DUALITY AND PRO-SPECTRA J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN Abstract.Cofiltered diagrams of spectra, also called pro-spectra, have ar* *isen in di- verse areas, and to date have been treated in an ad hoc manner. The purpo* *se of this paper is to systematically develop a homotopy theory of pro-spectra and t* *o study its relation to the usual homotopy theory of spectra, as a foundation for fut* *ure appli- cations. The surprising result we find is that our homotopy theory of pro* *-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. Thi* *s provides a convenient duality theory for all spectra, extending the classical noti* *on of Spanier- Whitehead duality which works well only for finite spectra. Roughly speak* *ing, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier* *-Whitehead duals of its finite subcomplexes. In the other direction, the duality fun* *ctor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whit* *ehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories* * by showing that both are equivalent to the category of ind-spectra (filtered diagram* *s of spectra). To construct our new homotopy theories, we prove a general existence theo* *rem for colocalization model structures generalizing known results for cofibrantl* *y generated model categories. 1. Introduction In recent years there have been many areas in which cofiltered diagrams of sp* *ectra have naturally arisen as a way to organize homotopical information. For example, see* * the work of Cohen, Jones and Segal [6] and Hurtubise [13] on Floer homology theory, Ando* * and Morava [1] on formal groups and free loop spaces, and unpublished work of Dwyer* * and Rezk and of Arone on Goodwillie calculus. Pro-spectra are also likely to be the* * target of an 'etale realization functor on stable motivic homotopy theory [16]. A cofiltered diagram is called a pro-spectrum (see Section 3) and in general * *it contains more information than its homotopy limit. Thus it is necessary to develop a hom* *otopy theory of pro-spectra, and our goal is to do this systematically and to study i* *ts relation to the usual homotopy theory of spectra. We expect that this framework will be use* *ful for many applications. Spanier-Whitehead duality was one of the reasons for which the stable homotop* *y cate- gory was invented [4] [19] [22]. The idea is that there is a contravariant func* *tor from the stable homotopy category to itself that induces an equivalence between the homo* *topy cat- egory of finite spectra and its own opposite. This functor is defined by taking* * a spectrum X to the function spectrum F (X, S0), where S0 is the sphere spectrum. ____________ 1991 Mathematics Subject Classification. 55P42 (Primary); 55P25, 18G55, 55U35* *, 55Q55 (Secondary). Key words and phrases. Spectrum, pro-spectrum, Spanier-Whitehead duality, clo* *sed model category, colocalization. The authors thank the SFB 343 at Universität Bielefeld, Germany. They also t* *hank Greg Arone for originally motivating the project and Stefan Schwede for useful conversatio* *ns. The first author was supported by an NSERC Research Grant and the second author was supported by an * *NSF Postdoctoral Research Fellowship. 1 2 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN One important property of Spanier-Whitehead duality is that the double dual o* *f an infinite spectrum is not weakly equivalent to the original spectrum. In some co* *ntexts, this is a useful property because it gives a method for producing new interesting sp* *ectra. On the other hand, it is sometimes inconvenient that Spanier-Whitehead duality doe* *s not give an equivalence between the whole stable homotopy category and its opposite. The same situation arises in linear algebra over a field k. The functional du* *al induces an equivalence of the category of finite dimensional k-vector spaces with its o* *wn opposite, but it does not extend to an equivalence on the whole category of k-vector spac* *es. One solution is to pass to the category of pro-finite k-vector spaces. In fact, the* * category of k-vector spaces is equivalent to the opposite of the category of pro-finite k-v* *ector spaces. Because of the strong analogy between the stable homotopy category and catego* *ries of chain complexes, it is natural to ask whether the same kind of solution works f* *or spectra. The main result of this paper is that it does. We define a homotopy theory for * *the category of pro-spectra and show that its opposite is equivalent to the usual homotopy t* *heory of spectra. However, the situation is significantly more complicated because of th* *e intricacies of homotopy categories. The most naive approach is to consider pro-objects in the homotopy category o* *f spectra. The unstable version of this approach appears in [3] and [23]. Homotopy theoris* *ts have learned through countless examples that considering diagrams in a homotopy cate* *gory is usually the wrong viewpoint. Rather, it is better to consider commutative diagr* *ams in a geometric category and then study the homotopy theory of these diagrams. Follow* *ing this philosophy, we consider the category of pro-objects in a geometric category of * *spectra and then equip it with a homotopy theory. More precisely, we construct a model structure on pro-spectra in which the we* *ak equiva- lences are detected by cohomotopy groups. This model structure is contravariant* *ly Quillen equivalent to a model structure on the category of ind-spectra (i.e., the categ* *ory of fil- tered systems of spectra) in which the weak equivalences are detected by homoto* *py groups. The model structure on ind-spectra is in turn Quillen equivalent to the usual s* *table model structure for spectra. The cofibrant pro-spectra in our new model structure are easy to describe. Th* *ey are the pro-spectra that are essentially levelwise cofibrant, that is, they are lev* *elwise cofibrant up to isomorphism. The fibrant pro-spectra are only slightly harder to describ* *e. They are the strictly fibrant pro-spectra that are essentially levelwise homotopy-fi* *nite, that is, the strictly fibrant pro-spectra (see Section 3.4) that are also levelwise weak* *ly equivalent to a finite complex, up to isomorphism. Dually, the fibrant ind-spectra are the* * essentially levelwise fibrant ind-spectra, and the cofibrant ind-spectra are the strictly c* *ofibrant ind- spectra that are essentially levelwise homotopy-finite. The importance of finit* *e complexes is no surprise since we are defining homotopy theories that work well with resp* *ect to Spanier-Whitehead duality. The description of fibrant pro-spectra in terms of homotopy-finite spectra al* *lows us to compute the total derived functor Rlim of the limit functor from pro-spectra to* * spectra. For a constant pro-spectrum X, RlimX is the Spanier-Whitehead double dual of X.