomorphism ring of the unit object of any stable homotopy category C in the Picard group of C . For a compact Lie group G, the isomorphism classes of invertible G-spectra form a group, Pic(HoGS ), under the smash product. Here HoGS is the stable homotopy category of G-spectra indexed on a complete G-universe, as defined in [21]. We shall prove the following theorem. Theorem 0.1. There is an exact sequence 0 -! Pic(A(G)) -! Pic(HoGS ) -! C(G): Here A(G) is the Burnside ring of G and C(G) is the additive group of continu* *ous functions from the space of subgroups of G to the integers, where subgroups are understood to be closed. In fact, we shall see that this is implicit in result* *s of tom Dieck and Petrie. Moreover, they and others have also studied the image of Pic(HoGS ) in C(G). In x1, we give some general results on Picard groups of categories, following* * up [17] and [24]. In particular, we prove the following theorem, which shows that * *the monomorphism of Picard groups displayed in Theorem 0.1 is formal. The notion of* * a "stable homotopy category" is axiomatized in [17]. There are examples in algebr* *aic topology, algebraic geometry, representation theory, and homological algebra. Theorem 0.2. Let C be a stable homotopy category, let S be the unit object, and let R = R(C ) be the ring of endomorphisms of S. There is a monomorphism of groups c : Pic(R) -! Pic(C ). The objects in the image of c are the invertible objects that are retracts of finite coproducts of copies of S. In practice, stable homotopy categories are usually constructed by localizing model categories so as to invert certain objects, thus forcing them to be eleme* *nts of Pic(C ). The theorem says that, on formal grounds, more objects must also be inverted. For example, the following parenthetical corollary is immediate from * *work of Morel [25] on the Morel-Voevodsky A1-stable homotopy category [26]; compare [24, 2.14, 4.11, 4.12]. ____________ Date: April 20, 2000. 1991 Mathematics Subject Classification. Primary 55P42, 55P91, 57S99; Second* *ary 18D10. 1 2 H. FAUSK, L.G. LEWIS, JR, AND J.P. MAY Corollary 0.3.Let k be a field, char k 6= 2, let C be the A1-stable homotopy ca* *te- gory of k, and let GW (k) be the Grothendieck-Witt ring of k. There is a monomo* *r- phism c : Pic(GW (k)) -! Pic(C ). Po Hu [18] has constructed various elements in Pic(C ). Her examples are gen- uinely exotic, in the sense that they are not in the image of Pic(GW (k)), since calculations in motivic cohomology show that they cannot be retracts of copies * *of the unit object. Returning to the equivariant stable homotopy category, in x2 we reduce the calculation of Pic(HoGS ) to the study of homotopy representations of G, starti* *ng with the following slightly nonstandard definition. We shall relate this defini* *tion to previous ones in x4. Definition 0.4.A generalized homotopy representation A is a finitely dominated based G-CW complex such that, for each subgroup H of G, AH is homotopy equiv- alent to a sphere Sn(H). A stable homotopy representation is a G-spectrum of the form -V 1 A, where V is a representation of G and A is a generalized homotopy representation. Theorem 0.5. Up to equivalence, the invertible G-spectra are the stable homoto* *py representations. In x3, we prove Theorem 0.1 by combining these algebraic and topological redu* *c- tions of the problem with some arguments from the work of tom Dieck and Petrie [4, 11, 12, 13]. Theorem 0.1 gives an appropriate conceptual setting and quick * *new proofs for some of the main results of [12, 13]. 1. The Picard group of a stable homotopy category We assume familiarity with [24, xx1-3]. As there, let C be a closed symmetric monoidal category with unit object S, product ^, and internal hom functor F . Recall that the dual of an object X is DX = F (X; S). Dualizable objects are discussed in [24, x2]. We are interested in invertible objects, and these are d* *ualizable by [24, 2.9]. We have the following observation. Lemma 1.1. An object X is invertible if and only if the functor (-)^X : C -! C is an equivalence of categories. If X is invertible, the canonical maps : S -! F (X; X), j : S -! X ^DX, and " : DX ^X -! S are