A GUIDE TO TELESCOPIC FUNCTORS NICHOLAS J. KUHN Abstract. In the mid 1980's, Pete Bousfield and I constructed certain p-local `telescopic' functors n from spaces to spectra, for each prime p, and each n 1. They are constructed using the full strength of the Nilpotence and Periodicity Theorems of Devanitz-Hopkins-Smith, and have some striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of peri* *odic phenomena. Here I offer a guide to their construction, characterization, application, and computation. 1. Introduction Let T and S be the categories of based spaces and spectra, localized at a fixed prime p, and 1 : T ! S and 1 : T ! S the usual adjoint pair. For n 1, in [K1 ], the author constructed functors between the homotopy categories Kn: ho(T ) ! ho(S) such that Kn( 1 X) ' LK(n)X, where LK(n) denotes Bousfield localization with respect to the Morava K- theory K(n). Thus the K(n)-localization of a spectrum depends only on its zero space. This mid 1980's result was modeled on the n = 1 version that had been newly established by Pete Bousfield in [B2 ], and heavily used the newly proved Nilpotence and Periodicity Theorems of Ethan Devanitz, Mike Hop- kins, and Jeff Smith [DHS , HS ]. Bousfield's main application was to proving uniqueness results about in- finite loopspaces; for example, he gives a `conceptual' proof of the Adams- Priddy theorem [AP ] that BSO(p)admits a unique infinite loopspace struc- ture up to homotopy equivalence. My main application was to note that the evaluation map ffl : 1 1 X ! X has a section after applying LK(n), and thus after applying other functors like K(n)*. The functors Knas described above allow for two important refinements. ____________ Date: February 2, 2008. 2000 Mathematics Subject Classification. Primary 55Q51; Secondary 55N20, 55P* *60, 55P65. This research was partially supported by a grant from the National Science F* *oundation. 1 2 KUHN Firstly, let T (n) be the mapping telescope of any vn-self map of a finite CW spectrum of type n. It is a well known application of the Periodicity Theorem that the associated localization functor LT(n) is independent of choices, and it is evident that K(n)-local objects are T (n)-local1. This suggests that Kn might refine to a functor Tn: ho(T ) ! ho(S) with LK(n) O Tn= Kn, such that Tn( 1 X) ' LT(n)X. Indeed, a careful reading of the arguments in [K1 ] shows that this is the case. One application of this refined functor is that there is a natural isomorphism of graded homotopy groups [B, Tn(Z)]* ' v-1 ss*(Z; B), for all spaces Z, where v : dB ! B is any unstable vn self map. Thus the spectrum Tn(Z) determines `periodic unstable homotopy'. Secondly, what one really wishes to have is a functor on the level of model categories, n : T ! S, inducing Tnon associated the homotopy categories. Once again, inspection of the papers [B2 , K1 ] suggests that this should be possible. However, it was not until Bousfield revisited these constructions in his 2001 paper [B5 ] that this was carefully worked out. One new consequence that emerged was Bousfield's beautiful theorem that every spectrum is naturally T (n)- equivalent to a suspension spectrum. Along with Bousfield's new application, there has been recent use of n by the author [K2 , K3 ] and C. Rezk [Re ], and new methods for computation available using the work of Arone and Mahowald in [AM ]. All of this suggests that the functors n have a fundamental role in the study of homotopy, both stable and unstable, as chromatically organized. Bousfield's detailed paper [B5 ] is not an easy read: the partial ordering on that paper's set of lemmas, propositions, and theorems induced by the logical flow of the proof structure is poorly correlated with the numerical total ordering. One could make a similar statement about [B3 ], on which [B5 ] relies in essential ways. By constrast, my paper [K1 ] offers a quite direct approach to the construc- tion of the Kn, while being admittedly short on detail. If one fills in detail* *s, and adds refinement as described above, it emerges that Bousfield and I have slightly different constructions. It turns out that both flavors satisfy basic characterizing properties, and thus they are naturally equivalent. Motivated by all of this, here we offer a guide to the n. This includes ____________ 1The Telescope Conjecture, open for n 2, asserts that the converse is also* * true, so that LT(n)= LK(n). TELESCOPIC FUNCTORS 3 o a listing of basic properties, and characterization of the functors by some of these, o a step by step discussion of their construction, including model cat- egory issues that arise, o Bousfield's `left adjoint' n : S ! T and its basic application, o a discussion of the uniqueness of the section to LT(n)(ffl), and o a discussion of calculations of n(Z) for various spaces Z including spheres. Though most of the results surveyed appear in the literature, a few haven't. Among those that have, I have tweaked the order in which they are `revealed'. For example, and most significantly, our Theorem 4.2, which describes important properties of the functor v (see just below) when v is a vn-self map, is proved in a direct manner here, enroute to proving our main theorem Theorem 1.1, which lists important properties of n. By constrast, in [B5 ], these properties of v first occur as consequences of the properties of n. We hope readers appreciate such unknotting of the logic. We end this introduction by stating a new characterization of the n. We need to briefly describe the basic construction on which the functors n are based. A self map of a space v : dB ! B with d > 0, induces a natural transformation v(Z) : Map T(B, Z) ! d Map T(B, Z) for all spaces Z 2 T . The map v(Z) then can be used to define a periodic spectrum v(Z) of period d, such that ss*( v(Z)) ' colim{[B, Z]* v-![B, Z]*+d v-!. .}.= v-1 ss*(Z; B). Theorem 1.1. For each n 1, there is a continuous functor n : T ! S satisfying the following properties. (1) n(Z) is T (n)-local, for all spaces Z. (2) There is a weak equivalence of spectra Map S(B, n(Z)) ' v(Z), for all unstable vn self maps v : dB ! B, natural in both Z and v. (3) There is a natural weak equivalence n( 1 X) ' LT(n)X, for all - spectra X. Furthermore, properties (1) and (2) characterize n, up to weak equiva- lence of functors. The rest of the paper is organized as follows. Background material, about both the model category of spectra and periodic homotopy, is given in x2. In x3, we present the basic theory of the telescopic functor v associated to a self map v : dB ! B, and, in x4, we study v when v is additionally a vn-self map. In x5, we define n, and the proof of Theorem 1.1 follows quickly from the previous results. The adjoint n is defined in x6, and using 4 KUHN it, we prove Bousfield's theorem that spectra are T (n)-equivalent to sus- pension spectra. A short discussion about the section to the T (n)-localized evaluation map is given in x7. Finally, in x8, we offer a brief guide to known computations of n(Z) and periodic homotopy groups. An outline of this material was presented in a talk at the special session on homotopy theory at the A.M.S. meeting held in Newark, DE in April, 2005. I would like to offer my congratulations to Martin Bendersky, Don Davis, Doug Ravenel, and Steve Wilson - the 60th birthday boys of that session and the March, 2007 conference at Johns Hopkins University - and thank them all for setting fine examples of grace and enthusiasm to we algebraic topologists who have followed. 