* * See Remark 6.9 for more details. Our chief tool for establishing the appropriate homotopy theories of pro-spec* *tra and ind-spectra is a general existence theorem for a certain kind of colocalization* * of model categories (see Theorem 2.6). This result is a generalization of [10, Thm. 5.1.* *1] because cofibrantly generated model structures satisfy our hypotheses, and our proof is* * very simi- lar. In our application, we begin with the strict model structure on pro-spectr* *a in which DUALITY AND PRO-SPECTRA 3 the weak equivalences are, up to isomorphism, the levelwise weak equivalences (* *see Sec- tion 3.4). Then we use mapping spaces into the spheres to determine the colocal* * weak equivalences of pro-spectra. Dually, for ind-spectra, we start with the strict * *structure and then use mapping spaces out of the spheres to determine the colocal weak equiva* *lences. In this paper, we need a model for spectra that has a well-behaved function s* *pectrum defined on the geometric category, not just on the homotopy category. We use sy* *mmetric spectra [12] for this model. Section 5 reviews the relevant ideas. It is also p* *ossible to work entirely in the category of S-modules [9]. In fact, we do not really need the full power of a general function spectrum * *construction. Rather, we only need to define function spectra of the form F (X, S0). It has b* *een suggested to us that this is probably possible on more naive categories of spectra such a* *s the one described in [5], but we have not checked the details. Our proofs are written in such a way that they can easily be applied to other* * situations involving pro-spectra. For example, if E is any generalized cohomology theory, * *then we can define a model structure on the category of pro-spectra in which the weak e* *quivalences are detected by E-cohomology groups. And the proof of our duality result extend* *s to a proof that this model category is Quillen equivalent to the opposite of the mod* *el category of End(E)-module spectra. We assume that the reader is familiar with the language and basic results of * *model categories. The original reference is [21], but we conform to the notations and* * terminology of [10]. See also [7] or [11]. 1.1. Organization. The paper is organized as follows. We begin with the general* * exis- tence theorem for K-colocal model structures. Next we review the theory of pro-* *categories and ind-categories. Then we study colocal model structures on pro-categories a* *nd ind- categories in general. The second part of the paper begins with a review of some details about symme* *tric spectra. Next we construct and study the model structures for pro-spectra and i* *nd-spectra. Finally, we prove that the various model structures are Quillen equivalent. 2. Colocalizations of Model Structures In this section, we prove a general theorem about colocalization of model str* *uctures. Much of what appears here is very similar to [10, Ch. 5]. One important differe* *nce is that we work with model structures that may not be cofibrantly generated. Start with a right proper model category C, that is, a model category in whic* *h the pullback of a weak equivalence along a fibration is always a weak equivalence. * *We refer to the cofibrations, weak equivalences, and fibrations of C as underlying cofib* *rations, weak equivalences, and fibrations. For convenience, assume that C is simplicial* * and write Map(., .) for the simplicial mapping space. In fact, the results of this sectio* *n carry over to the non-simplicial setting, but one must use the technical machinery of homotop* *y function complexes [10, Ch. 17]. Let K be a set of objects of C. Since we shall only use the homotopical prope* *rties of the objects in K, we may as well assume that each object in K is underlying cof* *ibrant. Definition 2.1. A map f : X ! Y in C is a K-colocal weak equivalence if for each A in K, the map Map(A, ^X) ! Map(A, ^Y) 4 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN is a weak equivalence of simplicial sets, where ^X! ^Yis a fibrant replacement * *for X ! Y , that is, there is a commuting square X ____//_^X | | | | fflffl| fflffl| Y ____//_^Y whose rows are underlying fibrant replacements. The idea is that we detect K-colocal weak equivalences by considering maps ou* *t of objects in K. Observe that the choice of fibrant replacements for X and Y does not matter; * *if the map Map(A, ^X) ! Map(A, ^Y) is a weak equivalence for one choice of fibrant replacements, then it is a weak* * equivalence for any other choice of fibrant replacements. Also note that underlying weak eq* *uivalences are automatically K-colocal weak equivalences. Definition 2.2. A map in C is a K-colocal fibration if it is an underlying fibr* *ation. A map in C is a K-colocal cofibration if it has the left lifting property with re* *spect to all K-colocal acyclic fibrations. Because there is no difference between underlying fibrations and K-colocal fi* *brations, we use the term "fibrationü nambiguously for maps in either class. We are defining a right Bousfield localization of the model category C in the* * sense of [10, Defn. 3.3.1]. We add more weak equivalences, keep the fibrations unchanged, and* * define the cofibrations to be what they must be. Theorem 2.6 states that under some general hypotheses on C, our definitions a* *re a model structure. However, the two-out-of-three axiom and the retract axiom are * *satisfied in general. This follows from an inspection of the definitions. The following results basically appear in [10, Ch. 5] with minor obvious chan* *ges in the proofs. Lemma 2.3. (a)The class of K-colocal acyclic