GOODWILLIE TOWERS AND CHROMATIC HOMOTOPY:
AN OVERVIEW
NICHOLAS J. KUHN
Abstract. This paper is based on talks I gave in Nagoya and Kinosaki
in August of 2003. I survey, from my own perspective, Goodwillie's work
on towers associated to continuous functors between topological model
categories, and then include a discussion of applications to periodic ho
motopy as in my work and the work of AroneMahowald.
1. Introduction
About two decades ago, Tom Goodwillie began formulating his calculus
of homotopy functors as a way to organize and understand arguments being
used by him and others in algebraic Ktheory. Though it was clear early on
that his general theory offered a new approach to the concerns of classical
homotopy, and often shed light on older approaches, it is relatively recently
that its promise has been begun to be realized. This has been helped by the
recent publication of the last of Goodwillie's series [G1 , G2 , G3 ], and by t*
*he
support of many timely new results in homotopical algebra and localization
theory allowing his ideas to be applied more widely.
At the Workshop in Algebraic Topology held in Nagoya in August 2002,
I gave a series of three talks entitled `Goodwillie towers: key features and
examples', in which I reviewed the aspects of Goodwillie's work that I find
most compelling for homotopy theory. A first goal of this paper is to offer a
written account of my talks. As in my talks, I focus on towers associated to
functors, i.e. the material of [G3 ]. As `added value' in this written version,
I include some fairly extensive comments about the general model category
requirements for running Goodwillie's arguments.
At the Conference on Algebraic Topology held in Kinosaki just previous
to the workshop, I discussed a result of mine [K5 ] that says that Goodwillie
towers of functors of spectra split after periodic localization. This is one of
a number of ways discovered so far in which Goodwillie calculus interacts
beautifully with homotopy as organized by the chromatic point of view;
another is the theorem of Greg Arone and Mark Mahowald [AM ]. A second
goal is to survey these results as well, and point to directions for the future.
____________
Date: October 13, 2004.
2000 Mathematics Subject Classification. Primary 55P43, 55P47, 55N20; Second*
*ary
18G55.
This research was partially supported by a grant from the National Science F*
*oundation.
1
2 KUHN
The paper is organized as follows.
In x2, I describe the major properties of Goodwillie towers associated
to continuous functors from one topological model category to another. In
x3, I discuss model category prerequisites. The basic facts about cubical
diagrams and polynomial functors are reviewed in x4. The construction of
the Goodwillie tower of a functor is given in x5, and I sketch the main ideas
behind the proofs that towers have the properties described in x2.
In x6, I discuss some of my favorite examples: Arone's model for the
tower of the functor sending a space X to 1 Map (K, Z) [A ], the tower for
the functor sending a spectrum X to 1 1 X, the tower of the identity
functor on the category of commutative augmented Salgebras, and tower
for the identity functor on the category of topological spaces as analyzed by
Brenda Johnson, Arone, Mahowald, and Bill Dwyer [J, AM , AD ]. Besides
organizing these in a way that I hope readers will find helpful, I have also
included some remarks that haven't appeared elsewhere, e.g. I note (in
Example 6.3) that the bottom of the tower for 1 1 X can be used to
prove the KahnPriddy Theorem, `up to one loop'.
The long x7 begins with a discussion of how Goodwillie towers interact
with Bousfield localization. Included is a simple example (see Example 7.4)
that shows that the composite of homogeneous functors between spectra
need not again be homogeneous. In the remainder of the section, I survey
three striking results in which the Goodwillie towers discussed in x6 interact
with chromatic homotopy theory: my theorems on splitting localized tow
ers [K5 ] and calculating the Morava Ktheories of infinite loopspaces [K4 ],
and Arone and Mahowald's work on calculating the unstable vnperiodic
homotopy groups of spheres [AM ]. All of these relate to telescopic functors
n from spaces to spectra constructed a while ago by Pete Bousfield and
me [B1 , K2 , B3] using the Nilpotence and Periodicity Theorems [DHS , HS ].
This suggests that Goodwillie calculus can be used to further explore these
curious functors. Included in this section, as an application of my work in
[K4 ], is an outline of a new way to possibly find a counterexample to the
Telescope Conjecture.
The Kinosaki conference was on the occasion of Professor Nishida's 60th
birthday, and I wish to both offer him my hearty congratulations, and thank
him for his kind interest in my research over the years. Many thanks also
to Noriko Minami and the other conference organizers for their hospitality.