isomorphisms. Conversely, if " is an isomorphism or if X is dualizable and j or is an isomorphism, then X is invertible. Proof.The first statement is clear. If X is invertible, the map C (-; S) -! C (-; F (X; X)) ~=C (- ^ X; X) induced by is the isomorphism (-) ^ X given by smashing maps with X, hence is an isomorphism by the Yoneda lemma. When X is dualizable, the definition of j in terms of given in [24, 2.3] shows that is an isomorphism if and only if * *j is an isomorphism; in turn, by [24, 2.6(ii)], j is an isomorphism if and only if "* * is_an isomorphism. Trivially, if " is an isomorphism, then X is invertible. * * |__| Now assume further that the category C is additive. Then R = R(C ) C (S; S) is a commutative ring and C is enriched over the category MR of R-modules, so that C (X; Y ) is naturally an R-module. Define functors ss0, ss0 : C -! MR by ss0(X) = C (S; X) and ss0(X) = C (X; S): THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY THEORY 3 The ^-product of maps gives a natural transformation OE : ss0(X) R ss0(Y ) -! ss0(X ^ Y ): The proof of Theorem 0.2 is based on application of ss0 to K"unneth objects of * *C , and we start with a characterization of the K"unneth objects. Observe that, by * *[24, 2.7] and adjunction, we have canonical isomorphisms ss0(F (X; Y ) ^ Z) ~=ss0(F (X; Y ^ Z)) ~=C (X; Y ^ Z) if X or Z is dualizable. In particular, if X is dualizable, ss0(DX) ~=ss0(X) and ss0(DX ^ Y ) ~=C (X; Y ): Recall that an R-module is finitely generated projective if and only if it is* * dual- izable [24, 2.4]. Proposition 1.2.The following conditions on a dualizable object X are equivalen* *t, and these conditions imply that ss0(X) is a finitely generated projective R-mod* *ule. A dualizable object satisfying these conditions is said to be a K"unneth object. (i)OE : ss0(DX) R ss0(X) -! ss0(DX ^ X) is an isomorphism. (ii)OE : ss0(Y ) R ss0(X) ~=ss0(DY ) R ss0(X) -! ss0(DY ^ X) ~=C (Y; X) is an isomorphism for all dualizable objects Y . (iii)OE : ss0(Y ) R ss0(X) -! ss0(Y ^ X) is an isomorphism for all objects Y . W n (iv)X is a retract of i=1S for some integer n. Proof.Clearly (iii) ) (ii) ) (i) by specialization of the given isomorphisms. By [21, III.1.9], (i) ) (iii), so that (i)-(iii) are equivalent; [21, III.1.9]Wals* *o shows that ss0(X) isQdualizable when these conditions hold. Write S_n = ni=1S; it * *is isomorphic to ni=1S. The implication (iv) ) (iii) is clear since the conclusi* *on of (iii) holds when X = S_n and is inherited by retracts of S_n. Thus it suffices * *to prove that (ii) ) (iv). Assuming (ii), ss0(X) is a finitely generated projectiv* *e R- module. Therefore ss0(X) is a direct summand and thus a retract of Rn ~=ss0(S_n) for some n. Moreover, (ii) gives natural isomorphisms ss0(Y ) R ss0(S_n) ~=C (Y; S_n) and ss0(Y ) R ss0(X) ~=C (Y; X): Since X and S_n are dualizable, we see by the Yoneda lemma that a retraction_ ss0(X) -! Rn -! ss0(X) induces a retraction X -! S_n -! X. |__| Of course, (iv) implies that X is dualizable, but the other conditions do not. The proposition shows that K"unneth objects in C are closely related to dualiza* *ble R-modules. In particular, when C = MR is the category of R-modules, (iv) says that X is finitely generated projective. Corollary 1.3.The K"unneth objects, the dualizable objects, and the finitely ge* *n- erated projectives coincide in the category of modules over a commutative ring. Let dC be the full subcategory of dualizable objects in C and let kC dC be t* *he full subcategory of K"unneth objects. These are closed symmetric monoidal addit* *ive subcategories of C , and D restricts to equivalences of categories D : dC op-! * *dC and D : kC op-! kC . Of course, kMR = dMR . From now on, we assume that dC is skeletally small. Let Iso(dC ) and Iso(kC ) denote the sets of isomorphism c* *lasses of objects in dC and kC ; these are both semi-rings under _ and ^. Corollary 1.4.ss0 : Iso(kC ) -! Iso(dMR ) is a monomorphism of semi-rings. 