2.Background 2.1. Categories of spaces and spectra. These days, it seems prudent to be precise about our categories of `spaces' and `spectra', and needed model category structures. We will let T denote the category of based compactly generated topolog- ical spaces, though one could as easily work instead with the category of based simplical sets, as Bousfield always does. Regarding spectra, we would like a single map of the form C ! dC to specify a spectrum. This suggests using the `plain vanilla' category of (pre)spectra (N -spectra in [MMSS ]). An object X in the category S is a sequence of spaces in T , X0, X1, . .,. together with a sequence of based maps oeXn: Xn ! Xn+1, or, equivalently, "oeXn: Xn ! Xn+1, for n 0. A morphism f : X ! Y in S is then a sequence of based maps fn : Xn ! Yn such that the diagram fn Xn _____//_ Yn |oeXn| oeYn|| fflffl|fn+1fflffl| Xn+1 _____//Yn+1 commutes for all n. The category S is a topological category; in particular Map S (X, Y ) is an object in T . It is also tensored and cotensored over T , with A ^ X and MapS (A, X) denoting the tensor and cotensor product of A 2 T with X 2 S. (See [MMSS , p.447] for more detail.) We let dX and dX denote Sd ^ X and MapS (Sd, X), as is usual. The adjoint pair 1 : T AE S : 1 is defined by letting ( 1 A)n = nA and 1 X = X0. For d 0, we let sd : S ! S be the d-fold shift functor with (sdX)n = Xn+d. This admits a left adjoint s-d : S ! S with ( Xn-d forn d (s-dX)n = * for0 n d. TELESCOPIC FUNCTORS 5 Composing these adjoints, we see that s-d O 1 : T ! S is left adjoint to the functor sending a spectrum X to its dth space Xd. 2.2. Model category structures. We describe model category structures on T and S. Our category T is endowed with the `usual' model category structure (see, e.g. [DS ]): the weak equivalences are the weak homotopy equivalences, the fibrations are the Serre fibrations, and the cofibrations are retracts of generalized CW inclusions. (We recall that f : A ! B in T is a weak homotopy equivalence if, for each point a 2 A, f* : ss*(A, a) ! ss*(B, f(a)) is a bijection, and is a Serre fibration if it has the right lifting property w* *ith respect to the maps Dn ,! Dn ^ I+ .) Starting from this model category structure on T , S is given its stable model category structure `in the usual way', as in [S, H2 , MMSS ], all of which follow the lead of [BF ]. Firstly, S has its `level' model structure2 in which the weak equivalences and fibrations are the maps f : X ! Y such that the levelwise maps fn : Xn ! Yn are weak equivalences and fibrations in T for all n. It is then easy to check that f is a cofibration exactly when the induced maps X0 ! Y0 and Xn+1 [ Xn Yn ! Yn+1 are cofibrations in T . When needed, we will write Sl for the category of spectra with the level model structure. Now we need to change the model structure to build in stability. Hovey's general method [H2 ] yields the following in our situation. We call a spectrum X an -spectrum if "oen: Xn ! Xn+1 is a weak equivalence in T for all n. Using our adjunctions, this rewrites as the statement that MapS(in, X) is a weak equivalence in T for all n, where in : s-(n+1) 1 S1 ! s-n 1 S0 is the canonical map in S. Let Q : Sl ! Sl denote Bousfield localization (as in [Hi]) with respect to the set of map {in, n 0}. Then [H2 , Thm.2.2] says that there is a stable model structure on S with cofibrations equal to level cofibrations, with weak equivalences the maps f : X ! Y such that Qf : QX ! QY is a level equivalence, and with fibrant objects the level fibrant -spectra. There are two alternative characterizations of the stable equivalences. It is formal to see that Qf : QX ! QY is a weak level equivalence if and only if f* : [Y, Z]l! [X, Z]l is a bijection for all -spectra Z, where [Y, Z]l denotes homotopy classes computed using the level model structure. True, but not formal, is the fact that weak equivalences are precisely maps of spectra inducing isomorphisms on ss*(X) = colimnss*+n(Xn): see [MMSS , Proposition 8.7] for a clear discussion of this point. It is easy to check that S is a topological model category in the sense of [EKMM , Definition 4.2], so that, for all spectra X and Y , [X, Y ] = ss0(MapS (Xcof, Y fib)), ____________ 2The name `level' model structure is used in [MMSS , x6]. Schwede [S] refer* *s to this as `strict', and Hovey [H2] uses `projective'. 6 KUHN where Xcof and Y fibare respectively cofibrant and fibrant replacements for X and Y . (Compare with [GJ , Proposition 3.10] for a nice presentation in the simplicial setting.) Handy observations include that the evident natural maps dX ! sdX and s-dX ! dX are stable equivalences. Also useful in calculation is that, if Xfibis a fibrant replacement for a spectrum X, then each of the evident maps 1 Xfib! hocolimn nXfibn hocolimn nXn is a weak equivalence of spaces. In one proof of ours - the proof of Theorem 5.2 - we make use of function spectra in the homotopy category of spectra3: these exist in ho(S) using well known `naive' constructions in S. To summarize our overuse of the notation MapS (X, Y ): o MapS (X, Y ) is in T for X, Y 2 S, o MapS (X, Y ) is in S for X 2 T and Y 2 S, and o MapS (X, Y ) is in ho(S) for X, Y 2 ho(S) We trust our meaning will be clear in context. We end this subsection with a useful lemma and corollary. Lemma 2.1 (Compare with [K1 , Lemma 3.3]). Given a diagram of spectra X(0) ! X(1) ! X(2) ! . . .and an increasing sequence of integers 0 d0 < d1 < d2 < . .,.the natural diagram of spectra s-d1 1 d1-d0X(0)d0W s-d2 1 d2-d1X(1)d1W s-d3 1 d3-d2X(2)d2Q offlffl| WWWWWWW++W o|fflffl WWWWWWW++W offlffl|QQ Q s-d0 1 X(0)d -d1 1 -d2 1 Q(( 0 s X(1)d1 s X(2)d2 fflffl| |fflffl fflffl| X(0) __________________//X(1)_________________//X(2)`` ` ` ``// induces a weak equivalence between the homotopy colimit of the top zig-zag and the homotopy colimit of the bottom. Sketch proof.One checks that the map induces an isomorphism on ss*. Al- ternatively, one can check that the map induces an isomorphism on [__, Y ] for all Y 2 S. Informally, this lemma says that there is a natural weak equivalence hocolims-dk 1 X(k)dk ~-!hocolimX(k). k k Corollary 2.2. Given a spectrum X and an increasing sequence of integers 0 d0 < d1 < d2 < . .,.the homotopy colimit of s-d1 1 d1-d0Xd0 s-d2 1 d2-d1Xd1 s-d3 1 d3-d2Xd2 UU UUUU PPP offlffl||UUUUUUU**U offlffl|| UUUUU**U offlffl||PPPPP -d 1 -d 1 -d 1 P((P s 0 Xd0 s 1 Xd1 s 2 Xd2 . . . ____________ 3Bousfield similarly