cofibrations is the same as the class of unde* *rlying acyclic cofibrations. (b)Let A be any object of K. For n 0, the map i : @ [n] A ! [n] A is a K-col* *ocal cofibration. (c)Let p : X ! Y be a fibration between fibrant objects X and Y . Then p is a K* *-colocal acyclic fibration if and only if it has the right lifting property with resp* *ect to every underlying cofibration @ [n] A ! [n] A in which A belongs to K and n * *0. Proof.For part (a), the proof of [10, Lem. 5.3.2] works word for word. For part* *s (b) and (c), the proofs of [10, Prop. 5.2.5] and [10, Prop. 5.2.4] also work. Although * *we do not have a set of generating acyclic cofibrations at our disposal, we have avoided * *this necessity by assuming that the map in part (c) is already a fibration. In order to prove the rest of the model structure axioms, we must add hypothe* *ses on C. Hypothesis 2.4. Let C be a right proper simplicial model category, and let K be* * a set of cofibrant objects in C. Suppose that there exists a regular cardinal ~ with * *the following DUALITY AND PRO-SPECTRA 5 properties. First, each object of K is ~-small relative to the underlying cofi* *brations. Second, if X0 ! X1 ! . .!.Xfi! . . . is a ~-sequence of underlying cofibrations and p : colimfiXfi! Y is a map such * *that the composition pfi: Xfi! Y is a fibration for each successor ordinal fi, then p is* * also a fibration. See Section 3.3 for a review of the notions of smallness and ~-sequences. The idea is that fibrations are closed under a certain kind of sufficiently l* *ong sequential colimit. If C is cofibrantly generated, then we may choose ~ such that the dom* *ains of each of the underlying generating acyclic cofibrations as well as the objects o* *f K are ~- small relative to the underlying cofibrations. Hence cofibrantly generated mode* *l categories always satisfy Hypothesis 2.4. However, in our intended application to pro-spec* *tra, C is not cofibrantly generated, but the above hypothesis is still satisfied. It is not usually a problem to find a single ~ for which each A in K is ~-sma* *ll. As long as each A is ~A-small for some ~A, we may choose ~ to be an upper bound for the* * ordinals ~A. The following lemma is not a factorization axiom for the K-colocal model stru* *cture that we are constructing. The problem is that K-colocal acyclic fibrations are not d* *etected by the right lifting property with respect to the maps @ [n] A ! [n] A. See * *[10, Ex. 5.2.7] for an example of this problem. Using this lemma, the factorization * *we want follows from [10, Prop. 5.3.5]. Lemma 2.5. Under Hypothesis 2.4, every map f : X ! Y has a factorization into a K-colocal cofibration i : X ! W followed by a fibration p : W ! Y that has the * *right lifting property with respect to all maps @ [n] A ! [n] A for A in K. Proof.We use a variation on the small object argument [10, x 10.5]. Let J0 be t* *he set of all squares @ [n] A____//_X | | | f| fflffl| fflffl| [n] A_____//_Y for which A belongs to K. Define Z0 to be the pushout _ ! a a [n] A X, J0 J`@ [n] A 0 and let j0 : X ! Z0 and q0 : Z0 ! Y be the obvious maps. Now factor the map q0 into an underlying acyclic cofibration i0 : Z0 ! W0 fol* *lowed by a fibration p0 : W0 ! Y . This finishes the first stage of the factorization. We build the whole factorization by a transfinite induction of length ~. If f* *i is a limit ordinal, then set Wfito be colimff> | "" | " fflffl|" P 0 in which the dotted arrow exists because P ! P 0is a strict acyclic cofibration* * and Y is strictly fibrant. Therefore, Y is also a retract of P 0. The class of pro-objec* *ts that belong to C essentially levelwise is closed under retract [15, Thm. 5.5], so it suffic* *es to consider P 0. For each s, there is a zig-zag Zs As ! Ps ! Ps0 of weak equivalences. Since Zs belongs to C, the assumption on C implies that e* *ach Ps0 belongs to C. Remark 3.8. Let C be a proper simplicial model category, and let C be any class* * of objects of C that is closed under weak equivalences. Then strict weak equivalences pres* *erve the class of objects in pro-C that belong to C essentially levelwise. The proof of * *this fact is basically the same as the proof of Proposition 3.7 but slightly shorter. 4.Colocalizations of Pro-Categories In this section, we apply Theorem 2.6 to get a general colocalization result * *for homotopy theories of pro-categories. Let C be a proper simplicial model category. Let * *K be any set of fibrant objects in C, and let cK be the set of constant pro-objects cA s* *uch that A belongs to K. Definition 4.1. A map in pro-C is a cofibration if it is an essentially levelwi* *se cofibration. These cofibrations are exactly the strict cofibrations of pro-spectra. Definition 4.2. A map f : X ! Y in pro-C is a cK-colocal weak equivalence if it induces a weak equivalence Map pro(X~, cA) = colimsMapC(X~s, A) ! colimtMapC(Y~t, A) = Mappro(Y~, cA) for all A in K, where ~Xand ~Yare strictly cofibrant replacements for X and Y . Definition 4.3. A map in pro-C is a cK-colocal fibration if it has the right li* *fting property with respect to all cK-colocal acyclic cofibrations. 12 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN Theorem 4.4. Let C be a proper simplicial model category, and let K be any set * *of fibrant objects in C. Definitions 4.1, 4.2, and 4.3 define a left proper simplicial mod* *el structure on pro-C. Proof.This is an application of a dual version of Theorem 2.6 in which the obje* *cts of K are required to be cosmall and cofibrations are preserved by sequential limit* *s. These hypotheses are proved in Proposition 3.3 and [15, Cor. 5.4]. We emphasize that it is not necessary that the model category C satisfies Hyp* *othesis 2.4 in Theorem 4.4. It is important that pro-C does satisfy this hypothesis. This h* *appens by general arguments about pro-categories, not by using specific properties of C. Theorem 4.4 could be stated even more generally. There is no need to colocali* *ze with respect only to constant pro-objects, since every pro-object is cosmall for som* *e cardinal (see Corollary 3.5). Theorem 6.5 below is one example of the situation in Theorem 4.4. See [18] fo* *r other examples. We really do need the dual of Theorem 2.6 in order to establish the cK-coloca* *l