4 H. FAUSK, L.G. LEWIS, JR, AND J.P. MAY Proof.Proposition 1.2 (ii) shows that if ss0(X) and ss0(Y ) are isomorphic, the* *n_the represented functors C (-; X) and C (-; Y ) are isomorphic. |_* *_| We now impose extra structure on C which ensures that ss0 is an isomorphism of semi-rings. The idea is to apply the Brown representability theorem [3]. Whi* *le this may not give maximal generality, we place ourselves in the context of [17] and assume that C is a stable homotopy category in the sense of [17, 1.1.4]. Th* *is amounts to the following conditions, details of which may be found in [17]. (a) C is triangulated and has arbitrary coproducts. (b) C is closed symmetric monoidal, compatibly with the triangulation. (c) C has a generating set of small dualizable objects. (d) Every cohomology functor on C is representable. Here a "cohomology functor" is an exact additive contravariant functor from C to Abelian groups that carries coproducts to products. Proposition 1.5.If C is a stable homotopy category, ss0 : Iso(kC ) -! Iso(dMR ) is an isomorphism of semi-rings. In particular, ss0 induces an isomorphism of Abelian groups Pic(kC ) -! Pic(dMR ) = Pic(R): Proof.As one can check directly, if P is a finitely presented R-module, then the functor (-) R P commutes with arbitrary products. Of course, a projective R- module P is flat, so that the functor (-) R P is exact. Thus, if P is a finite* *ly generated projective R-module, then (-) R P is an exact additive functor that carries products to products. The functor ss0 is exact by standard properties * *of triangulated categories and it carries coproducts to products. Therefore the co* *m- posite functor on C that sends an object Y to ss0(Y )R P is a cohomology functo* *r. It can be represented by an object X, so that ss0(Y ) R P ~=C (Y; X). Since the action of R on ss0(Y ) is given by composition of maps in C , this is an isomor* *phism of R-modules by naturality. In particular, taking Y = S, ss0(X) ~=P . Arguing as in the last step of the proof of Proposition 1.2, we see that X is a retract of* * some_ S_n and is therefore a K"unneth object. This proves that ss0 is an epimorphism.* * |__| Proof of Theorem 0.2.Let c : Pic(R) ~=Pic(kC ) -! Pic(dC ) Pic(C ) be induced by the inclusion kC -! dC . Since the homomorphism of Picard groups associated to any full embedding of symmetric monoidal categories is a monomorphism, c is * *a_ monomorphism. Its image consists of the invertible K"unneth objects. |* *__| Parenthetically, we relate these Picard groups to the evident groups of units* * in Grothendieck rings. Let L(C ) and K(C ) be the Grothendieck rings associated to Iso(kC ) and Iso(dC ). Write K(R) = K(MR ) and note that K(R) = L(MR ). The inclusion kC -! dC induces a homomorphism of rings L(C ) -! K(C ) and thus a homomorphism of rings c : K(R) ~=L(C ) -! K(C ). Letting Ax denote the units of a ring A, we have the commutative diagram (1.6) Pic(R) ~=Pic(kC_)_c__//Pic(C ) fi|| |fi| fflffl| fflffl| K(R)x ~=L(C )x __c__//_K(C )x : THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY THEORY 5 The maps fi in (1.6) are considered in [24, x3]. The left arrow fi is a monom* *or- phism for any R, by [24, 3.8]. We do not know whether or not the bottom arrow c is a monomorphism in general. However, we can prove that this is often the case. Let G(R) denote the Grothendieck group of finitely generated R-modules. Proposition 1.7.Let C be a unital algebraic stable homotopy category. If ss0(X) is a finitely generated R-module for all dualizable objects X and the natural m* *ap : K(R) -! G(R) is a monomorphism, then c : K(R) ~=L(C ) -! K(C ) is a monomorphism. Proof.Let X and Y be K"unneth objects of C such that X _ Z ~=Y _ Z for some dualizable object Z. Then ss0(X) ss0(Z) ~=ss0(Y ) ss0(Z) as R-modules. Since is a monomorphism, there is a finitely generated projective R-module P such that ss0(X) P ~=ss0(Y ) P . Let W be a K"unneth object such that ss0(W ) ~=P . The* *n_ ss0(X _ W ) ~=ss0(Y _ W ) and thus X _ W ~=Y _ W by Corollary 1.4. |__| 2. Dualizable and invertible G-spectra To prove Theorem 0.5, we must characterize the invertible G-spectra in terms * *of G-spaces, and we first characterize the dualizable G-spectra. Here we are compa* *ring the homotopy category HoGT of based G-spaces to the homotopy category HoGS of G-spectra, and we may restrict attention to based G-CW complexes and to G-CW spectra. We write 1 for the suspension G-spectrum functor HoGT - ! HoGS . We write SV for the one-point compactification of a representation V , by whi* *ch we understand a finite dimensional real G-inner product space. We continue to write SV for 1 SV . These linear sphere spectra are invertible elements of HoGS* * , this being the essential point of the construction of HoGS . We write S-V for * *the inverse of SV . We have desuspension functors -V given by smashing with S-V . Up to equivalence, the finite G-CW spectra are those of the form -V 1 B for a finite G-CW complex B and a representation V of G [21, I.8.16]. We have a simil* *ar space level characterization of dualizable G-spectra. Proposition 2.1.Up to equivalence, the dualizable G-spectra are the G-spectra of the form -V 1 A, where A is a finitely dominated based G-CW complex and V is a representation of G. Proof.By an argument due to Greenlees [23, XVI.7.4], the dualizable G-spectra are the retracts up to homotopy of the finite G-CW spectra, that is, the finite* *ly dominated G-CW spectra. Since the functors -V 1 preserve retracts, it is clear that the G-spectra of the statement are dualizable. We must prove conversely th* *at every retract of a finite G-CW spectrum is obtained by applying one of the func* *tors -V 1 to a finitely dominated G-CW complex. Let X = X1 be a retract of a finite G-CW spectrum -V 1 B, where B is a finite G-CW complex and V is a representation of G. Since HoGS is triangulated, retracts split. Thus there is a G-spectrum X2 such that X1 _ X2 ' -V 1 B. Projection and inclusion give idempotent maps ei: -V 1 B -! -V 1 B such that e1e2 = 0 = e2e1 and e1 + e2 = id. Explicitly, ei is the composite -V 1 B ' X1 _ X2 -'!X1 x X2 ssi-!Xi-i! X1 _ X2 ' -V 1 B: By the Freudenthal suspension theorem [23, IX.1.4], we can suspend by V W for W sufficiently large that 1 gives a bijection from the homotopy classes of 6 H. FAUSK, L.G. LEWIS, JR, AND J.P. MAY self-maps of the G-space W B to the homotopy classes of self-maps of the G- spectrum 1 W B ~= W 1 B ~= V W -V 1 B. Moreover, we may as well assume that W R2, so that W B is simply G-connected. Now V W ei= 1 fi for idempotent G-maps fi : W B - ! W B such that f1f2 = 0 = f2f1 and f1+f2 = id. Taking the fito be cellular maps, let Aibe the telescope of countab* *ly many iterates of fi. The composite of the pinch map W B -! W B _ W B and the wedge of the canonical maps W B -! Ai gives a map : W B -! A1 _ A2. On passage to fixed points and homology, H*realizes the evident isomorphism H*((W B)H ) ~=f1*H*((W B)H ) f2*H*((W B)H ): Since these fixed point spaces are simply connected, each H is a weak equivalen* *ce and thus is a G-equivalence by the Whitehead theorem. The evident composites V W Xi- ! W 1 B ~=1 W B -! 1 Aj are 0 if i 6= j, and the sum of the composites with i = j is an equivalence. Th* *us the composite with i = j = 1 is an equivalence. This displays X1 as -(V W) 1_A1,_ where A1 is a wedge summand of the finite G-CW complex W B. |__| We will prove Theorem 0.5 by using the geometric fixed point functors H : HoGS -! HoS of [21, IIx9] to compare invertible G-spectra to invertible spectra. By [21, II.9.9 and II.9.12], for based G-spaces A and for G-spectra X and Y , we have natural equivalences H 1 A ' 1 AH and H (X ^ Y ) ' H (X) ^ H (Y ). By [21, III.1.9], this implies formally that if X is a dualizable G-spectrum, t* *hen H X is a dualizable spectrum and H DX ~=DH X. Moreover, by a variant of the Whitehead theorem [23, XVIx6], a map f of G-spectra is an equivalence if and only if each H f is an equivalence of spectra. Recall our notion of a stable homotopy representation from Definition 0.4. Proof of Theorem 0.5.We must characterize the invertible G-spectra X. Since invertible G-spectra are dualizable, we may assume that X = -V 1 A, where A is a finitely dominated based G-CW complex and V is a representation of G. By suspending and desuspending by R2, we may as well assume that A is simply G-connected. By Lemma 1.1, X is invertible if and only if the evaluation map " : DX ^ X -! SG is an equivalence, where SG is the sphere G-spectrum. This holds if and only if each H " is a nonequivariant equivalence. By the results c* *ited above, the map H " is isomorphic to the map " : DH (X) ^ H (X) -! S. This map is an equivalence if and only if H (X) is an invertible spectrum. By an elementary argument using the Hurewicz and Whitehead theorems (or see [27] or [16]), the only invertible spectra are the spheres 1 Sn for integers n. Sin* *ce H (-V 1 A) ' -V H1 AH , X is invertible if and only if, for each H, 1 AH ' Sn(H)for some integer n(H). Since we have assumed that A is simply G-connected, n(H) 2. Thus AH has the same homology as Sn(H) and is therefore equivalent to Sn(H) by the Hurewicz and Whitehead theorems. We conclude that the G- spectrum X is invertible if and only if the G-space A is a generalized homotopy* *__ representation, which means that X is a stable homotopy representation. |* *__| By easy inspections, smash products and duals (= inverses) of stable homotopy representations are stable homotopy representations. This also follows directly* * from Theorem 0.5. THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY THEORY 7 Corollary 2.2.Pic(HoGS ) is the group of isomorphism classes in HoGS of sta- ble homotopy representations. 3.The exact sequence for Pic(HoGS ) We prove Theorem 0.1 by combining the formal algebraic considerations of x1, the topological reduction from G-spectra to G-spaces of x2, and a lemma from the work of tom Dieck and Petrie [12, 13] on space level homotopy representations. Theorem 0.2 applies since HoGS is a stable homotopy category. Here R(HoGS ) is the Burnside ring A(G), which is a well studied ring. In particular, its pr* *ime ideals and its localizations at prime ideals are understood [4, 21]. For an invertible G-spectrum X = -V 1 A, where A is a generalized homotopy representation, let dH (X) = n(H) - dim(V H), where AH is equivalent to Sn(H). Thus H X is a sphere spectrum Sd(H); d(H) depends only on the conjugacy class (H) of H, and dH (X ^ Y ) = dH (X) + dH (Y ) for invertible G-spectra X and Y . Let (G) denote the space of conjugacy classes of (closed) subgroups of G. It * *is a totally disconnected compact metric space [4, 21]. Let C(G) denote the additi* *ve group of continuous functions (G) -! Z, where Z has the discrete topology. It is more usual to restrict attention to subgroups H of finite index in their normal* *izers, but that is not appropriate for the present purposes. Definition 3.1.Define the dimension homomorphism d : P ic(HoGS ) -! C(G) by letting d(X) : (G) -! Z send (H) to dH (X). We must check that d(X) is continuous. Certainly d(X) depends only on the homotopy type of X and thus only on its isomorphism class in HoGS . By [19, 1.4* *], a finitely dominated G-CW complex is homotopy equivalent to a G-CW complex that has finitely many orbit types. The dimension function of a G-CW complex having finitely many orbit types is continuous [11, IVx3], and it follows that * *d(X) is continuous. As we will discuss in x4, much is known about the image of d, but it is not f* *ully understood. Consider the sequence 0 -! P ic(A(G)) -c!P ic(HoGS ) d-!C(G): We know that c is a monomorphism, hence the following result completes the proof of Theorem 0.1. Theorem 3.2. An invertible G-spectrum X is a K"unneth object if and only if d(X) = 0. Therefore the kernel of d is equal to the image of c. Proof.The last clause follows from the definition of the map c in terms of K"un* *neth objects of HoGS . Let X = -V 1 A for a representation V of G and a generalized homotopy representation A. As usual, we may as well assume that A is simply G- connected. Suppose firstWthat X is a K"unneth object. Then, by Proposition 1.2(v), X is a retract of ni=1SG for some n. Suspending by V W for a sufficiently large W and arguingWas in the proof of Proposition 2.1, we find that the G-space W A is* * a retract of ni=1SV W . Passing to H-fixed point spaces and observing that a sp* *here that is a retract of a wedge of m-spheres must be an m-sphere, we see that AH is equivalent to Sn(H), where n(H) = dim(V H). Thus dH (X) = 0 and X, regarded as an element of P ic(HoGS ) is in the kernel of d. 8 H. FAUSK, L.G. LEWIS, JR, AND J.P. MAY Conversely, suppose that X is in the kernel of d. This means that AH is equiv* *a- lent to Sn(H), where n(H) = dim(V H). Equivalently, d(1 A) = d(SV ). We must prove that X is a K"unneth object. Write