needs this: see [B5, Thm. 11.9]. TELESCOPIC FUNCTORS 7 is naturally weakly equivalent to X. Proof.Apply the lemma to the case when X(k) = X for all k. Informally, this corollary says that there is a natural weak equivalence hocolim s-dk 1 Xdk ~-!X. k 2.3. Periodic homotopy. We recall some of the terminology and big the- orems used when one studies homotopy from the chromatic point of view. A good general reference for this material is Doug Ravenel's book [Ra ]. We let C ho(S) denote the stable homotopy category of p-local finite CW spectra, and then we let Cn C be the full subcategory with objects the K(n - 1)*-acylic spectra. The categories Cn are properly nested [Mi ]: C = C0 C1 C2 . ... An object F 2 Cn - Cn+1 is said to be of type n. For finite spectra, the remarkable work of Ethan Devanitz, Mike Hopkins, and Jeff Smith [DHS ] tells us the following. Theorem 2.3 (Nilpotence Theorem [HS , Thm.3]). Given F 2 C, a map v : dF ! F is nilpotent if and only if K(n)*(v) is nilpotent for all n 0. The next two consequences were proved by Hopkins and Smith. Theorem 2.4 (Thick Subcategory Theorem [HS , Ra ]). A nonempty full subcategory of C that is closed under taking cofibers and retracts is Cn for some n. Given F 2 C, a map v : dF ! F is called a vn-self map if K(n)*(v) is an isomorphism, while K(m)*(v) is nilpotent for all m 6= n. Theorem 2.5 (Periodicity Theorem [HS , Ra ]). (a) F 2 Cn if and only if F has a vn-self map. (b) Given F, F 02 Cn with vn-self maps u : cF ! F and v : dF 0! F 0, and f : F ! F 0, there exist integers i, j such that ic = jd and the diagram icf icF ____//_ jdF 0 |vi| vj|| fflffl|f fflffl| F ________//F 0 commutes. Given F 2 C of type n, we let T (F ) denote the mapping telescope of a vn-self map. An immediate consequence of the Periodicity Theorem is that T (F ) is independent of choice of self map. Furthermore, one deduces that the Bousfield class of T (F ) is independent of the choice of type n spectrum F . In other words, If F and F 0are both of type n, then T (F ) ^ Y ' * if and only ifT (F 0) ^ Y ' *. 8 KUHN It is usual to let T (n) ambiguously denote T (F ) for any particular type n finite spectrum F . Another consequence of the Periodicity Theorem was proved by the au- thor in [K1 ]. Proposition 2.6 ([K1 , Cor.4.3]). There exists a diagram in C, f(1) f(2) F (1)_____//F (2)___//Fk(3)k__//. . . | wwww kkkkkkk | wwwkkkk fflffl|--wwuukkkkk S0 such that each F (k) 2 Cn, and hocolim F (k) ! S0 induces an T (m)*- k isomorphism for all m n. Remark 2.7. The statement of this proposition deserves some comment, as homotopy colimits of general diagrams in a triangulated category like ho(S) are not always defined. However, the hocolimit of a sequence as above is defined (as the cofiber of an appropriate map between coproducts of the F (k)). We note also that only this construction is used in the proof of the proposition given in [K1 ]; in other words, the proposition is proved working solely in the triangulated homotopy category. We give standard names to some localization functors. Let Lfn: S ! S denote localization with respect to T (0)_. ._.T (n), and then define functors Cfn-1, Mfn: S ! S by letting Cfn-1X be the homotopy fiber of X ! Lfn-1X and MfnX be the homotopy fiber of LfnX ! Lfn-1X. These functors are all smashing, e.g. X ^ LfnS0 ' LfnX for all X, and from this one can quite easily deduce that LT(n) and Mfndetermine each other. More precisely, there are natural equivalences LT(n)MfnX ' LT(n)X and Cfn-1LT(n)X ' MfnLT(n)X ' MfnX An alternative proof of Proposition 2.6 occurs in [B5 , proof of Thm. 12.1], where Bousfield notes that Cfn-1S0 can be written in the form hocolimF (k) k with each F (k) 2 Cn. This same result also was proved by Mahowald and Sadofsky in [MS , Proposition 3.8]. We end this section with characterizations of spectra that are T (n)-local or in the image of Mfn. Lemma 2.8. Consider the following three properties that a spectrum X might satisfy. (i) [F, X] = 0 for all F 2 Cn+1. (ii) [Y, X] = 0 whenever F (n)^Y ' * for some type n finite spectrum F (n). TELESCOPIC FUNCTORS 9 (iii) T (i) ^ X ' * for 0 i n - 1. Properties (i) and (ii) hold if and only if X is T (n)-local. Properties (i) and (iii) hold if and only if X ' MfnX. Proof.We can assume that T (n) is the telescope of a vn-self map v : dF (n) ! F (n). Condition (i) is equivalent to the statement that X is Lfn-local, while property (ii) says that X is F (n)-local. Thus if X is T (n)-local, both (i) and (ii) are true. If condition (i) holds, so that X is Lfn-local, we observe that v : F (n) ^ X ! -dF (n) ^ X is an equivalence, as the cofiber is null, since it can be written (using S-duality) in the form MapS (F, X) with F of type n + 1. It follows that F (n) ^ X ' T (n) ^ X, and thus F (n)*(X) ' T (n)*(X) ' T (n)*(LT(n)X) ' F (n)*(LT(n)X). Thus if condition (ii) also holds, so that X, as well as LT(n)X, is F (n)-local, we conclude that X ' LT(n)X, i.e. X is T (n)-local. Finally, property (iii) says that Lfn-1X ' *, so that MfnX ' LfnX. 3. Telescopic functors associated to a self map of a space 3.1. The basic construction. Given a space B and a map v : dB ! B with d > 0, we define a functor v : T ! S as follows. If n -e mod d, with 0 e d - 1, we let v(Z)n = eMapT (B, Z). The structure maps v(Z)n ! v(Z)n+1 identify with the identity unless n 0 mod d, in which case it equals the map * d d v(Z) : MapT (B, Z) v-!MapT ( B, Z) = MapT (B, Z). The construction is functorial in v in the following sense: a commutative diagram df dA _____// dB |u| |v| |fflfflf |fflffl A _______//_B induces a natural transformation f* : v(Z) ! u(Z). We list some basic properties of v(Z) in the next omnibus lemma. Lemma 3.1. (a) ss*( v(Z)) = v-1 ss*(Z; B). (b) If a map of spaces Y ! Z induces an isomorphism on ss* for * 0, then v(Y ) ! v(Z) is a stable equivalence. In particular, the r-connected 10 KUHN covering map Z ! Z induces a stable equivalence v(Z) ! v(Z) for all r. (c) v* : v(Z) ! dv(Z) is a stable equivalence. (d) For all spaces A, v(MapT (A, Z)) = 1A^v(Z) = MapS (A, v(Z)). In particular, cv(Z) = c v(Z) for all c. (e) v takes weak equivalences to level weak equivalences (and thus stable equivalences), fibrations to level fibrations, and homotopy pullbacks to level homotopy pullbacks (and thus stable homotopy pullbacks). (f) Given a commutative diagram df dg dA _____// dB_____// dC |u| |v| |w| |fflfflf |fflfflg |fflffl A _______//_B______//C, if A f-!B -g!C is a homotopy cofiber sequence of spaces, then the induced sequences * f* w (Z) g-! v(Z) -! u(Z) are homotopy fibration sequences of spectra for all Z. (g) If B is a finite CW complex, there is a natural stable equivalence hocolim v(Zd) ' v(hocolim Zd) d d for all diagrams Z1 ! Z2 ! Z3 ! . .o.f spectra. (h) If v0, v1 : dB ! B are homotopic maps, then v0(Z) is naturally stably equivalent to v1(Z). (i) There is a natural stable equivalence v(Z) -~! vr(Z), where vr : rdB ! B denotes the evident r-fold composition of v with its various suspensions. Proof.All of this is quite easily verified. Part (a) is clear, and then parts (* *b) and (c) follow by check of homotopy groups. Part (d) follows by inspection, since MapT (A ^ B, Z) = MapT (A, MapT (B, Z)). Parts (e) and (f) follow from the fact that MapT (B, Z) takes cofibrations in the B-variable and fibrations in the Z-variable to fibrations. Similarly, part (g) follows from the fact that hocolimnMapT (B, Zn) ' MapT (B, hocolimnZn) if B is a finite complex. To check part (h), suppose v0, v1 : dB ! B are homotopic maps. If H : dB ^ I+ ! B is a homotopy from v0 to v1, let V : dB ^ I+ ! B ^ I+ TELESCOPIC FUNCTORS 11 be defined by V (x, t) = (H(x, t), t). Then there is a commutative diagram _di0//_d odi1o_ dB B ^ I+ dB |v0| |V| v1|| fflffl|i0 fflffl|i1 fflffl| B _______//_B ^ I+oo___ B, which induces natural equivalences i*0 i*1 v0(Z) -~ V (Z) -!