2. Properties of Goodwillie towers
The basic problem that Goodwillie calculus is designed to attack is as
follows. One has a homotopy functor
F : C ! D
between two categories in which one can do homotopy. One wishes to un
derstand the homotopy type of F (X), perhaps for some particular X 2 C.
CALCULUS AND CHROMATIC HOMOTOPY 3
Goodwillie's key idea is to use the functoriality as X varies, to construct
a canonical polynomial resolution of F (X) as a functor of X.
The first thing to specify is what is meant by categories in which one can
do homotopy theory. In Goodwillie's papers, these are T , the category of
pointed topological spaces, or S, an associated category of spectra (e.g. the
Smodules of [EKMM ]), or variants of these, e.g. TY , the category of spaces
over and under a fixed space Y . But the arguments and constructions of
[G3 ] are written in a such a manner that they apply to situations in which
C and D are suitably nice based model categories: in x3, we will spell out
precisely what we mean.
Among all functors F : C ! D, some will be dexcisive (or polynomial of
degree at most d). This will be carefully explained in x4.2; we note that a 0
excisive functor is one that is homotopically constant, a functor is 1excisive
if it sends homotopy pushout squares to homotopy pullback squares, and a
(d  1)excisive functor is also dexcisive.
Goodwillie's first theorem says that any F admits a canonical polynomial
resolution.
Theorem 2.1. [G3 , Thm.1.8] Given a homotopy functor F : C ! D there
exists a natural tower of fibrations under F (X),
..
.


fflffl
P2F:(X):
tt
tttt p2
e2 tttt fflffl
tttt jP1F4(X)4
ttt jjjj
tttjjje1jjjjjt p1
tjjjjjjtte0 fflffl
F (X) ________________//_P0F (X),
such that
(1) PdF is dexcisive, and
(2) ed : F ! PdF is the universal weak natural transformation to a d
excisive functor.
Let us explain what we mean by property (2). By a weak natural transfor
mation f : F ! G, we mean a pair of natural transformation F g!H h G
such that H(X) h G(X) is a weak equivalence for all X. Note that a weak
natural transformation induces a well defined natural transformation be
tween functors taking values in the associated homotopy category. Property
(2) means that, given any dexcisive functor G, and natural transformation
4 KUHN
f : F ! G, there exists a weak natural transformation g : PdF ! G such
that, in the homotopy category of D,
ed(X)
F (X) _____//PdF (X)
tt
f(X)  ttttt
fflfflg(X)yytt
G(X)
commutes for all X 2 C, and any two such g agree.
A very useful property of the Pd construction is the following.
Lemma 2.2. Given natural transformations F ! G ! H, if
F (X) ! G(X) ! H(X)
is a fiber sequence for all X, then so is
PdF (X) ! PdG(X) ! PdH(X).
Let DdF : C ! D be defined by letting DdF (X) be the homotopy fiber of
PdF (X) ! Pd1F (X). The lemma and theorem formally imply that DdF
is homogeneous of degree d: it is dexcisive, and Pd1DdF (X) ' * for all X.
When D is T , Goodwillie discovered a remarkable fact: these fibers are
canonically infinite loopspaces. For a general D, we let S(D) be the associ
ated category of `Dspectra' (see x3), and Goodwillie's second theorem then
goes as follows.
Theorem 2.3. [G3 , Thm.2.1] Let F : C ! D be homogeneous of degree d.
Then there is a naturally defined homogeneous degree d functor F st: C !
S(D), such that, for all X 2 C, there is a weak equivalence
F (X) ' 1 (F st(X)).
The category S(D) is an example of a stable model category. In a manner
similar to results in the algebra literature, Goodwillie relates homogenous
degree d functors landing in a stable model category to symmetric multilin
ear ones. A functor L : Cd ! D is dlinear if it is homogeneous of degree
1 in each variable, and is symmetric if L is invariant under permutations of
the coordinates of Cd. Goodwillie's third theorem goes as follows.
Theorem 2.4. [G3 , Thm.3.5] Let F : C ! D be a homogeneous functor of
degree d with D a stable model category. Then there is a naturally defined
symmetric dlinear functor LF : Cd ! D, and a weak natural equivalence
(LF (X, . .,.X))h d ' F (X).