model structure of Theorem 4.4; the dual of [10, Thm. 5.1.1] is not strong enough. Th* *e problem is that the strict model structure for pro-C is not fibrantly generated in general* *. See [17, x5] for a proof that the strict model structure for pro-simplicial sets is not fibr* *antly generated. Remark 4.5. Suppose that C is a stable model category in the sense that the loo* *ps and suspension functors are inverse Quillen equivalences of C with itself. Let K be* * a set of fibrant objects of C such that for every A in K, A and A~are weakly equivalen* *t to elements of K, where ~Ais a cofibrant replacement for A. In other words, K is c* *losed, up to homotopy, under suspensions and loops. Then it can be proved that the cK-co* *local model structure is also stable. We will not need this result, but the model str* *ucture of Theorem 6.5 is an example of this situation. We do not know whether the cK-colocal model structure on pro-C is always righ* *t proper, even though we are always assuming that C is proper. If we assume in addition t* *hat C is stable, then we can show that the model structure is right proper. Proposition 4.6. If C is a stable proper simplicial model category and K is a s* *et of fibrant objects of C, then the cK-colocal model structure is right proper. Proof.Suppose given a pullback square q W ____//_Z g|| f|| fflffl| fflffl| X __p_//_Y in pro-C in which p is a cK-colocal fibration and f is a cK-colocal weak equiva* *lence. We want to show that g is also a cK-colocal weak equivalence. Let F be the homotopy fibre of p with respect to the strict model structure, * *which is also the homotopy fibre of q. Now F~is the homotopy cofibre of both p and q, w* *here ~F is a cofibrant replacement for F . We have a diagram Map (W~, cA)___//_Map(Z~, cA)_//_Map( F~, cA) | | | | | | fflffl| fflffl| fflffl| Map (X~, cA)___//_Map(Y~, cA)_//_Map( F~, cA) DUALITY AND PRO-SPECTRA 13 of simplicial sets in which the rows are fibre sequences. Here ~W, ~Z, ~X, and * *~Yare cofibrant replacements for W , Z, X, and Y , and A is any object of K. The second and thi* *rd vertical maps are weak equivalences, which means that the first vertical map is also. 4.1. Fibrant Pro-Objects. Later we shall need a more explicit description of cK* *-colocal fibrant pro-objects. This section contains this description. We are still assum* *ing that C is a proper simplicial model category and that each A in K is fibrant. Definition 4.7. The class of K-nilpotent objects of C is the smallest class of * *fibrant objects such that: (1)the terminal object of C is K-nilpotent; (2)weak equivalences between fibrant objects preserve K-nilpotence; (3)and if X is K-nilpotent, A belongs to K, and X ! A@ [n]is any map, then t* *he fibre product X xA@ [n]A [n]is again K-nilpotent. In other words, an object is K-nilpotent if and only if it can be built, up t* *o weak equivalence, from the terminal object by a finite sequence of base changes of m* *aps of the form A [n]! A@ [n]with A in K. The terminology arises from the connection with nilpotent spaces when C is the category of simplicial sets and K is the co* *llection of Eilenberg-Mac Lane spaces. See [18] for details. In this paper, the only important example occurs with C the category of spect* *ra and K the set of spheres. In this specific case, we give in Proposition 6.6 a more* * concrete description of K-nilpotent spectra. Lemma 4.8. Let f : X ! Y be any cK-colocal weak equivalence between cofibrant p* *ro- objects. Then the map Map (f, cZ) : Map(Y, cZ) ! Map(X, cZ) is a weak equivalen* *ce for all K-nilpotent objects Z of C. Proof.The map Map (f, cZ) is a weak equivalence (even an isomorphism) when Z = * **. Since X and Y are cofibrant and cZ is strictly fibrant, the weak homotopy type* *s of Map(Y, cZ) and Map (X, cZ) do not depend on the choice of Z up to weak equivale* *nce. It only remains to consider condition (3) of Definition 4.7. Suppose for indu* *ction that Z is K-nilpotent and that Map(Y, cZ) ! Map(X, cZ) is a weak equivalence. Let Z0* *be the fibre product Z xA@ [k]A [k]for some object A in K. Since A [k]! A@ [k]is a fib* *ration, this fibre product is actually a homotopy fibre product, which means that Map (* *Y, cZ0) is the homotopy fibre product of the top row in the diagram Map (Y, cZ)____//Map(@ [k] Y, cA)___//Map( [k] Y, cA) | | | | | | fflffl| fflffl| fflffl| Map (X, cZ)____//Map(@ [k] X, cA)__//_Map( [k] X, cA), and Map (X, cZ0) is the homotopy fibre product of the bottom row. The left vert* *ical map is a weak equivalence by the induction assumption, and the other two vertical m* *aps are weak equivalences because the cK-colocal model structure is simplicial and beca* *use f is a cK-colocal weak equivalence. Thus, the induced map on homotopy fibre products i* *s also a weak equivalence. Proposition 4.9. An object X of pro-C is cK-colocal fibrant if and only if it i* *s strictly fibrant and essentially levelwise K-nilpotent. That X is essentially levelwise K-nilpotent means that X is isomorphic to a p* *ro-object Y such that each Ys is K-nilpotent. 14 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN Proof.First suppose that X is cK-colocal fibrant. Every cK-colocal fibration is* * a strict fibration, so X is strictly fibrant. It remains to show that X is essentially l* *evelwise K- nilpotent. By applying a functorial cofibrant replacement construction levelwis* *e to X, we get a map ~X! X which is a levelwise weak equivalence such that ~Xis levelwise * *cofibrant (and, in particular, strictly cofibrant). If we can show that ~Xis essentially * *levelwise K- nilpotent, then we can use Proposition 3.7 to conclude that X is also essential* *ly levelwise K-nilpotent. In other words, we might as well assume that X is strictly cofibra* *nt. Consider the factorization X ! W ! c* of the map X ! c* as described in the d* *ual to the proof of Lemma 2.5, so W is cK-colocal fibrant. Note that each cA is !