~ v1(Z). Finally the stable equivalence of part (i) is defined as follows. Write n in the form n = mrd - sd - e, with 0 s r - 1 and 0 e d - 1. Then let v(Z)n ! vr(Z)n be the map e(vs)* e sd e+sd eMapT (B, Z) -----! MapT ( B, Z) = MapT (B, Z). Corollary 3.2. v(Z) is a periodic spectrum with period d: v(Z) ' d v(Z). Furthermore, the induced functor v : ho(T ) ! ho(S) is de- termined by the stable homotopy class of vr for any r. Proof.Combine properties (c), (d), (h), and (i) of the lemma. Our last property needs some notation. Given an unstable map u : cA ! A and X 2 S, let u-1 MapS (A, X) denote the homotopy colimit of the diagram * c u* 2c MapS (A, X) u-!MapS ( A, X) -! MapS ( A, X) ! . ... Lemma 3.3. Given maps u : cA ! A and v : dB ! B, there are natural stable equivalences u-1 MapS (A, v(Z)) ' u^v(Z) ' v-1 MapS (B, u(Z)). Sketch proof.By symmetry, we need just verify the first of these equiva- lences. By Lemma 3.1(d), u-1 MapS (A, v(Z)) is equal to * u* u* hocolim{ A^v (Z) u-! cA^v(Z) -! 2cA^v(Z) -! . .}.. By Lemma 2.1, this is stably equivalent to hocolim s-kd 1 MapT ( kcA ^ B, Z). k This, in turn, maps to hocolim s-k(c+d) 1 MapT (A ^ B, Z), k using evident natural maps of the form 1 rW ! s-r 1 W , and a check of homotopy groups shows this map between homotopy colimits is an equiva- lence. Finally, by Lemma 2.1 again, this last homotopy colimit is equivalent to u^v(Z). 12 KUHN 3.2. Identifying v( 1 Z). Recall that, for X 2 S, 1 X = X0. The following elementary `swindle' is critical to our arguments. Note that it says that the functor that assigns v-1 MapS (B, X) to a spectrum X depends only on the space X0. Proposition 3.4 (Compare with [K1 , Prop.3.3(4)]). If X 2 S is fibrant (i.e. is an -spectrum), then, given v : dB ! B, there is a natural weak equivalence v( 1 X) ' v-1 MapS (B, X). Proof.We have natural equivalences: v-1 MapS (B, X) -~ hocolimrs-rd 1 MapS ( rdB, X)rd = hocolimrs-rd 1 MapT ( rdB, Xrd) = hocolimrs-rd 1 MapT (B, rdXrd) -~ hocolim s-rd 1 Map (B, X ) r T 0 -~! 1 v( X). Here the first equivalence follows from Lemma 2.1, the last equivalence sim- ilarly follows from Corollary 2.2, and the second to last equivalence holds because X is fibrant. Remark 3.5. It is not easy to spot the analogue of this proposition in [B5 ], but [B5 , Thm.11.9] is a more elaborate result of this type, and its proof, given in [B5 , xx11.10-11.11] uses arguments very similar to our proof of Lemma 2.1. Remark 3.6. Using the proposition, we can give an alternative proof of part of Lemma 3.3: that u-1 MapS (A, v(Z)) ' v-1 MapS (B, u(Z)). If we let fibu(Z) be a fibrant replacement for u(Z), then 1 fibu(Z) ' hocolimrMapT ( rcA, Z). Thus v-1 MapS (B, u(Z)) ' v(hocolimrMapT ( rcA, Z)) (by the proposition) ' hocolimr v(MapT ( rcA, Z)) = hocolimrMapS ( rcA, v(Z)) = u-1 MapS (A, v(Z)). 4. v when v is a vn-self map of a space Note that if v : dB ! B is nilpotent, v(Z) will be contractible for all Z. So that this might not be the case, in this section, we study the case when v is a vn-self map of a finite CW complex B of type n. First we discuss a construction in the homotopy category of spectra. Given F 2 Cn, it is convenient to let (F, Z) 2 ho(S) denote t u(Z), where u : cA ! A is an unstable vn-map of a finite CW complex A such that tF ' 1 A. Similarly, given a map f : F ! F 0between finite spectra TELESCOPIC FUNCTORS 13 in Cn, we define f* : (F 0, Z) ! (F, Z) to be tff* : t v(Z) ! t u(Z) where __dff//_ dA dB |u| |v| |fflfflff |fflffl A _______//_B is a commutative diagram of spaces with vn-self maps, and tf ' 1 ff. Lemma 4.1. : Copnxho(T ) ! ho(S) is a well defined functor and satisfies the next two properties. (a) takes cofibration sequences in the Cn-variable to fibration sequences in ho(S). (b) MapS (F, (F 0, Z)) ' (F 0^ F, Z) ' MapS (F 0, (F, Z)). Proof.This follows from the Periodicity Theorem and the results in the last section. Now we prove that, when v is a vn-self map, v : T ! S satisfies versions of the properties listed in Theorem 1.1. Theorem 4.2. Let v : dB ! B is an unstable vn-self map. (1) v(Z) ' Mfn v(Z) and is also T (n)-local, for all spaces Z. (2) v( 1 X) ' MapS (B, LT(n)X) for all fibrant X 2 S. Proof of Theorem 4.2(1).We need to verify that v(Z) satisfies the three properties listed in Lemma 2.8. Property (i) says that [F, v(Z)] = 0 for all F 2 Cn+1. To see this, we first note that, since v(Z) is periodic, we can assume that F = 1 A for some finite CW complex A of type at least n + 1. But then [F, v(Z)] = ss0(MapS (A, v(Z)) = 0, because MapS (A, v(Z)) = 1A^v(Z) ' *, as A ^ B will have type greater than n, so that 1A ^ v : dA ^ B ! A ^ B will be nilpotent. Property (ii) says that, with F (n) a fixed finite spectrum of type n, [Y, v(Z)] = 0 whenever F (n) ^ Y ' *. To prove this, we make use of the properties of the functor listed in Lemma 4.1. So suppose that F (n) ^ Y ' *. Let CY = {F 2 Cn | [Y, (F, Z)]* = 0 for allZ}. Using the Thick Subcategory Theorem, we check that CY = Cn, thus ver- ifying property (ii). Firstly, CY is a thick subcategory by Lemma 4.1(a). Secondly, it contains at least one type n complex, as it contains all type 14 KUHN n complexes of the form F (n) ^ F , with F of type n. To see this, using Lemma 4.1(b), we have [Y, (F (n)^F, Z)]* = [Y, MapS (F (n), (F, Z)]* = [Y ^F (n), (F, Z)]* = 0. Property (iii) says that T (i)^ v(Z) ' * for i n-1. We can assume that T (i) is the telescope of the S-dual of an unstable v(i)-map u : cA ! A, where A is a finite CW complex of type i. Then T (i) ^ v(Z)' u-1 MapS (A, v(Z)) ' v-1 MapS (B, u(Z)) (by Lemma 3.3) = hocolimr rd u^1B(Z)) ' *, as A ^ B has type greater than i, so that u ^ 1B : cA ^ B ! A ^ B is nilpotent, and thus u^1B(Z) ' *. Proof of Theorem 4.2(2).This is similar to the author's proof of [K1 , Prop. 3.4]. Thanks to Proposition 3.4, we just need to show that, if v : dB ! B is a vn-map, then there is a weak equivalence v-1 MapS (B, X) ' MapS (B, LT(n)X). This is easy to do. We claim that each of the maps v-1 MapS (B, X) ! v-1 MapS (B, LT(n)X) MapS (B, LT(n)X) is an equivalence. If we let T (n) be modeled by the telescope of the dual of v, then the first map identifies with the equivalence T (n)^X -~!T (n)^LT(n)X. The second map is an equivalence as v is a T (n)*-isomorphism, so that each map in the diagram * d v* 2d MapS (B, LT(n)) v-!MapS ( B, LT(n)) -! MapS ( B, LT(n)) ! . . . is an equivalence. 