A GUIDE TO TELESCOPIC FUNCTORS
NICHOLAS J. KUHN
Abstract. In the mid 1980's, Pete Bousfield and I constructed certain
plocal `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 DevanitzHopkinsSmith,
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 vnself 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)]* ' v1 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
vnself 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!. .}.= v1 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 vnself 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
fflfflfn+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 dfold shift functor
with (sdX)n = Xn+d. This admits a left adjoint sd : S ! S with
(
Xnd forn d
(sdX)n =
* for0 n d.
TELESCOPIC FUNCTORS 5
Composing these adjoints, we see that sd 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 ! sn 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 sdX ! 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
sd1 1 d1d0X(0)d0W sd2 1 d2d1X(1)d1W sd3 1 d3d2X(2)d2Q
offlffl WWWWWWW++W offlffl WWWWWWW++W offlfflQQ Q
sd0 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 zigzag
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
hocolimsdk 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
sd1 1 d1d0Xd0 sd2 1 d2d1Xd1 sd3 1 d3d2Xd2
UU UUUU PPP
offlfflUUUUUUU**U offlffl UUUUU**U offlfflPPPPP
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 sdk 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 plocal 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 vnself 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 vnself map.
(b) Given F, F 02 Cn with vnself 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
fflfflf fflffl
F ________//F 0
commutes.
Given F 2 C of type n, we let T (F ) denote the mapping telescope of a
vnself 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
fflfflwwuukkkkk
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
Cfn1, Mfn: S ! S by letting Cfn1X be the homotopy fiber of X ! Lfn1X
and MfnX be the homotopy fiber of LfnX ! Lfn1X.
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 Cfn1LT(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 Cfn1S0 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 vnself map v :
dF (n) ! F (n).
Condition (i) is equivalent to the statement that X is Lfnlocal, 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 Lfnlocal, we observe that v : F (n) ^
X ! dF (n) ^ X is an equivalence, as the cofiber is null, since it can be
written (using Sduality) 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 Lfn1X ' *, 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)) = v1 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 rconnected
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 rfold 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 Bvariable and
fibrations in the Zvariable 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
fflffli0 fflffli1 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 u1 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 u1 MapS (A, v(Z)) ' u^v(Z) ' v1 MapS (B, u(Z)).
Sketch proof.By symmetry, we need just verify the first of these equiva
lences. By Lemma 3.1(d), u1 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 skd 1 MapT ( kcA ^ B, Z).
k
This, in turn, maps to
hocolim sk(c+d) 1 MapT (A ^ B, Z),
k
using evident natural maps of the form 1 rW ! sr 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 v1 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) ' v1 MapS (B, X).
Proof.We have natural equivalences:
v1 MapS (B, X) ~ hocolimrsrd 1 MapS ( rdB, X)rd
= hocolimrsrd 1 MapT ( rdB, Xrd)
= hocolimrsrd 1 MapT (B, rdXrd)
~ hocolim srd 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 , xx11.1011.11] 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 u1 MapS (A, v(Z)) ' v1 MapS (B, u(Z)). If
we let fibu(Z) be a fibrant replacement for u(Z), then 1 fibu(Z) '
hocolimrMapT ( rcA, Z). Thus
v1 MapS (B, u(Z)) ' v(hocolimrMapT ( rcA, Z)) (by the proposition)
' hocolimr v(MapT ( rcA, Z))
= hocolimrMapS ( rcA, v(Z)) = u1 MapS (A, v(Z)).
4. v when v is a vnself 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 vnself 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 vnmap 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 vnself 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 Cnvariable 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 vnself map, v : T ! S satisfies versions
of the properties listed in Theorem 1.1.
Theorem 4.2. Let v : dB ! B is an unstable vnself 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 n1. We can assume that
T (i) is the telescope of the Sdual of an unstable v(i)map u : cA ! A,
where A is a finite CW complex of type i. Then
T (i) ^ v(Z)' u1 MapS (A, v(Z))
' v1 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 vnmap, then there is a weak equivalence
v1 MapS (B, X) ' MapS (B, LT(n)X).
This is easy to do. We claim that each of the maps
v1 MapS (B, X) ! v1 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 vnmap 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
fflfflwwuukkkkk
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
vnself 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 Lfncospectrum' 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
st(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 Lfnlocal spectrum, then 1 X is Lfnlocal 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 Lfnlocal spectra admit
no nontrivial maps from objects in Cn+1. Thus MapT ( A, 1 X) ' *, and
so 1 X is Lfnlocal.
Lemma 6.3. If Z is a Lfnlocal space, then it is also Lvlocal for all unstable
vnself 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 Lfnlocal.
Thus the middle map is an equivalences, and so Z is Lvlocal.
A deeper property of Lfngoes as follows.
Proposition 6.4. If v is a vnself 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 Lfnlocal. 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 vnself 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
vnself 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 Lfnequivalence, and each fi(k)* is itself
a natural zigzag 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 zigzag
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)st(k+1)X,/Z)/_MapT( v(k)r(k) e(k)st(k+1)X,MZ)apT(ov(k)r(k)st(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 Lfnlocal
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 Lfnlocal.
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) fflffl1
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 JamesHopf 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) ' Cfn1S0.
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 vnself 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 vnperiodic homotopy groups of a space Z are defined
to be
v1nss*(Z) = colimv1 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,
v11ss*(Z) = colimv1 ss*+1(Z; Z=pk).
k
Lemma 8.3. The groups v1nss*(Z) can be rewritten in terms of n(Z) in
various ways:
v1nss*(Z)= colimss*(MapS (DF (k), n(Z)))
k
= colimss*(F (k) ^ n(Z))
k
= ss*(Cfn1 n(Z))
= ss*(Mfn n(Z)).
The next lemma relates nequivalences 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* : v1nss*(W ) ! v1nss*(Z) is an isomorphism.
26 KUHN
(c) f* : v1 ss*(W ; B) ! v1 ss*(Z; B) is an isomorphism for all unstable
vnself maps v : dB ! B.
(d) f* : v1 ss*(W ; B) ! v1 ss*(Z; B) is an isomorphism for some unstable
vnself 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 v1 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
v1 limss*(Zk; B) ! limv1 ss*(Zk; B)
k k
is an isomorphism for some unstable vnself 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 E2term 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 v1 E2 = 0 if v is a vnself 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 plocal 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 mk 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
v1 limss*(Rk(Sm ); B) ! limv1 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* : v1 ss*(Sm ; B) ! v1 ss*(Rn(Sm ); B)
is an isomorphism for any vnself 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) mk L(k, m) for k = 0, . .,.n.
With a little diagram chasing, one can do better than this. Let L(k)m11
be the fiber of the natural map of spectra L(k, 1) ! L(k, m). The fibration
sequence of spectra
L(k)m11! 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 !
m1 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+1k L(k)m11 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 JamesHopf map 2kS2k+1 ! Q RP 2kinduces an isomor
phism on v1periodic 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 v11ss*(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 1equivalences [B3 ].
In summary, for appropriate spaces Z, v11ss*(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 2primary homotopy equivalence
1(DI(4)) ' LK(1) 725019T ^ M,
where DI(4) is the DywerWilkerson exotic 2compact group, T is the three
cell finite spectrum S0 [j e2 [2 e3, and M is a mod 221 Moore spectrum.
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Department of Mathematics, University of Virginia, Charlottesville, VA
22903
Email address: njk4x@virginia.edu