C = HoGS and identify A(G) with R(C ). It suffices to prove that the canonical map ss0(Y ) A(G)ss0(X) -! C (Y; X) displayed in Proposition 1.2(ii) is an isomorphism for all dualizable G-spectra* * Y . This holds if, for all maximal ideals q of A(G), (3.3) ss0(Y )q A(G)qss0(X)q -! C (Y; X)q is an isomorphism. The maximal ideals of A(G) are of the form q(H; p) where p is a prime number and H is a subgroup of G with finite Weyl group W H = NH=H of order prime to p. The ideal q(H; p) consists of all maps OE : SG - ! SG such that degH (OE) 0 mod p, where degH (OE) is the degree of the fixed point map fH : (SV )H - ! (SV )H of a space level representative f : SV - ! SV of OE. We have the following key lemma, which generalizes an observation of tom Dieck and Petrie [12, x2]. We defer its proof to the end of the section (where we indicat* *e how to be precise about the relevant degrees of maps). Lemma 3.4. Let X 2 Ker(d) and let W H have finite order prime to p. Then there are maps f : SG -! X and k : X -! SG such that deg(fH ) 6 0 mod p and deg(kH ) 6 0 mod p. Returning to the proof of Theorem 3.2, fix a maximal ideal q = q(H; p) of A(G* *), and let f and k be as in the lemma. The composite f O k : SG -! SG is a unit in A(G)q since deg((f O k)H ) = deg(fH )deg(kH ) 6 0 mod p, so that f O k is not in q. Smashing maps SG -! SG with X gives an isomorphism of rings C (SG ; SG ) ~= C (X; X), and k O f is a unit in C (X; X)q. Thus f* : C (SG ; SG )q -! C (SG ; * *X)q is an isomorphism with inverse k*. Changing back to the notations in (3.3), the vertical arrows in the following naturality diagram are isomorphisms. ss0(Y )q A(G)qA(G)q_____//ss0(Y )q 1f* || f*|| fflffl| fflffl| ss0(Y )q A(G)qss0(X)q___//_C (Y; X)q Since the top arrow is clearly an isomorphism, so is the bottom arrow. Thus X_is a K"unneth object and the proof is complete. |__| In view of Proposition 1.5 and [24, 2.11], Theorem 3.2 has the following imme- diate consequence. Theorem 3.5. If X is a stable homotopy representation such that d(X) = 0, then ss0(X) is a finitely generated projective A(G)-module of rank 1. Remark 3.6.For finite groups, Theorem 3.2 is a version of [13, 6.5] of tom Dieck and Petrie; for compact Lie groups, it is a version of [10, 1.6] of tom Dieck. * *Related information about Pic(A(G)) is given in tom Dieck's papers [8, 9]. Theorem 3.5 generalizes [12, Thm.1] of tom Dieck and Petrie, which gives the result for sph* *ere G-spectra SW-V . We have a homomorphism Sph from the real representation ring RO(G), regarded as an abelian group under addition, to Pic(Ho GS ). It THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY THEORY 9 sends W - V to SW-V , and W - V is in the kernel of Sph if and only if SV is stably G-homotopy equivalent to SW . A necessary condition for this to hold is that W - V be in the subgroup RO0(G) of RO(G) generated by those W - V with dimV H = dimW H for all H. Define jO(G) to be the image of the restriction Sph : RO0(G) -! Pic(Ho GS ). Clearly jO(G) is contained in the kernel Pic(A(G)) of d. The group jO(G) is studied in [4, 12]. Proof of Lemma 3.4.We first observe that we need only construct f : SG -! X. Indeed, if we can do this, then we can construct f0 : SG -! DX in the same fash* *ion. Taking the smash product of f0 with the identity map of X and composing with the equivalence " : DX ^ X -! SG , we obtain the desired map k : X -! SG . Suspending maps SG - ! X by a sufficiently large representation V W , we reduce the problem to consideration of space level maps SV W -! W A. Chang- ing notations, it suffices to consider maps SV -! A, where V is a representa- tion and A is a generalized homotopy representation such that AH ' SV H for all H G. We may as well assume that A and SV are G-simply connected, so that n(H) dim(V H) 2 for all H. We must make sense of deg(fH ) for a G-map f : SV -! A. There is a standard way of doing this, due to Laitinen [20, x2] and discussed in detail by tom Diec* *k [11, pp 169-173]. Tom Dieck considers maps between generalized homotopy representa- tions A and B with the same dimension functions {n(H)}, and he assumes that the fixed point spaces AH and BH both have topological dimension n(H). However, the use