5. The construction of n and the proof of Theorem 1.1 5.1. The construction on the level of homotopy categories. Recall that we have a functor : Copnx ho(T ) ! ho(S) defined by letting (F, Z) denote t u(Z), where u : cA ! A is an un- stable vn-map of a finite CW complex A such that tF ' 1 A. Now consider a resolution of S0 as in Proposition 2.6: a diagram f(1) f(2) (5.1) F (1)_____//F (2)___//Fk(3)k__//. . . |q(1)wwwq(2)wkkkq(3)kkkk | wwwkkkk fflffl|--wwuukkkkk S0 TELESCOPIC FUNCTORS 15 such that each F (k) 2 Cn, and such that the map q = hocolimq(k) : hocolimF (k) ! S0 k k induces an isomorphism in T (m)* for all m n. Definition 5.1. Define Tn: ho(T ) ! ho(S) by the formula Tn(Z) = holim (F (k), Z) k We have the following theorem, which is Theorem 1.1 on the level of homotopy categories. Theorem 5.2. Tnsatisfies the following properties. (1) Tn(Z) is T (n)*-local for all Z 2 ho(T ). (2) MapS (F, Tn(Z)) ' (F, Z) 2 ho(S) for all F 2 C and Z 2 ho(T ). (3) Tn( 1 X) ' LT(n)X, for all X 2 ho(S). Proof.By Theorem 4.2(1), each (F (k), Z) is T (n)*-local. Since the homo- topy limit of local objects is again local, statement (1) follows. To see that (2) is true, given F 2 C, we compute in ho(S): MapS (F, Tn(Z))' MapS (F, holim (F (k), Z)) k ' holimMapS (F, (F (k), Z)) k ' holimMapS (F (k), (F, Z)) k ' MapS (hocolim F (k), (F, Z)) k -~ Map 0 S (S , (F, Z)) = (F, Z). Here the third equivalence is an application of Lemma 4.1(b), while the last map is an equivalence because it is induced by the T (n)*-isomorphism q and (F, Z) is T (n)*-local (by Theorem 4.2(1)). The proof that (3) is true is similar: Tn( 1 X) = holim (F (k), 1 X) k ' holimMapS (F (k), LT(n)X) (by Theorem 4.2(2)) k = MapS (hocolim F (k), LT(n)X) k -~ Map S (S, LT(n)X) = LT(n)X. 16 KUHN 5.2. Rigidifying the construction. Definition 5.3. A rigidification of diagram (5.1) consists of the following data. (i) Finite complexes B(k) of type n. (ii) Natural numbers d(k) such that d(k)|d(k + 1) together with unstable vn-self maps v(k) : d(k)B(k) ! B(k). (iii) Natural numbers t(k) such that t(k) t(k+1) together with maps p(k) : B(k) ! St(k)and fi(k) : e(k)B(k) ! B(k +1), where e(k) = t(k +1)-t(k). These are required to satisfy three properties: (a) 1 B(k) 2 S represents t(k)F (k) 2 ho(S), 1 p(k) represents t(k)q(k), and 1 fi(k) represents t(k+1)f(k). (b) With r(k) = d(k + 1)=d(k), the diagram d(k+1)fi(k) e(k)+d(k+1)B(k) __________//_ d(k+1)B(k + 1) ||e(k)v(k)r(k) |v(k+1)| fflffl| fi(k) fflffl| e(k)B(k)_________________//_B(k + 1) commutes in T . (c) The diagram fi(k) e(k)B(k)L________________//_B(k + 1) LLLe(k)p(k)Lp(k+1)ssss LLL ssss L&&L yyss St(k+1) commutes. Lemma 5.4. Rigidifications exist. Sketch proof.This is basically Bousfield's construction of `an admissible spectral Lfn-cospectrum' given in [B5 , Thm.12.1]. One proceeds by induc- tion on k. Having constructed B(k), v(k), and p(k), using the Periodicity Theorem in the stable range, one chooses e(k) so large that there exists B(k + 1), v(k + 1), p(k + 1)), and fi(k) making property (a) hold, and so that the diagrams in (b) and (c) commute up to homotopy. Then one re- places B(k + 1) and fi(k), so that the new fi(k) is a cofibration. Finally, one uses the homotopy extension property of cofibrations (applied to both fi(k) TELESCOPIC FUNCTORS 17 and d(k+1)fi(k)) to replace v(k + 1) and p(k + 1) by homotopic maps so that the new diagrams (b) and (c) strictly commute. Given a rigidification of (5.1), we will make use of two families of induced natural maps. The maps fi(k) : e(k)B(k) ! B(k + 1) induce a natural maps v(k+1)(Z) ! e(k)v(k)r(k)(Z) = e(k) v(k)r(k)(Z). Adjointing these, and suspending t(k)-times, yield natural maps fi(k)* : t(k+1) v(k+1)(Z) ! t(k) v(k)r(k)(Z). The `top cell' maps p(k) : B(k) ! St(k)induce maps s-t(k)X ! t(k)X = MapS (St(k), X) ! MapS (B(k), X). Adjointing these, yield natural maps p(k)* : X ! st(k)MapS (B(k), X). The last maps are compatible as k varies, and so induce a natural map p* : X ! holimst(k)MapS (B(k), X). k Lemma 5.5. p* is an equivalence if X is T (n)-local. Proof.In the homotopy category, p(k)* corresponds to q(k)* : X ! MapS (F (k), X) so that p* corresponds to q* : X = MapS (S0, X) ! MapS (hocolim F (k), X). k This is an equivalence if X is T (n)-local. Definition 5.6. Given a rigidification of (5.1), we define n : T ! S by letting n(Z) be the homotopy limit of the diagram t(3) v(3)r(3)(Z) t(2) v(2)r(2)(Z) t(1) v(1)r(1)(Z) p88p OO kk55 OO kk55 OO ppp |o kkkk o| kkkk o| ppp | kkkfi(2)* | kkkfi(1)* | . .p. t(3) v(3)(Z) t(2) v(2)(Z) t(1) v(1)(Z). Informally, we write n(Z) = holim t(k) v(k)(Z). k Proof of Theorem 1.1.By construction, in ho(S), n(Z) represents the holimit of the diagram f(2)* f(1)* . .!. (F (3), Z) ---! (F (2), Z) ---! (F (1), Z), i.e. Tn(Z). The various properties of n stated in Theorem 1.1 are verified by giving proofs similar to those given in proving the analogous properties of Tnlisted in Theorem 5.2, with the constructions of natural equivalences `rigidifying' as needed in straightforward ways. 18 KUHN We run through some details. Property (1) is clear: n(Z) is the homotopy limit of T (n)-local spectra, thus is itself T (n)-local. For property (2), we have MapS (B, n(Z)) = MapS (F, holim t(k) v(k)(Z)) k = holimMapS (B, t(k) v(k)(Z)) k ~- holim t(k)Map (B, (Z)) k S v(k) ' holim t(k)MapS (B(k), v(Z)) (by Lemma 3.3) k ~-!holimst(k)Map (B(k), (Z)) k S v ~- v(Z), since v(Z) is T (n)-local. For property (3), we have n( 1 X) = holim t(k) v(k) 1 X) k -~!holim st(k) 1 X) k v(k) ' holimst(k)MapS (B(k), LT(n)X) (by Theorem 4.2(2)) k -~ L T(n)X (by Lemma 5.5) . Finally suppose that a functor 0n: T ! S satisfies the next two proper- ties, analogues of properties (1) and (2). (10) 0n(Z) is T (n)-local, for all spaces Z. (20) There is a weak equivalence of spectra Map S(B, 0n(Z)) ' v(Z), for all unstable vn self maps v : dB ! B, natural in both Z and v. Then we have: 0n(Z) ~-!holim st(k)MapS (B(k), 0n(Z)) (by (10)) k -~ holim t(k)Map (B(k), 0(Z)) k S n ' holim t(k) v(k)(Z)) (by (20)) k = n(Z). Thus properties (1) and (2) characterize n. 6. Bousfield's adjoint n In [B5 ], Bousfield constructs a functor n : S ! T , which serves as a left adjoint of sorts to n. In this section, we run through how this works. TELESCOPIC FUNCTORS 19 6.1. The construction of v. Given a self map of a space v : dB ! B, the functor v : T ! S admits a left adjoint v : S ! T , defined as follows. Given a spectrum X with iterated structure maps oer : dXrd ! X(r+1)d, v(X) is defined to be the coequalizer of the two maps ` v ` dB ^ Xrd -!