If F : C ! D is a homotopy functor with C and D either T or S, let
CF (d) = L(DdF )st(S, . .,.S), a spectrum with daction. Goodwillie refers
to CF (d) as the dth Taylor coefficient of F due the following corollary of the
last theorem.
CALCULUS AND CHROMATIC HOMOTOPY 5
Corollary 2.5. In this situation, there is a weak natural transformation
(CF (d) ^ X^d)h d ! (DdF )st(X)
that is an equivalence if either X is a finite complex, or F commutes with
directed homotopy colimits up to weak equivalence.
As will be illustrated in the examples, these equivariant spectra have often
been identified.
The theorems above are the ones I wish to stress in these notes, but I
should say a little about convergence. In [G2 ], Goodwillie carefully proves
a generalized BlakersMassey theorem, and uses it to study questions that
are equivalent to the convergence of these towers in the cases when C is T
or S. In particular, many functors can be shown to be `analytic', and an
analytic functor F admits a `radius of convergence' r(F ) with the property
that the tower for F (X) converges strongly for all r(F )connected objects
X. The number r(F ) is often known, as will be illustrated in the examples.
A nice result from [G2 ] reads as follows.
Proposition 2.6. [G2 , Prop.5.1] Let F ! G be a natural transformation
between analytic functors, and let r be the maximum of r(F ) and r(G).
If F (X) ! G(X) is an equivalence for all X that are equivalent to high
suspensions then it is an equivalence for all rconnected X.
3.Model category prerequisites
References for model categories include Quillen's orginal 1967 lecture
notes [Q ], Dwyer and Spalinski's 1995 survey article [DS ], and the more
recent books by Hovey and Hirschhorn [H1 , Hi].
3.1. Nice model categories. We will assume that C and D are either
simplicial or topological based model categories. `Based' means that the
initial and final object are the same: we denote will this object by *.
As part of the structure of a based topological (or simplicial) model cat
egory C, given K 2 T and X 2 C, one has new objects in C, X K and
Map (K, X) satisfying standard properties. This implies that C supports
canonical homotopy limits and colimits: given a functor X : J ! C from
a small category J , hocolimJ X and holimJ X are defined as appropriate
coends and ends:
hocolim X = X(j) j2J EJ (j)+ , and
J
Z
holim X = Map(EJ (j)+ , X(j))
J j2J
With such canonical homotopy limits and colimits, C will support a sen
sible theory of homotopy Cartesian and coCartesian cubes, as discussed in
[G2 ]: see x4.1 below. To know that certain explicit cubes in C are homotopy
coCartesian, one also needs that C be left proper, and it seems prudent to
6 KUHN
require both C and D to be proper: the pushout of a weak equivalence by a
cofibration is a weak equivalence, and dually for pullbacks.
D then needs a further axiom ensuring that the sequential homotopy
colimit of homotopy Cartesian cubes is again homotopy Cartesian: assuming
that D admits the (sequential) small object argument does the job: see [Sch,
x1.3].
Examples 3.1. The following categories satisfy our hypotheses:
o TY , the category of spaces over and under Y ,
o R  Mod , the category of Rmodules, where R is an E1 ring spec
trum, a.k.a. commutative Salgebra [EKMM ],
o R  Alg, the category of augmented commutative Ralgebras,
o simplicial versions of all of these, e.g. spectra as in [BF ].
3.2. Spectra in model categories. Let D be a model category as above,
and let X denote X S1. Trying to force the suspension : D ! D
to be `homotopy invertible' leads to a model category of spectra S(D) in
the `usual way': this has been studied carefully by Schwede [Sch] (following
[BF ]), Hovey [H2 ], and BasterraMandell [BMa ]. Roughly put, an object in
S(D) will consist of a sequence of objects X0, X1, X2, . .i.n D, together with
maps Xn ! Xn+1. The point of this construction is that the model cate
gory structure S(D) has the additional property that it is stable: homotopy
cofibration sequences in S(D) agree with the homotopy fibration sequences.
The associated homotopy category will be triangulated.
As in the familiar case when D = T , there are adjoint functors
1 : D ! S(D) and 1 : S(D) ! D.
If D is already stable these functors form a Quillen equivalence. For an
arbitrary D, this adjoint pair can take a surprising form, as the following
example illustrates.