of this hypothesis is to obtain restrictions on the dimensions {n(H)}, * *and it therefore suffices to assume that either A or B has this property. Since SV has* * this property, the discussion applies in our situation. The conclusion is that SV a* *nd A have the same orientation behavior and admit coherent choices of fundamental classes in the homologies of their fixed point spaces. Use of these fundamental classes fixes the degrees {deg(fH )}. An elementary obstruction theory argument shows that we can extend a non- equivariant map SV H -! AH of degree one to an H-map e : SV - ! A; see e.g. [11, II.4.11(ii)]. To obtain the required G-map f : SV - ! A, we apply a transf* *er argument. Suspending further if necessary, we can assume that G=H embeds as a sub G-space of V . Using a tubular neighborhood of the embedding and the Pontryagin-Thom construction, we obtain a G-map SV -! G+ ^H SW , where W is the complement in V of the tangent space of G=H at eH (see e.g. [21, II.5.1]* *). Using the inclusion W V there results a G-map t : SV -! G+ ^H SV . We define f to be the composite SV -t!G+ ^H SV -id^He---!G+ ^H A -!A; where is given by the action of G on A. The W (H)-space (G+ ^H A)H is the wedge of |W (H)| copies of A, with W (H) permuting the wedge summands, and similarly with A replaced by SV . By virtue of the coherent choices of orientat* *ions,_ we see that the degree of fH is |W (H)|, which is prime to p. * *|__| 4. Remarks on homotopy representations Since our definition of a generalized homotopy representation differs slightl* *y from the usual one, we give a comparison. In the literature, homotopy representations are defined as unbased spaces, and joins are used instead of smash products. We shall reinterpret the classical definitions in the based context appropriate to* * stable 10 H. FAUSK, L.G. LEWIS, JR, AND J.P. MAY homotopy theory, and we require AH to have the homotopy type of a sphere Sn(H) for each (closed) subgroup H of G. With this understanding, tom Dieck's definition [11, II.10.1] of a generalized homotopy representation A replaces our condition that the G-CW complex A be finitely dominated by the conditions that A have finite dimension and finitely * *many orbit types. We are interested in G-homotopy types, and our definition, unlike * *tom Dieck's, is homotopy invariant. We have the following comparison. Proposition 4.1.Let A be a G-CW complex such that AH has the homotopy type of a sphere Sn(H) for each H G. If A is finitely dominated, then A is homotopy equivalent to a finite dimensional G-CW complex B having finitely many orbit ty* *pes. Conversely, if A is finite dimensional and has finitely many orbit types, then * *A is finitely dominated. Proof.By [19, Thm D] or [22, 14.9], if A is finitely dominated, then it is homo* *topy equivalent to a finite dimensional G-CW complex A0. Then, by [19, 1.4], A0 is homotopy equivalent to a G-CW complex B having finitely many orbit types. With the proof of the cited result, B is still finite dimensional. The converse is p* *roven_ (although only stated for actual homotopy representations) by L"uck [22, 20.2].* * |__| Thus our definition of a generalized homotopy representation is just a homoto* *py invariant modification of the usual one. Homotopy representations are restricted kinds of generalized homotopy represe* *n- tations. The crucial restriction is the requirement that AH be an n(H)-dimensio* *nal space, and that is required in all definitions in the literature. This restrict* *ion gives control on the possible values taken by the image of d, as we used implicitly i* *n the obstruction theory step of the proof of Lemma 3.4. In [11, II.10.1], but not in* * [13] and most other sources, two further restrictions are required on A for it to qu* *alify as a homotopy representation, namely (i)The set Iso(A) of isotropy groups of A is closed under intersection. (ii)If H 2 Iso(A) is a proper subgroup of K, then n(H) > n(K). Observe that both conditions can be arranged by smashing A with SV for a well chosen representation V . Thus, for stable purposes, we can assume these condit* *ions without loss of generality. We have the following comparison. Proposition 4.2.Let G be finite or a torus. For any generalized homotopy repre- sentation A, there is a representation V such that A^SV is equivalent to a homo* *topy representation B. Therefore every element of Pic(HoGS ) can be represented as -W 1 B for some homotopy representation B and representation W . Proof.First assume that G is finite. Under conditions on A specified in [13, 6.