-! B ^ Xrd, r oe r where, on the rdth wedge summand, v is dB ^ Xrd v^1--!B ^ Xrd, while oe is dB ^ Xrd ' B ^ dXrd 1^oer---!B ^ X(r+1)d. It is easy and formal to check that v and v form an adjoint pair. However, to be homotopically meaningful, one would like these functors to form a Quillen pair, so that they induce an adjunction on the associated homotopy categories. For this to be true, it is necessary and sufficient to check that v preserves trivial fibrations, and also fibrations between fibrant objects. (See, e.g. [B5 , Lem.10.5] or [Hi, Prop.8.5.4].) v certainly preserves trivial fibrations, as it takes a trivial fibration to* * a levelwise trivial fibration, which will then be a stable trivial fibration. Suppose that W ! Z is a fibration in T . In the stable model category structure, v(W ) ! v(Z) will be a fibration between fibrant objects only if the obvious necessary condition holds: v(W ) and v(Z) must both be fibrant, i.e. -spectra. Unravelling the definitions, v(Z) will be an -spectrum if and only if the map v* : Map (B, Z) ! Map ( dB, Z) is a weak equivalence. One can force this condition to be true as follows. Let Lv : T ! T denote localization with respect to the map f, and then let Tv denote T with the associated model category structure in which weak equivalences Tv are maps f so that Lvf is a weak equivalence in T . (See e.g. [Hi] for these constructions and many references to the literature.) We recover a variant of [B5 , Lem.10.6]. Lemma 6.1. For any v : dB ! B, we have the following. (a) v : S ! Tv and v : Tv ! S form a Quillen pair. (b) For all X 2 S and Z 2 T , there is a natural bijection [ v(X), LvZ] ' [X, v(LvZ)]. 20 KUHN 6.2. Periodic localization of spaces. In [B5 , x4.3], Bousfield defines Lfn: T ! T to be localization with respect to the map A ! *, where A is chosen so that 1 A is equivalent to a finite spectrum of type n + 1, and the connectivity of H*(A; Z=p) is chosen to be as low as possible. His proof that this is independent of choice appears in [B3 , Thm.9.15], and depends on the Thick Subcategory Theorem. For our purposes, Lfn: T ! T satisfies two elementary properties that we care about. Lemma 6.2. If X is a Lfn-local spectrum, then 1 X is Lfn-local space. Proof.Let A be chosen as in the definition of Lfn: T ! T . For all t, sst(MapT ( A, 1 X)) = [ 1 t+1A, X] = 0, since Lfn-local spectra admit no nontrivial maps from objects in Cn+1. Thus MapT ( A, 1 X) ' *, and so 1 X is Lfn-local. Lemma 6.3. If Z is a Lfn-local space, then it is also Lv-local for all unstable vn-self maps v : dB ! B that are double suspensions. Proof.Since v is a double suspension, it fits into a cofibration sequence of the form C ! dB v-!B ! 2C, where 1 C has type n + 1. This induces a fibration sequence * d MapT ( 2C, Z) ! MapT (B, Z) v-!MapT ( B, Z) ! MapT ( C, Z), in which the first and last of these mapping spaces are null if Z is Lfn-local. Thus the middle map is an equivalences, and so Z is Lv-local. A deeper property of Lfngoes as follows. Proposition 6.4. If v is a vn-self map, the natural map v(Z) ! v(LfnZ) is a stable equivalence. Proof.We wish to show that the map of spaces hocolimrMapT ( rdB, Z) ! hocolimrMapT ( rdB, LfnZ) induces an isomorphism on homotopy groups (in high dimensions). This is pretty much [B3 , Theorem 11.5], and we sketch how the proof goes. The map we care about factors in the homotopy category: hocolimrMapT ( rdB, Z)_______//hocolimrMapTO(OrdB, LfnZ) | | | | fflffl| ~ | LfnhocolimrMapT ( rdB, Z) oo___hocolimr LfnMapT ( rdB, Z), where the indicated equivalence is [B3 , Lemma 11.6]. TELESCOPIC FUNCTORS 21 The right vertical arrow induces an isomorphism on homotopy groups in high dimensions, due to [B3 , Theorem 8.3], a general result which describes to what extent functors like L C preserve fibrations. Applied to the case in hand, one learns that there is a number ffi such that the natural map LfnMapT (C, Z) ! MapT (C, LfnZ) will induce an isomorphism on ssi for i ffi for all finite complexes C. Thus our right vertical map will induce isomorphisms on ssi in the same range. It follows that we need just check that the left vertical map is an equiva- lence, or, otherwise said, that hocolimrMapT ( rdB, Z) is Lfn-local. Recalling that Lfn= L A for a well chosen A of type (n + 1), we have MapT ( A, hocolimrMapT ( rdB, Z))' hocolimrMapT ( A, MapT ( rdB, Z) = hocolimrMapT ( A ^ rdB, Z) ' *, since 1 A ^ v will be nilpotent by the Nilpotence Theorem. Remark 6.5. It would interesting to have a proof of this proposition that avoided the use of [B3 , Theorem 8.3]. Combining this proposition with Lemma 6.1 and Lemma 6.3 yields the next theorem. Theorem 6.6. If v : dB ! B is a vn-self map and a double suspension, there is a natural bijection [ v(X), LfnZ] ' [X, v(Z)], for all Z 2 T and X 2 S. 6.3. The definition of n. Let the following data make up a rigidification of diagram (5.1), as used in the definition of n: (i) Finite complexes B(k) of type n. (ii) Natural numbers d(k) such that d(k)|d(k + 1) together with unstable vn-self maps v(k) : d(k)B(k) ! B(k). (iii) Natural numbers t(k) such that t(k) t(k+1) together with maps p(k) : B(k) ! St(k)and fi(k) : e(k)B(k) ! B(k +1), where e(k) = t(k +1)-t(k). By double suspending everything, we can also assume that each v(k) is a double suspension. 22 KUHN Definition 6.7. Given this data, we define n : S ! T by letting n(Z) be the homotopy colimit of the diagram v(1)r(1)( t(1)X) v(2)r(2)( t(2)X) v(3)r(3)( t(3)X) | TTTfi(1)*TTTT | TTTfi(2)*TTTT| OOOfi(3)*OO fflffl| TT)) fflffl| T))T fflffl| OOOO'' v(1)( t(1)X) v(2)( t(2)X) v(3)( t(3)X) . .,. where each vertical map will be an Lfn-equivalence, and each fi(k)* is itself a natural zig-zag diagram fi(k)* t(k) v(k)r(k)( t(k)X) -~ v(k)r(k)( e(k) t(k+1)X) ---! v(k+1)( X). Informally, we write n(X) = hocolim v(k)( t(k)X). k From Theorem 6.6, we deduce Theorem 6.8 ([B5 , Theorem 5.4(iii)]). There is a natural bijection [ n(X), LfnZ] = [X, n(Z)] for all Z 2 T and X 2 S. Proof.The idea is that, since n is the limit of functors of the form v, and n is the colimit of their adjoints v, the theorem should follow from Theorem 6.6. The only detail needing a careful check is that the zig-zag natural map fi(k)* t(k+1) v(k)r(k)( t(k)X) -~ v(k)r(k)( e(k) t(k+1)X) ---! v(k+1)( X) used in the definition of n above really is adjoint to the more directly defined map fi(k)* t(k) t(k+1) v(k+1)(Z) ---! v(k)r(k)(Z) used in the definition of n. To see this we have a commutative diagram: * ~ MapT( v(k+1) t(k+1)X,fZ)i(k)//_MapT( v(k)r(k) e(k)Mt(k+1)X,aZ)pT(ov(k)r(k)ot(k* *)X,_Z) | |o| o|| |o| fflffl| fi(k)* fflffl| ~ fflffl| MapT ( v(k+1)s-t(k+1)X,/Z)/_MapT( v(k)r(k) e(k)s-t(k+1)X,MZ)apT(ov(k)r(k)s-t(k)* *X,oZ)_ || || || || || || || || fi(k)* || ~ || MapS(X, st(k+1)Ov(k+1)Z)//_MapS(X,Ost(k+1) e(k)ov(k)r(k)Z)MapS(X,ost(k)_v(k)r(* *k)Z) | OO OO |o| o|| |o| | fi(k)* | | MapS(X, t(k+1) v(k+1)Z)_//MapS(X, t(k) v(k)r(k)Z)MapS(X,__t(k) v(k)r(k)Z). TELESCOPIC FUNCTORS 23 Remark 6.9. The lack of elegance in the proof of the `detail' checked above reflects the fact, though our functor n : T ! S is intuitively the homotopy limit of right adjoint