Example 3.2. In [BMa ], the authors show that the category S(R  Alg) is
Quillen equivalent to RMod so that 1 : RAlg ! S(RAlg ) identifies
with the Topological Andr'eQuillen Homology functor1 T AQ : R  Alg !
RMod , and 1 : S(RAlg ) ! RAlg identifies with the functor sending
an Rmodule M to the trivial augmented Ralgebra R _M. (Partial results
along these lines were also proved in [BMc , Sch].)
3.3. Functors between model categories. Suppose C and D are nice
topological model categories. There are couple of useful properties that a
functor
F : C ! D
might have.
____________
1To be precise, by TAQ(B) we mean the Topological Andr'eQuillen Homology of*
* B
with coefficients in the Bbimodule R.
CALCULUS AND CHROMATIC HOMOTOPY 7
Firstly F will usually be continuous: for all X and Y in C, the function
F : Map C(X, Y ) ! Map D(F (X), F (Y ))
should be continuous.
If F is continuous, given X 2 C and K 2 T , there is a natural assembly
map
(3.1) F (X) K ! F (X K)
defined by means of various adjunctions. The existence of these assembly
maps implies that F will be a homotopy functor: a weak equivalence between
fibrant cofibrant objects in C is carried by F to a weak equivalence in D.
The second property that some functors F satisfy is that F commutes
with filtered homotopy colimits, up to weak equivalence. A functor having
this property has sometimes also been termed `continuous', but Goodwillie
[G3 ] more cautiously uses the term finitary and so will we.
One implication of being finitary is that the assembly map (3.1) will be an
equivalence. Thus there are many interesting functors that are not finitary,
as the next example shows.
Example 3.3. Let LE : S ! S be Bousfield localization of spectra with
respect to a spectrum E. Then LE is finitary exactly when the assembly
map
LE (S) ^ X ! LE (X)
is a weak equivalence for all spectra X. In other words, LE is finitary exactly
when it is smashing, a property that many interesting LE 's do not have.
Just to confuse the issue, we note that if LE is regarded as taking val
ues in the topological model category LE S, in which equivalences are E*
isomorphisms and fibrant objects are E*local [EKMM , Chap.VIII], then
LE : S ! LE S is finitary.
Finally, lets say a word about maps between functors. If C is not small,
then it seems a bit daunting (set theoretically) to impose a model cate
gory structure on the class of functors F : C ! D. As an adequate fix
for calculus purposes, we use the following terminology. Call a natural
transformation f : F ! G a weak equivalence, and write F f!~G, if
f(X) : F (X) ! G(X) is a weak equivalence for all X in C. By a weak
natural transformation f : F ! G we mean a pair of natural tranformations
of the form F g~ H h!G or F h! H g~ G. We say that a diagram of
weak natural transformations commutes if, after evaluation on any object
X, the associated diagram commutes in the homotopy category of D. Fi
nally, we say that a diagram of functors F ! G ! H a fiber sequence if
F (X) ! G(X) ! H(X) is a (homotopy) fiber sequence for all X.
8 KUHN
4.Cubical diagrams and polynomial functors
4.1. Cubical diagrams. We review some of the theory of cubical diagrams;
a reference is [G2 , x1].
Let S be a finite set. The power set of S, P(S) = {T S} , is a
partially ordered set via inclusion, and is thus a small category. Let P0(S) =
P(S)  {;} and let P1(S) = P(S)  {S}.
Definitions 4.1. (a) A dcube in C is a functor X : P(S) ! C with S = d.
(b) X is Cartesian if the natural map
X (;) ! holim X (T )
T2P0(S)
is a weak equivalence.
(c) X is coCartesian if the natural map
hocolim X (T ) ! X (S)
T2P1(S)
is a weak equivalence.
(d) X is strongly coCartesian if X P(T) : P(T ) ! C is coCartesian for all
T S with T  2.
Often S will be the concrete set d = {1, . .,.d}.
Example 4.2. A 0cube X (0) is Cartesian if and only if it is coCartesian
if and only if X (0) is acyclic (i.e. weakly equivalent to the initial object **
*).
Example 4.3. A 1cube f : X (0) ! X (1) is Cartesian if and only if it is
coCartesian if and only if f is an equivalence.
Example 4.4. A 2cube
X (0)_____//_X (1)
 
 
fflffl fflffl
X (2)____//_X (12)
is Cartesian if it is a homotopy pullback square, and coCartesian if it is a
homotopy pushout square.