-c* *osmall by Proposition 3.3. Therefore, we may take ~ to be !, and there are no limit ordin* *als in the construction of W . Since X is strictly cofibrant, the dual of Lemma 2.3(c) tells us that the map* * X ! W is a cK-colocal acyclic cofibration. Hence X is a retract of W because X is cK-* *colocal fibrant. The class of pro-objects having any property essentially levelwise is * *closed under retracts [15, Thm. 5.5], so it suffices to consider W . But the class of pro-ob* *jects having any property essentially levelwise is also closed under cofiltered limits [15, * *Thm. 5.1], so it suffices to consider each Wfi. Assume for induction that the pro-object Wfi-1is levelwise K-nilpotent. We ma* *y take a level representation for the diagram Q [n] JficA | | Q fflffl| Wfi-1____//_JficA@ [n], and we construct Zfiby taking the levelwise fibre product. In fact, it is possi* *ble to construct the level representation in such a way that the replacement for Wfi-1is a diagr* *am of objects that already appeared in the original Wfi-1. This means that the new Wf* *i-1is still levelwise K-nilpotent. The construction of arbitrary products in pro-categories [14, Prop. 11.1] sho* *ws that the map Y Y cA [n]! cA@ [n] Jfi Jfi is levelwise a finite product of maps of the form A [n]! A@ [n]. It follows immediately that Zfiis levelwise K-nilpotent. Now Wfi! Zfiis a leve* *lwise weak equivalence, so Wfiis also levelwise K-nilpotent. This finishes one implic* *ation. Now suppose that X is essentially levelwise K-nilpotent and strictly fibrant.* * We may assume that each Xs is K-nilpotent. Using the lifting property characterization* * of cK- colocal fibrant pro-objects, we must show that the map f : Map(B, X) ! Map(A, X) is an acyclic fibration of simplicial sets for every cK-colocal acyclic cofibra* *tion i : A ! B in pro-C. In fact, by [10, Prop. 13.2.1], we may further assume that A and B ar* *e cofibrant pro-objects (using that the cK-colocal model structure is left proper). We alre* *ady know that f is a fibration because of the strict model structure. Since X is strictl* *y fibrant, f is weakly equivalent to holimsfs, where fs is the map fs : Map(B, cXs) ! Map(A, cXs). DUALITY AND PRO-SPECTRA 15 Therefore, we need only show that each fs is a weak equivalence. This is true * *by Lemma 4.8. Remark 4.10. In Definition 4.7, one might also require that the class of K-nilp* *otent objects is closed under retracts. Strangely, this makes no difference in Proposition 4* *.9. The statement of that proposition is true exactly as worded, whether or not K-nilpo* *tent objects are closed under retracts. This surprising phenomenon arises from the surprisin* *g way in which retracts interact with essentially levelwise properties of pro-objects [1* *5, Thm. 5.5]. 4.2. Ind-Categories. All the results of this section dualize to ind-categories.* * More specif- ically, let K be a set of cofibrant objects in a proper simplicial model catego* *ry C. A map f : X ! Y in ind-C is a cK-colocal weak equivalence if it induces a weak equiva* *lence Map ind(cA, ^X) = colimsMapC(A, ^Xs) ! colimtMapC(A, ^Yt) = Mapind(cA, ^Y) for all A in K, where ^Xand ^Yare strictly fibrant replacements for X and Y . F* *ibrations are just strict fibrations, and cK-colocal cofibrations are defined by a liftin* *g property. These definitions give a right proper cK-colocal model structure on ind-C. If C* * is stable, then this model structure is also left proper. An ind-object is cK-colocal cofi* *brant if and only if it is strictly cofibrant and, up to isomorphism, it is levelwise weakly* * equivalent to an object that can be built out of the initial object by a finite sequence o* *f cofibrant homotopy pushouts of maps of the form @ [n] A ! [n] A. 5.Preliminaries on Spectra Now we review some definitions and results about spectra. We work in the cate* *gory of symmetric spectra [12]. This category has a proper simplicial cofibrantly gener* *ated stable model structure. We take this structure as the "standard" model structure for s* *pectra. Whenever we write "spectrum", we always mean "symmetric spectrum". We write Sn * *for a fixed cofibrant and fibrant model for the nth sphere spectrum. The category of symmetric spectra is closed symmetric monoidal. This means th* *at it has an associative commutative unital smash product ^ and an internal function * *object F (., .) such that F (Z, .) is right adjoint to .^Z. That is, there is a biject* *ion between maps X ! F (Z, Y ) and maps X ^ Z ! Y . We shall use the following model theoretic property of the functor F (., Y ) * *when Y is an arbitrary fixed fibrant spectrum [12, Cor. 5.3.9]. Namely, F (., Y ) takes c* *ofibrations to fibrations. More precisely, if i : A ! B is a cofibration, then F (i, Y ) : F (B, Y ) ! F (A, Y ) is a fibration. The weak equivalences in the category of spectra are defined in [12, Defn. 3.* *1.3]. The stable homotopy category is the category obtained by inverting these maps, whic* *h are also called stable equivalences. We will not repeat the definition of weak equi* *valence here, but using the following definition of homotopy groups we will state an equivale* *nt condition below. Definition 5.1. For any spectrum X, let ßnX be the set [Sn, X] of maps in the s* *table homotopy category. When X is a fibrant spectrum, ßnX is isomorphic to the traditional nth stable* * homotopy group colimkßn+kXk. More generally, we can calculate ßnX by considering the tra* *ditional nth stable homotopy group of a fibrant replacement for X. Weak equivalences ar* *e not defined in terms of homotopy groups because the definition of homotopy groups d* *epends 16 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN on the prior existence of the stable homotopy category. Nevertheless, the homot* *opy groups do detect stable weak equivalences of symmetric spectra in the sense that a map* * f : X ! Y is a stable equivalence if and only if ßnf is an isomorphism for every n 2 Z. Filtered colimits preserve fibrant spectra since the generating acyclic cofib* *rations have compact domains [12, Defn. 3.4.9] (recall that an object C of a category is com* *pact if Hom (C, .) commutes with filtered colimits). Since filtered colimits also p* *reserve the traditional stable homotopy groups, it follows that ßn(colimsXs) ~=colims(ßnXs) for every filtered diagram X of spectra. 