functors v(k), it does not make up the right part of an adjoint pair. Note that our official definition involves the use of : S ! S, which induces an equivalence of homotopy categories, but is a left adjoint, not a right one. It doesn't seem possible to somehow replace t(k)by st(k) (which is a right adjoint) in Definition 5.6. This same problem shows up in Bousfield's construction: see the paragraph before [B5 , Theorem 11.7]. Corollary 6.10. In ho(S), there is a natural equivalence LT(n) 1 n(X) ' LT(n)X. Proof.In the last theorem, let Z be the space 1 LT(n)Y , which is Lfn-local by Lemma 6.2. We see that, for all X, Y 2 S, there are natural isomorphisms [ 1 n(X), LT(n)Y ]' [ n(X), 1 LT(n)Y ] ' [X, n( 1 LT(n)Y )] ' [X, LT(n)Y ]. The corollary then follows from Yoneda's lemma. Remark 6.11. The careful reader will note that this corollary is not depen- dent on Proposition 6.4 (and thus not dependent on Bousfield's careful study of the behavior of localized fibration sequences), as we have derived it by only applying our other results to a space Z that is Lfn-local. Remark 6.12. From the corollary, it follows that the Telescope Conjecture is equivalent to the statement that if a space is K(n)*-acyclic, then it is T (n)*-acyclic. 7. The section jn One of the main applications of the functor n, is that it leads to the construction of a natural transformation jn(X) : LT(n)X ! LT(n) 1 1 X which is a natural homotopy section of the T (n)-localization of the evalua- tion map ffl(X) : 1 1 X ! X. The construction is immediate: jn is defined by applying n to the natural map j( 1 X) : 1 X ! Q 1 X. This section is both used and studied in [K3 , Re]. It seems plausible that jn is the unique section of LT(n)ffl. We have a couple of partial results along these lines. The first was discussed in [K4 ]. 24 KUHN Proposition 7.1. jn is unique up to `tower phantom' behavior in the Good- willie tower for 1 1 in the following sense: for all d, the composite jn(X) 1 1 LT(n)ed(X) 1 LT(n)X ----! LT(n) X -------! LT(n)Pd (X) LT(n)pd(X) is the unique natural section of LT(n)Pd1(X) -------! LT(n)X. ed(X) 1 th Here 1 1 X - --! Pd (X) is the d stage of the Goodwillie tower, and pd is the canonical natural transformation such that ffl = pd O ed. The uniqueness asserted in the proposition is an immediate consequence of the main theorem of [K2 ]. Our second observation is in the spirit of observations by Rezk in [Re ]. As usual, we let QZ denote 1 1 Z. Proposition 7.2. Any natural transformation f(X) : X ! LT(n) 1 1 X will be determined by f(S0) : S0 ! LT(n) 1 QS0. Proof.We begin by showing that we can reduce to the case when X = 1 Z, a suspension spectrum. As the range of f is T (n)-local, we can extend the domain to LT(n)X. In the diagram f( 1 1 X) 1 1 LT(n) 1 1 X ___________//LT(n) Q X |ffl*| |ffl*| fflffl| f(X) fflffl| LT(n)X _______________//LT(n) 1 1 X, the left vertical map has a section given by jn(X). Thus the bottom map is determined by the top map. Next we observe that any continuous functor G : T ! S comes with a natural transformation Z ^G(W ) ! G(Z ^W ), and this structure is natural in G. Applied to our situation, for any space Z, we have a commutative diagram 1Z^f(S0) 1 0 Z ^ 1 S0 __________//_Z ^ LT(n) QS | o|| || fflffl| f(Z) fflffl|1 1 Z _______________//LT(n) QZ. Thus the bottom map is determined by the top. Question 7.3. Is it true that the map Y LT(n) 1 QS0 ! LT(n) 1 B d+, d arising from the James-Hopf maps QS0 ! QB d+, is monic on ss0? If so, then the last propositions combine to show that jn is the unique natural section to the localized evaluation map. TELESCOPIC FUNCTORS 25 8.A guide to computations In this section we briefly survey calculations that have been made of n(Z) for various pairs (n, Z). Before jumping into this, we first explain that there is also interest in explicit calculations of ss*( n(Z)), or variants thereof. 8.1. Periodic homotopy groups of spaces. Recall that there is a se- quence of spectra F (1) ! F (2) ! F (3) ! . . . such that each F (k) is finite of type n and hocolim F (k) ' Cfn-1S0. k Dualizing this in the stable homotopy category, one gets a diagram DF (1) DF (2) DF (3) . ... Furthermore, each DF (k) comes with a vn-self map which we will generically call `v', these are compatible in the usual way, and any given finite part of this data `eventually' desuspends to spaces. Thus the following definition makes sense. Definition 8.1. The vn-periodic homotopy groups of a space Z are defined to be v-1nss*(Z) = colimv-1 ss*(Z; DF (k)). k Example 8.2. When n = 1, one takes F (k) to be a Moore spectrum of type Z=pk, and the self maps are `Adams maps'. Using traditional notation, v-11ss*(Z) = colimv-1 ss*+1(Z; Z=pk). k Lemma 8.3. The groups v-1nss*(Z) can be rewritten in terms of n(Z) in various ways: v-1nss*(Z)= colimss*(MapS (DF (k), n(Z))) k = colimss*(F (k) ^ n(Z)) k = ss*(Cfn-1 n(Z)) = ss*(Mfn n(Z)). The next lemma relates n-equivalences to isomorphisms on localized homotopy groups. Lemma 8.4. Given a map f : W ! Z between spaces, the following condi- tions are equivalent. (a) n(f) : n(W ) ! n(Z) is a weak equivalence. (b) f* : v-1nss*(W ) ! v-1nss*(Z) is an isomorphism. 26 KUHN (c) f* : v-1 ss*(W ; B) ! v-1 ss*(Z; B) is an isomorphism for all unstable vn-self maps v : dB ! B. (d) f* : v-1 ss*(W ; B) ! v-1 ss*(Z; B) is an isomorphism for some unstable vn-self map v : dB ! B. Proof.Let g = n(f). Then condition (a) says that g is an equivalence, (b) says that Mfng is an equivalence, and conditions (c) and (d) say that DB ^ g is an equivalence for appropriate B's. As g is a map between T (n)- local spectra, the three conditions are all equivalent: clearly (a) implies all the other statements, LT(n)Mfng ' g so that (b) implies (a), (c) obviously implies (d), and finally (d) implies that v-1 DB ^ g is an equivalence, so that g is a T (n)*-isomorphism and thus (a) holds. 8.2. Basic observations. From properties of v listed in Lemma 3.1, one deduces the next two useful basic calculational rules. Lemma 8.5. MapS (A, n(Z)) ' n(MapT (A, Z)) for all A, Z 2 T . Lemma 8.6. n takes homotopy pullbacks in T to homotopy pullbacks in S. One might wonder to what extent n might take the homotopy limit (`microscope') of a sequence Z1 Z2 Z3 . . .to the corresponding holimit in S. Unfortunately this will not always be the case; the correct statement can be formally deduced from Theorem 6.8. Lemma 8.7. Given a sequence of spaces Z1 Z2 Z3 . .,.we have holim n(Zk) ' n(holim LfnZk). k k Since n(holimk Zk) ' n(LfnholimkZk), one sees that the failure of n to commute with microscopes is caused by the failure of Lfn: T ! T to commute with microscopes. More constructively, one has the following consequence of Lemma 8.4. Lemma 8.8. Given a sequence of spaces Z1 Z2 Z3 . .,.the natural map n(holim Zk) ! holim n(Zk) k k is an equivalence if and only if v-1 limss*(Zk; B) ! limv-1 ss*(Zk; B) k k is an isomorphism for some unstable vn-self map v : dB ! B. Since n(Z) can be `calculated' as