Example 4.5. Strongly coCartesian dcubes are equivalent to ones con
structed as follows. Given a family of cofibrations f(t) : X(0) ! X(t) for
1 t d, let X : d ! C be defined by X (T ) = the pushout of{f(t)  t 2
T }. (Note that X (T ) can be interpreted as the coproduct under X(0) of
X(t), t 2 T .)
Critical to Goodwillie's constructions, is a special case of this last exampl*
*e.
Definition 4.6. If T is a finite set, andaX is an object in C, let X * T be
the homotopy cofiber of the folding map X ! X.
T
CALCULUS AND CHROMATIC HOMOTOPY 9
For T d, the assignment T 7! X * T is easily seen to define a strongly
coCartesian dcube X : if X ! * factors as X i!CX p!*, with i a cofi
bration and p an acyclic fibration, then X agrees with the cube of the last
example with f(t) = i : X ! CX for all t.
In the special case when C = T , X * T is the (reduced) join of X and T :
the union of T  copies of the cone CX glued together along their common
base X.
There is a very useful way to inductively identify Cartesian cubes. Note
that the fibers of the vertical maps in a Cartesian 2cube as in Example 4.4
form a Cartesian 1cube as in Example 4.3. This generalizes to higher di
mensional cubes as we now explain.
Regard d as the obvious subset of d + 1. Given an (d + 1)cube X :
P(d + 1) ! C, we define three associated dcubes
Xtop, Xbottom, @X : P(d) ! C
as follows. Let Xtop(T ) = X (T ) and Xtop(T ) = X (T [ {n + 1}). Then define
@X (T ) by taking homotopy fibers of the evident natural transformation
between these:
@X (T ) = hofib{Xtop(T ) ! Xbottom(T )}.
Lemma 4.7. X is Cartesian if and only if @X is Cartesian.
Lemma 4.8. If Xtopand Xbottomare Cartesian, so is X .
Remark 4.9. Dual lemmas hold for coCartesian cubes. One application of
this is that if C is a stable model category, so that homotopy fibre sequences
are the same as homotopy cofiber sequences, then X is Cartesian if and only
if X is coCartesian.
4.2. Polynomial functors. Let C and D be topological or simplicial model
categories as in x3.1.
Definition 4.10. F : C ! D is called dexcisive or said to be polynomial
of degree at most d if, whenever X is a strongly coCartesian (d + 1)cube in
C, F (X ) is a Cartesian cube in D.
Example 4.11. F has degree 0 if and only if F (X) ! F (*) is an equivalence
for all X 2 C, i.e. F is homotopy constant.
Example 4.12. F : C ! D is 1excisive means that F takes pushout
squares to pullback squares.
In the classical case C and D are spaces or spectra, this implies that the
functor sending X to ss*(F (X)) satisfies the MayerVietoris property.
If F is also finitary, then Milnor's wedge axiom holds as well. Then there
are spectra C0 and C1 such that F (X) ' C0 _ (C1 ^ X) if D = S and
F (X) ' 1 (C0 _ (C1 ^ X)) if D = T .
10 KUHN
Remark 4.13. Without the finitary hypothesis, classifying 1excisive func
tors seems very hard. Examples of 1excisive functors of X from spectra
to spectra include the localization functors LE X and functors of the form
MapS (C, X) where C 2 S is fixed.
The following proposition of Goodwillie constructs dexcisive functors out
of dvariable 1excisive functors.
Proposition 4.14. [G2 , Prop.3.4] If L : Cd ! D is 1excisive in each of
the dvariables, then the functor sending X to L(X, . .,.X) is dexcisive.
Corollary 4.15. In this situation, if L is symmetric, and D is a stable
model category, then, given any subgroup G of the dth symmetric group n,
the functor sending X to L(X, . .,.X)hG is dexcisive.
The various lemmas about identifying Cartesian cubes can be used to
prove the next two useful lemmas.
Lemma 4.16. If F is dexcisive, then F is cexcisive for all c d.
Lemma 4.17. If F ! G ! H is a fiber sequence of functors, and G and
H are both dexcisive, then so is F .
5. Construction of Goodwillie towers and the proof of the
main properties
5.1. Construction of the tower and the proof of Theorem 2.1. If
one is to construct a dexcisive functor PdF , then PdF (X ) needs to be
Cartesian for for all strongly coCartesion (d + 1)cubes X . The idea behind
the construction of PdF is to force this condition to hold for certain strongly
coCartesion (d + 1)cubes X .