* *1], [13, 6.6] proves that A is equivalent to a homotopy representation. When each AH is 2-connected, as can be arranged by smashing with S3, the conditions are vers* *ions of (i) and (ii) above, and they can be arranged by smashing with a suitable SV . When G is a torus, the result is proven in [7, p. 463], where it is shown tha* *t_V and W can be found such that A ^ SV is equivalent to SW . |__| The following definition and results help to compare our work with the litera* *ture. Definition 4.3.Define V (G) to be the Grothendieck group associated to the monoid M(G) under smash product of equivalence classes of homotopy represen- tations, with [S0] as unit. Note that [A] = [A0] in V (G) if and only if A ^ B* * is THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY THEORY 11 equivalent to A0^ B for some B. An isomorphic group is obtained using unbased homotopy representations and the join operation. Define V 0(G) similarly, but u* *sing generalized homotopy representations. In these groups, inverses are adjoined formally, whereas actual inverse topol* *ogical objects are present in Pic(HoGS ). Proposition 4.2 implies the following result. Corollary 4.4.If G is finite or a torus, the canonical map V (G) -! V 0(G) is an isomorphism. As far as we know, there is no information in the literature about the relati* *onship between homotopy representations and generalized homotopy representations for more general compact Lie groups. It is natural to hope for the following conjec* *ture. Conjecture 4.5. The canonical map V (G) -! V 0(G) is an isomorphism for any compact Lie group G. Proposition 4.6.There is a canonical isomorphism V 0(G) -! Pic(HoGS ). Proof.The functor 1 gives a map of monoids M(G) -! Pic(HoGS ), which extends uniquely to a map of groups V (G) -! Pic(HoGS ). This map is an epi- morphism by Theorem 0.5. It is a monomorphism since if A and B are generalized homotopy representations such that 1 A is equivalent to 1 B, then there is a __ representation V such that A ^ SV is equivalent to B ^ SV . |_* *_| The image of d : V (G) -! C(G) has been studied extensively; see [6, 7, 11, 13, 14, 15] for finite groups and [1, 2] for general compact Lie groups. It is * *rarely an epimorphism, although it is so trivially if G is cyclic of order 2. For fin* *ite nilpotent groups G, in particular for p-groups, the image of d is realized by l* *inear representations. In more detail, there are necessary conditions, called the Bo* *rel- Smith conditions, for an element f 2 C(G) to be the dimension function of a homotopy representation, and when G is nilpotent every such f is d(Sff) for some virtual representation ff; see [11, IIIx5]. In [2], Bauer defines a subgroup D(* *G) of C(G) and displays a short exact sequence 0 -! Pic(A(G)) -! V (G) -! D(G) -! 0: It refines the exact sequence of Theorem 0.1 to a short exact sequence when G is finite or a torus, and this will remain true for general compact Lie groups * *G if Conjecture 4.5 holds. References [1]S. Bauer. Dimension functions of homotopy representations for compact Lie g* *roups. Math. Ann. 280(1988), 247-265. [2]S. Bauer. A linearity theorem for group actions on spheres with application* *s to homotopy representations. Comment. Math. Helv. 64 (1989), no. 1, 167-172. [3]E. H. Brown, Jr. Abstract homotopy theory. Trans. Amer. Math. Soc. 119 (196* *5), 79-85. [4]T. tom Dieck. 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In Ad* *ams Memorial Symposium on algebraic topology Vol. 2. London Math. Soc. Lecture Notes No. * *176, 1992, 45-54. Department of Mathematics, The University of Chicago, Chicago, IL 60637 E-mail address: fausk@math.uchicago.edu Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150 E-mail address: lglewis@mailbox.syr.edu Department of Mathematics, The University of Chicago, Chicago, IL 60637 E-mail address: may@math.uchicago.edu