6. ß*-Model Structure on Pro-Spectra In this section we specialize Theorem 4.4 to describe a model structure on th* *e category of pro-spectra that involves cohomotopy. Later we compare the associated homot* *opy theory to ordinary stable homotopy theory. Definition 6.1. A map of pro-spectra is a cofibration if it is an essentially l* *evelwise cofibration. These cofibrations are identical with strict cofibrations of pro-spectra. Recall that the cohomotopy group ßnX of a spectrum X is the group of stable w* *eak homotopy classes [X, Sn]. Thus ß* is the cohomology theory represented by the s* *phere spectrum S0. Definition 6.2. A map f : X ! Y of pro-spectra is a ß*-weak equivalence if it i* *nduces an isomorphism colimsßnYs ! colimtßnXt for every n 2 Z. It is important that we are not requiring that ßnX ! ßnY be an ind-isomorphis* *m. Non- isomorphic ind-abelian groups may have isomorphic colimits when one allows infi* *nitely generated abelian groups. It is also important that we are not using the groups* * ßn limsXs or ßn holimsXs. In general, very little can be said about these groups in term* *s of the groups ßnXs. Proposition 6.3. Let f : X ! Y be a map of pro-spectra, and let ~f: ~X! ~Ybe a * *strictly cofibrant replacement for f. The following conditions are equivalent: (1)f is a ß*-weak equivalence; (2)Map (f~, cSn) : Map (Y~, cSn) ! Map (X~, cSn) is a weak equivalence of si* *mplicial sets for every n 2 Z. (3)colimtF (Y~t, S0) ! colimsF (X~s, S0) is a weak equivalence of spectra. Proof.For any cofibrant pro-spectrum Z and any k 0, ßkMap (Z, cSn) = ßkcolimtMap(Zt, Sn) = colimt[Zt, Sn-k]. The case k = 0 tells us that condition (2) implies condition (1). Now suppose that f is a ß*-weak equivalence. From the computation of the prev* *ious paragraph, we know that Map (f~, cSn) induces an isomorphism on all homotopy gr* *oups at the canonical basepoints. Since Map (f~, cSn) is weakly equivalent to the l* *oop space Map (f~, cSn+1), we only need to compute homotopy groups at one basepoint. Th* *is shows that condition (1) implies condition (2). For condition (3), note that ßkcolimtF (X~, S0) is isomorphic to colimtß-kXt * *(and similarly for ~Y). This shows that condition (3) is equivalent to condition (1). DUALITY AND PRO-SPECTRA 17 Definition 6.4. A map of pro-spectra is a ß*-fibration if it has the right lift* *ing property with respect to all ß*-acyclic cofibrations. Theorem 6.5. Definitions 6.1, 6.2, and 6.4 define a proper simplicial model str* *ucture on the category of pro-spectra. We call this the ß*-model structure on pro-spectra. Proof.Proposition 6.3 tells us that we are discussing a cK-colocalization, wher* *e K is the set of spheres. Therefore, Theorem 4.4 gives us everything but right properness* *. Since the model category of spectra is stable (see the last paragraph of Section 3), Prop* *osition 4.6 gives us right properness. Now we will identify the ß*-fibrant pro-spectra. We say that a spectrum is ho* *motopy- finite if it is weakly equivalent to a finite complex, i.e. if its image in the* * stable homotopy category is in the thick subcategory generated by S0. Proposition 6.6. Let K be the set of spheres. A spectrum is K-nilpotent (see D* *efini- tion 4.7) if and only if it is fibrant and homotopy-finite. Proof.First suppose that X is K-nilpotent. We work by induction over the number* * of pullbacks in the construction of X, noting that the terminal object is homotopy* *-finite and weak equivalences preserve homotopy-finiteness. To do the inductive step, assume that X equals Y x(Sk)@ [n](Sk) [n], where Y * * is homotopy-finite. We have to show that X is also homotopy-finite. Each of the sp* *ectra Y , (Sk)@ [n], and (Sk) [n]is homotopy-finite, and the map (Sk) [n]! (Sk)@ [n]i* *s a fibration. Therefore, X is a homotopy fibre product of three homotopy-finite sp* *ectra, so X is also homotopy-finite. Now assume that X is fibrant and homotopy-finite. We have to show that X is * *K- nilpotent. We induct on the number of cells in X. If X is weakly contractible, * *then it is K-nilpotent by definition. To do the inductive step, suppose that there is a fi* *bre sequence X ! Y ! Sk, where X has one more cell than Y and Y is K-nilpotent. (This is dual to the usu* *al way of attaching cells, but produces the same class of finite complexes because of * *Spanier- Whitehead duality.) We claim that there is a homotopy pullback square *______//(Sk) [1] | | | | | fflffl| fflffl| Sk ____//_(Sk)@ [1], where the bottom horizontal map is inclusion into the first factor. See the fol* *lowing lemma for the proof. In the diagram X _____//_*____//(Sk) [1] | | | | | | | | fflffl| fflffl| fflffl| Y ____//_Sk___//(Sk)@ [1], the left square is also a homotopy pullback square since X is the homotopy fibr* *e of Y ! Sk. Thus the composite square is also a homotopy pullback square, which means that * *X is K-nilpotent. 18 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN Lemma 6.7. Let f : Sk ! (Sk)@ [1]= Sk x Sk be the inclusion into the first fact* *or. Then there is a homotopy pullback square *______//(Sk) [1] | | | | | fflffl| fflffl| Sk __f_//_(Sk)@ [1]. Proof.Consider the pushout diagram @ [1]+_____// [1]+ | | | | fflffl| fflffl| [0]+ _____//_ [1] of pointed simplicial sets, where the top horizontal arrow is the obvious inclu* *sion and the left vertical arrow takes 0 to 0 and takes both 1 and the basepoint to the base* *point. Note that [1] is pointed at 1. If we apply the functor Map ( 1 (-), Sk) to this diagram, we obtain a pullbac* *k square Map ( 1 [1], Sk)___//(Sk) [1] | | | | fflffl| fflffl| (Sk) [0]________//(Sk)@ [1] of spectra in which the upper right corner is contractible. This diagram is a h* *omotopy pullback diagram because the right vertical map is a fibration. Proposition 6.8. A pro-spectrum X is ß*-fibrant if and only if it is essentiall* *y levelwise homotopy-finite and strictly fibrant. This means that each Xs has the weak homotopy type of a finite complex. There* * is no compatibility requirement for the weak equivalences between the spectra Xs a* *nd the finite complexes. Proof.This follows immediately from Proposition 4.9 together with Proposition 6* *.6. Remark 6.9. Using the above description of ß*-fibrant objects, it is possible t* *o describe explicitly the total derived functor Rlimof the limit functor from pro-spectra * *to spectra. This functor is computed by taking the limit of a ß*-fibrant replacement. With * *the strict structure on pro-spectra, Rlim is just the homotopy limit functor. However, wi* *th the ß*-model structure, Rlimis related to Spanier-Whitehead duality. Let X be a spectrum; we shall calculate Rlim(cX). Take a ß*-fibrant replaceme* *nt ^Xfor cX. From Proposition 4.9, we know that each spectrum in ^Xis homotopy-finite; t* *hus, ^Xis levelwise weakly equivalent to F (F (X^, S0), S0). Therefore, Rlim(cX) is weakl* *y equivalent to holimsF (F (X^s, S0), S0), which is equivalent to F (colimsF (X^s, S0), S0).