Fix an object X 2 C. As discussed above, for T d + 1, the assignment
T 7! X * T defines a strongly coCartesian d + 1cube X . For example, when
d + 1 = 2, one gets the pushout square
X ______//CX
 
 
fflffl fflffl
CX _____// X.
Definition 5.1. Let TdF : C ! D be defined by
TdF (X) = holim F (X * T ).
T2P0(d+1 )
Note that there is an evident natural transformation td(F ) : F ! TdF ,
and that this is an equivalence if F is dexcisive.
Definition 5.2. Let PdF : C ! D be defined by
td(F) td(TdF)
PdF (X) = hocolim{F (X) ! TdF (X) ! TdTdF (X) ! . .}..
CALCULUS AND CHROMATIC HOMOTOPY 11
Example 5.3. T1F (X) is the homotopy pullback of
F (CX)


fflffl
F (CX) _____//F ( X),
Suppose that F (*) ' *. Then F (CX) ' *, so that T1F (X) is equivalent to
the homotopy pullback of
*


fflffl
* _____//F ( X),
which is F ( X). It follows that there is a natural weak equivalence
P1F (X) ' hocolimn!1 nF ( nX).
Example 5.4. Specializing the last example to the case when F is the
identity functor Id : D ! D, we see that
P1(Id)(X) ' 1 1 X.
If D = T , topological spaces, we see that P1(Id)(X) = QX.
If D = R  Alg, we see that P1(Id)(B) ' R _ T AQ(B) for an augmented
commutative Ralgebra B.
The proof of Theorem 2.1 amounts to checking that the Pd construction
just defined has the two desired properties: PdF should always be dexcisive,
and F ! PdF should be universal. Checking the first of these is by far the
more subtle, and follows from the next lemma.
Lemma 5.5. [G3 , Lemma 1.9] If F : C ! D is a homotopy functor, and X
is strongly coCartesian (d + 1)cube in C, then there is a Cartesian (d + 1)
cube Y in D, such that F (X ) ! TdF (X ) factors through Y.
The construction of Y is very devious. Y is (roughly) constructed to be
the homotopy limit of (d + 1)cubes in D that are each seen to be Cartesian
for the following reason: they are constructed by applying F to (d + 1)
cubes in C formed by means of evident objectwise equivalences between two
dcubes.
In contrast, proving that F ! PdF is appropriately universal is much
easier. Once one knows that PdF is dexcisive, universality amounts to
checking the following two things:
(a) If F is dexcisive, then ed(F ) : F ! PdF is a weak equivalence, and
(b) Pd(ed(F ))) : PdF ! PdPdF is a weak equivalence.
12 KUHN
These follow immediately from Tdversions of these statements:
(a') If F is dexcisive, then td(F ) : F ! TdF is a weak equivalence, and
(b') Pd(td(F ))) : PdF ! PdTdF is a weak equivalence.
As was noted above, the first of these is clear. The second admits a fairly
simple proof based on the commutativity of iterated homotopy inverse limits.
Similar reasoning verifies the next lemma, which in turn implies Lemma 2.2,
which said that Pd preserves fiber sequences.
Lemma 5.6. Given natural transformations F ! G ! H, if
F (X) ! G(X) ! H(X)
is a fiber sequence for all X, then so is
TdF (X) ! TdG(X) ! TdH(X).
5.2. Delooping homogeneous functors and Theorem 2.3. The most
surprising property of Goodwillie towers is stated in Theorem 2.3. This says
that, for d > 0, homogeneous dexcisive functors are infinitely deloopable.
To show this, Goodwillie proves his beautiful key lemma, which says that
PdF (X) ! Pd1F (X) is always a principal fibration if F is reduced: F (*) '
*.
Lemma 5.7. [G3 , Lemma 2.2] Let d > 0, and let F : C ! D be a reduced
functor. There exists a homogeneous degree d functor RdF : C ! D fitting
into a fiber sequence of functors
PdF ! Pd1F ! RdF.
Iteration of the Rd construction leads to Theorem 2.3: if F is homogeneous
of degree d, then we can let F st(X) be the spectrum with rthspace RrdF (X).