LT(n)X if Z = 1 X, the following strategy for computing n(Z) emerges: try to `resolve' Z by towers of fi- brations with fibers which are infinite loopspaces, and hope that Lemma 8.8 can be applied when needed. TELESCOPIC FUNCTORS 27 8.3. n(Sm ) when m is odd. The strategy just described was implimented in beautiful work of Arone and Mahowald [AM ] on the Goodwillie tower of the identity functor. It allows for the identification of a short resolution of n(Z) with `known' composition factors, when Z is an odd dimensional sphere. We will be brief here; for a slightly different overview of how this goes, see the last sections of our survey paper [K4 ]. We need some notation. Let maek denote the direct sum of m copies of the reduced real regular representation of Vk = (Z=p)k. Then GLk(Z=p) acts on the Thom space BVkmaek. Let ek 2 Z(p)[GLk(Z=p)] be any idempotent in the group ring representing the Steinberg module, and then let L(k, m) be the associated stable summand of BVkmaek: L(k, m) = ek 1 BVkmaek. The spectra L(k, 0) and L(k, 1) agree with spectra called M(k) and L(k) in the literature from the early 1980's: see e.g. [MP , KP ]. Two properties of the L(k, m) play a crucial roles for our purposes: o When m is odd, the cohomology H*(L(k, m); Z=p) is free over the finite subalgebra A(k - 1) of the Steenrod algebra A. o Fixing m, the connectivity of L(k, m) has a growth rate like pk. The first fact here implies that L(k, m) is T (n)*-acylic for k > n. Indeed, the E2-term of the Adams spectral sequence which computes [B, L(k, m)]* for any finite B will have a vanishing line of small enough slope so that one can immediately deduce that v-1 E2 = 0 if v is a vn-self map of B. Arone and Mahowald's analysis in [AM ], supported by [AD ], shows that, for odd m, there is a tower of fibrations under the p-local sphere Sm : .. . | | fflffl| R2(Sm:): uu uuuu p2|| e2 uuuu fflffl| uuuu jR1(Sm5)5 uuu jjjj uuuujje1jjjjj p1|| jjjjjjjuue0u fflffl| Sm ________________//_R0(Sm ), such that Sm ' holimRk(Sm ), R0(Sm ) = QSm , and, for k 1, the fiber of k pk is equivalent to 1 m-k L(k, m). (The space Rk(Sm ) is the pkth stage of the Goodwillie tower of the identity functor applied to Sm .) Using the two properties of the L(k, m) bulleted above, Arone and Ma- howald then deduce that v-1 limss*(Rk(Sm ); B) ! limv-1 ss*(Rk(Sm ); B) k k 28 KUHN is an isomorphism for any self map v : dB ! B. See [AM , x4.1]. It follows that en* : v-1 ss*(Sm ; B) ! v-1 ss*(Rn(Sm ); B) is an isomorphism for any vn-self map v : dB ! B. One deduces the following about n(Sm ). Theorem 8.9 (see [K4 , Theorem 7.18]). Let m be odd. The map n(en) : n(Sm ) ! n(Rn(Sm )) is an equivalence. Thus the spectrum n(Sm ) admits a finite decreasing filtration with fibers LT(n) m-k L(k, m) for k = 0, . .,.n. With a little diagram chasing, one can do better than this. Let L(k)m-11 be the fiber of the natural map of spectra L(k, 1) ! L(k, m). The fibration sequence of spectra L(k)m-11! L(k, 1) ! L(k, m) induces a short exact sequence in mod p cohomology, and is thus split as A(k - 1)-modules. By applying n to the fiber sequence m0Sm ! S1 ! m-1 Sm and applying the previous theorem, one deduces an improved re- sult. Theorem 8.10 ([K4 , Theorem 7.20]). Let m be odd. The spectrum n(Sm ) admits a finite decreasing filtration with fibers LT(n) m+1-k L(k)m-11 for k = 1, . .,.n. Example 8.11. When p = 2, L(1)m1 = RP m. Specializing to n = 1, we learn that, for there is a weak equivalence 1(S2k+1) ' LT(1) 2k+1RP 2k. Specializing to n = 2, we learn that there is a fibration sequence of spectra 2(S2k+1) ! LT(2) 2k+1RP 2k! LT(2) 2kL(2)2k1. The first of these is equivalent to an older theorem of Mahowald [Ma ] that said that the James-Hopf map 2kS2k+1 ! Q RP 2kinduces an isomor- phism on v1-periodic homotopy groups. The odd prime version of this is due to Rob Thompson [T ]. 8.4. 1(Z) for many Z. There is a huge amount known about v-11ss*(Z) thanks to the prodigious efforts of Bousfield, together with Don Davis and his collaborators. A survey article by Davis [D1 ] describes computations known by the mid 1990's. In recent years, beginning with [B4 ], there has been an explosion of new, more elegantly organized, computations, often explictly describing 1(Z) enroute: see the references below for entries into the recent literature. Ingredients special to the n = 1 case that enter the story include the following. o The identification of 1(S2k+1) as described above. TELESCOPIC FUNCTORS 29 o The fact that LT(1)= LK(1). o The identification of LK(1)S0 as the fiber of an appropriate map of the form r - 1 : KO ! KO [B1 ]. o A tight relationship between a maps of spaces being K(1)*-isomorphisms and being a 1-equivalences [B3 ]. In summary, for appropriate spaces Z, v-11ss*(Z) is essentially determined by KO*(Z; Zp), together with Adams operations. Bousfield's recent careful study [B6 ] is state of the art in this area. Davis [D2 ] gives many complete calculations when Z is a compact Lie group, with calculations beginning with knowledge of the Lie group's represenation ring. A very recent amusing result in this spirit is due to Martin Bendersky and Davis [BD ], and says that there is a 2-primary homotopy equivalence 1(DI(4)) ' LK(1) 725019T ^ M, where DI(4) is the Dywer-Wilkerson exotic 2-compact group, T is the three cell finite spectrum S0 [j e2 [2 e3, and M is a mod 221 Moore spectrum. References [AP]J. F. Adams and S. B. Priddy, Uniqueness of BSO , Math. Proc. Camb. Phil. S* *oc. 80 (1976), 475-509. [AD]G. Z. Arone and W. G. Dwyer, Partition complexes, Tits buildings and symmet* *ric products, Proc. London Math. Soc. 82 (2001), 229-256. [AM] G. Arone and M. Mahowald, The Goodwillie tower of the identity functor and* * the unstable periodic homotopy of spheres , Invent. Math. 135(1999), 743-788. [BD]M. Bendersky and D. M. Davis, v1-periodic homotopy groups of the Dywer- Wilkerson space, preprint, 2007. [B1]A. K. Bousfield, The localization of spectra with respect to homology, Topo* *logy 18 (1979), 257-281. [B2]A. K. 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[MMSS] M.A.Mandell, J.P.May, S.Schwede, B.Shipley, Model categories of diagram * *spec- tra, Proc. London Math. Soc. 82 (2001), 441-512. [MP] S. A. Mitchell and S. B. Priddy, Stable splittings derived from the Steinb* *erg module, Topology 22(1983), 285-298. [Mi]S. A. Mitchell, Finite complexes with A(n)-free cohomology, Topology 24(198* *5), 227- 248. [Ra]D. C. Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals* * of Mathematics Study 128, 1992. [Re]C. Rezk, The units of a ring spectrum and a logarithmic cohomology operatio* *n, J. Amer. Math. Soc. 19 (2006), 969-1014 [S]S. Schwede, Spectra in model categories and applications to the algebraic co* *tangent complex, J.P.A.A. 120 (1997), 77-104. [T]R. D. Thompson, The v1-periodic homotopy groups of an unstable sphere at odd primes, Trans.A.M.S. 319(1990), 535-560. Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: njk4x@virginia.edu