* * A computa- tion of homotopy groups shows that ßn colimsF (X^s, S0) equals ß-nX, so colimsF* * (X^s, S0) is weakly equivalent to the ordinary Spanier-Whitehead dual F (X, S0) of X. Thu* *s, we have shown that Rlim(cX) is equivalent to F (F (X, S0), S0). A similar analysis shows that if X is an arbitrary pro-spectrum, then RlimX i* *s equiv- alent to F (colimsF (Xs, S0), S0), i.e., the Spanier-Whitehead dual of the coli* *mit of the levelwise Spanier-Whitehead dual of X. DUALITY AND PRO-SPECTRA 19 7. ß*-Model Structure on Ind-Spectra Now we proceed to ind-spectra. All of the following definitions and results a* *re dual to analogous results in the previous section. We skip the proofs because they are * *no different. Definition 7.1. A map of ind-spectra is a fibration if it is an essentially lev* *elwise fibra- tion. These fibrations are identical with strict fibrations of ind-spectra. Definition 7.2. A map of ind-spectra X ! Y is a ß*-weak equivalence if for every n 2 Z, the map colimsßnXs ! colimtßnYt is an isomorphism. Proposition 7.3. Let f : X ! Y be a map of ind-spectra, and let ^f: ^X! ^Ybe a * *strictly fibrant replacement for f. The following conditions are equivalent: (1)f is a ß*-weak equivalence; (2)Map (cSn, ^f) : Map (cSn, ^Y) ! Map (cSn, ^X) is a weak equivalence of si* *mplicial sets for every n 2 Z; (3)colimsXs ! colimtYt is a stable weak equivalence of spectra. Definition 7.4. A map of ind-spectra is a ß*-cofibration if it has the left lif* *ting property with respect to all ß*-acyclic fibrations. Theorem 7.5. Definitions 7.1, 7.2, and 7.4 define a proper simplicial model str* *ucture on the category of ind-spectra. We call this the ß*-model structure on ind-spectra. Proof.Everything but left properness is an application of the dual version of T* *heorem 4.4. Left properness follows in a manner dual to the proof of Proposition 4.6. Proposition 7.6. Consider the smallest class of cofibrant spectra such that: (1)* belongs to the class; (2)the class is closed under weak equivalences between cofibrant spectra; (3)and if X belongs`to the class and @ [n] Sk ! X is any map, then the pus* *hout [n] Sk @ [n] SkX also belongs to the class. This class coincides with the class of cofibrant homotopy-finite spectra. Proposition 7.7. An ind-spectrum X is ß*-cofibrant if and only if it is essenti* *ally level- wise homotopy-finite and strictly cofibrant. 8.Comparison of Homotopy Theories This section contains the main results of this paper. Namely, the homotopy ca* *tegory of pro-spectra is the opposite of the ordinary stable homotopy category. First,* * we study the homotopy theory of ind-spectra. Theorem 8.1. The constant functor c from spectra to ind-spectra is right adjoin* *t to the functor colim. These functors form a Quillen equivalence when considering the ß* **-model structure on ind-spectra. Proof.Let X be an ind-spectrum, and let Y be a spectrum. By direct calculatio* *n, Hom ind(X, cY ) ~=Hom (colimX, Y ). Thus c and colimare adjoint. The functor c preserves fibrations and weak equivalences. Therefore, c and co* *limare a Quillen pair. By Proposition 7.3, X ! cY is a ß*-weak equivalence if and only if colimsXs !* * Y is a weak equivalence of spectra. Hence, c and colimform a Quillen equivalence. 20 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN According to Theorem 8.1, the homotopy category of ind-spectra is equivalent * *to the ordinary stable homotopy category. We suspect that this model structure is not * *cofibrantly generated, but we have not been able to prove it. By Proposition 7.7, every ind-spectrum X is ß*-weakly equivalent to an ind-sp* *ectrum whose objects are finite cell complexes. Therefore, the ordinary stable homotop* *y category is equivalent to a homotopy category of ind-(finite cell complexes), but this l* *atter homotopy category does not arise from a model structure. The next step is to compare the categories of pro-spectra and ind-spectra. Lemma 8.2. The contravariant functor F (., Y ) from spectra to spectra is its o* *wn adjoint. Proof.A map X ! F (Z, Y ) corresponds to a map X ^ Z ! Y . This corresponds to a map Z ! F (X, Y ) because X ^ Z and Z ^ X are isomorphic. Let Y be a fixed spectrum. By acting levelwise, the functor F (., Y ) induce* *s a con- travariant functor from pro-spectra to ind-spectra. It also induces a contravar* *iant functor from ind-spectra to pro-spectra. Proposition 8.3. Let Y be an arbitrary fixed spectrum. The contravariant func* *tors F (., Y ) from pro-spectra to ind-spectra and from ind-spectra to pro-spectra a* *re adjoint in the sense that Hom pro(X, F (Z, Y )) ~=Hom ind(Z, F (X, Y )) for every pro-spectrum X and every ind-spectrum Z. Proof.This follows from direct computation and Lemma 8.2. Proposition 8.4. Let Y be a homotopy-finite fibrant spectrum. Then the contrava* *riant functors F (., Y ) from pro-spectra to ind-spectra and from ind-spectra to pro-* *spectra are a Quillen pair between the ß*-model structure on pro-spectra and the opposite of * *the ß*-model structure on ind-spectra. Proof.We already know that the functors are an adjoint pair by Proposition 8.3.* * In order to show that they are a Quillen pair, we must prove that F (., Y ) takes * *cofibrations (resp., ß*-acyclic cofibrations) of pro-spectra to fibrations (resp., ß*-acycli* *c fibrations) of ind-spectra. Since Y is fibrant, F (., Y ) takes cofibrations of spectra to fibrations of * *spectra. There- fore, F (., Y ) takes levelwise cofibrations of pro-spectra to levelwise fibrat* *ions of ind- spectra. It follows that F (., Y ) takes essentially levelwise cofibrations of* * pro-spectra to essentially levelwise fibrations of ind-spectra. Now let i : A ! B be a ß*-acyclic cofibration of pro-spectra. Fix k 2 Z and l* *et Z be a fibrant model for the spectrum kY . Since Z is again homotopy-finite, the cons* *tant pro- spectrum cZ is ß*-fibrant. Therefore, the map Map (B, cZ) ! Map (A, cZ) is an a* *cyclic fibration of simplicial sets. In particular, ß0colimsMap (Bs, Z) ! ß0colimtMap(* *At, Z) is an isomorphism. This means that ß0colimsF (Bs, Z) ! ß0colimtF (At, Z) is also * *an isomorphism (because ß0F (C, D) = ß0Map (C, D) for any spectra C and D and be- cause ß0 commutes with filtered colimits). Since ß0colimsF (Bs, Z) is isomorph* *ic to ßkcolimsF (Bs, Y ) (and similarly for A), it follows that colimsF (Bs, Y ) ! co* *limtF (At, Y ) is a weak equivalence of spectra. Proposition 7.3 tells us that the map F (i, Y* * ) : F (B, Y ) ! F (A, Y ) is a ß*-weak equivalence of ind-spectra. Theorem 8.5. The contravariant functors F (., S0) from pro-spectra to ind-spect* *ra and from ind-spectra to pro-spectra form a Quillen equivalence between the ß*-model* * structure on pro-spectra and the opposite of the ß*-model structure on ind-spectra. DUALITY AND PRO-SPECTRA 21 Proof.We already showed in Proposition 8.4 that the functors are a Quillen pair* *. Note that the hypothesis of Proposition 8.4 is satisfied because S0 is homotopy-fini* *te. Let X be a cofibrant pro-spectrum, let Z be a cofibrant ind-spectrum, and let* * f : X ! F (Z, S0) be a map of pro-spectra. Our goal is to show that f is a ß*-weak equi* *valence if and only if the adjoint map Z ! F (X, S0) is a ß*-weak equivalence of ind-spect* *ra. By Proposition 6.3, the map f is a ß*-weak equivalence if and only if the map colimsF (F~(Zs, S0), S0) ! colimtF (X, S0) is a weak equivalence of spectra. Here ~F(C, D) refers to a cofibrant replaceme* *nt for the function spectrum F (C, D). By Proposition 7.7, we may assume that each Zs is homotopy-finite. Since the * *Spanier- Whitehead double dual of a finite complex is itself, the map Z ! F (F~(Z, S0), * *S0) is a levelwise weak equivalence. In particular, the map colimsZs ! colimsF (F~(Zs, S0), S0) is a weak equivalence of spectra. The previous two paragraphs imply that f is a weak equivalence if and only if* * the composition colimsZs ! colimsF (F~(Zs, S0), S0) ! colimtF (X, S0) is a weak equivalence of spectra. By Proposition 7.3, this last map is a weak e* *quivalence if and only if the map Z ! F (X, S0) is a ß*-weak equivalence of ind-spectra. Corollary 8.6 (Main result). The category of pro-spectra with its ß*-model stru* *cture is Quillen equivalent to the opposite of the category of symmetric spectra with it* *s usual stable model structure. The equivalence is a composite of two Quillen pairs going in * *opposite directions: pro-spectra! ind-spectraop spectraop. We have indicated the directions of the left adjoints. 22 J. DANIEL CHRISTENSEN AND DANIEL C. ISAKSEN References [1]M. Ando and J. Morava, A renormalized Riemann-Roch formula and the Thom isom* *orphism for the free loop space, in Topology, geometry, and algebra: interactions and new di* *rections (Stanford, CA, 1999), 11-36, Contemp. Math., 279, Amer. Math. Soc., Providence, RI, 2001, m* *ath.AT/0101121. [2]M. Artin, A. Grothendieck, and J. L. Verdier, Th'eorie des topos et cohomolo* *gie 'etale des sch'emas, Lecture Notes in Mathematics, vol. 269, Springer Verlag, 1972. [3]M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 10* *0, Springer Verlag, 1969. [4]J. C. Becker and D. H. Gottlieb, A history of duality in algebraic topology,* * History of topology, North-Holland, Amsterdam, 1999, pp. 725-745. [5]A. K. Bousfield and E. M. Friedlander, Homotopy theory of -spaces, spectra,* * and bisimplicial sets, Geometric Applications of Homotopy Theory, vol. II (Proc. Conf., Evanston, I* *L, 1977), Lecture Notes in Mathematics, vol. 658, Springer Verlag, 1978, pp. 80-130. [6]R. L. Cohen, J. D. S. Jones and G. B. Segal, Floer's infinite dimensional Mo* *rse theory and homotopy theory, in The Floer memorial volume, 297-325, Progr. Math., 133, Birkhäuser* *, Basel, 1995. [7]W. G. Dwyer and J. Spali'nski, Homotopy theories and model categories, Handb* *ook of algebraic topology, North-Holland, 1995, pp. 73-126. [8]D. A. Edwards and H. M. Hastings, Cech and Steenrod homotopy theories with a* *pplications to geo- metric topology, Lecture Notes in Mathematics, vol. 542, Springer Verlag, 19* *76. [9]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and * *algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Math* *ematical Society, 1997. [10]P. S. Hirschhorn, Model categories and their localizations, Mathematical Su* *rveys and Monographs, Vol. 99, American Mathematical Society, 2003. [11]M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, A* *merican Mathematical Society, 1999. [12]M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Amer. Math. Soc. * *13 (2000), no. 1, 149-208. [13]D. E. Hurtubise, The Floer homotopy type of height functions on complex Gra* *ssmann manifolds, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2493-2505. [14]D. C. Isaksen, A model structure for the category of pro-simplicial sets, T* *rans. Amer. Math. Soc. 353 (2001), 2805-2841. [15]D. C. Isaksen, Calculating limits and colimits in pro-categories, Fund. Mat* *h. 175 (2002), no. 2, 175-194. [16]D. C. Isaksen, Etale realization on the A1-homotopy theory of schemes, Adv.* * Math., to appear. [17]D. C. Isaksen, Strict model structures for pro-categories, in Algebraic Top* *ology: Categorical Decom- position Techniques (Skye, 2001), Progress in Mathematics, Vol. 215, Birkhau* *ser, 2003, 179-198. [18]D. C. Isaksen, Completions of pro-spaces, in preparation. [19]J. P. May, Stable algebraic topology, 1945-1966, History of topology, North* *-Holland, Amsterdam, 1999, pp. 665-723. [20]C. V. Meyer, Approximation filtrante de diagrammes finis par Pro-C, Ann. Sc* *i. Math. Qu'ebec 4 (1980), no. 1, 35-57. [21]D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, * *Springer Verlag, 1967. [22]E. H. Spanier and J. H. C. Whitehead, Duality in homotopy theory, Mathemati* *ka 2 (1955), 56-80. [23]D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of * *Math. (2) 100 (1974), 1-79. Department of Mathematics, University of Western Ontario, London, Ontario, Ca* *nada E-mail address: jdc@uwo.ca Department of Mathematics, Wayne State University, Detroit, MI 48202, USA E-mail address: isaksen@math.wayne.edu