The proof of Lemma 5.7 is yet another clever manipulation of categories
related to cubes. As an indication of how this might work, we sketch how
one can construct a homotopy pullback square
TdF (X) ______//KdF (X)
 
 
fflffl fflffl
Td1F (X) _____//QdF (X)
with KdF (X) ' *, in the case when d = 2.
One needs to look at how one passes from P0(2) to P0(3). In pictures,
P0(2) looks like
1
"""
2____12,
CALCULUS AND CHROMATIC HOMOTOPY 13
while P0(3) looks like
v1
vvv 
2 ______12 
  
 3 _______13
 """  vvv
23 _____123.
Now we decompose the poset P0(3) as
(5.1) A \ B ___________//B
 
 
fflffl fflffl
A _______//A [ B = P0(3),
where A is
3 ________13
""" www
23 _____123,
and B is
v 1
vvv 
2 _____ 12 
  
  w13
  ww
23 ____123,
so that A \ B is
13
vvv
23 ____123.
The decomposition of posets (5.1) induces a homotopy pullback diagram
holimT2P0(3)F (X * T )_____//_holimT2AF (X * T )
 
 
fflffl fflffl
holim T2BF (X * T )______//_holimT2A\BF (X * T ).
The top left corner is T3F (X), by definition. As P0(2) is cofinal in B,
the bottom left corner is equivalent to T2F (X). Finally A has initial object
{3}, so that the upper left corner is contractible:
holimF (X * T ) ' F (X * {3}) = F (CX) ' *.
T2A
5.3. Cross effects and the proof of Theorem 2.4.
Definition 5.8. Let F : C ! D be a functor. We define crdF : Cd ! D,
the dth cross effect of F , to be the the functor of d variables given by
` `
(crdF )(X1, . .,.Xd) = hofib{F ( Xi) ! holim F ( Xi)}.
i2d T P0(d) i2dT
14 KUHN
`
The dcube sending T to Xi is easily seen to be strongly coCarte
i2dT
sian; letting d = 2 for example, the square
X1 _ X2 ____//_X2
 
 
fflffl fflffl
X1 ________//*
is weakly equivalent to the evidently coCartesian square
X1 _ X2 _______//CX1 _ X2
 
 
fflffl fflffl
X1 _ CX2 _____//CX1 _ CX2.
It follows that if F is (d  1)excisive, then crdF (X1, . .,.Xd) ' * for all
Xi. A similar argument [G3 , Lemma 3.3] shows that if F is dexcisive, then
crdF is 1excisive in each of its variables.
Another property of crdF : Cd ! D that is easy to see is that it is reduced:
crdF (X1, . .,.Xd) ' * if Xi' * for some i.
A permutation of d, oe 2 d, induces an evident isomorphism
oe* : crdF (X1, . .,.Xd) ! crdF (Xoe(1), . .,.Xoe(d)),
satisfying (oe O o)* = oe* O o*: a functor of d variables with this structure is
called symmetric.
Definition 5.9. Let LdF : Cd ! D be the functor obtained from crdF by
applying P1 to each variable. Thus we have
LdF (X1, . .,.Xd) ' hocolimn n1+...+ndcrdF ( n1X1, . .,. ndXd).
i!1
LdF will always be symmetric and dlinear, and if F is dexcisive, then
the natural map crdF ! LdF is an equivalence.
If G is a finite group, let G  D denote the category of objects in D with
a Gaction. Given Y 2 G  D, we let YhG and Y hG denote the associated
homotopy quotient and fixed point objects in D.
Definition 5.10. Let dF : C ! d  D be defined by
dF (X) = LdF (X, . .,.X).
A more precise version of Theorem 2.4 is the following.
Theorem 5.11. Let F : C ! D be a homotopy functor, with D a stable
model category. Then there is a natural weak equivalence
dF (X)h d ' DdF (X).
If F is dexcisive then dF (X) can be identified with (crdF )(X, . .,.X).
In this case, one gets a natural transformation
ffd(X) : ( dF )(X)h d ! F (X)
CALCULUS AND CHROMATIC HOMOTOPY 15
defined to be the composite
`d
( dF )(X)h d ! F ( X)h d ! F (X).
i=1
W d
Here the second map is induced by the fold map i=1X ! X. The theorem
is then proved by verifying that crd(ffd) is an equivalence.
We indicate how Corollary 2.5 follows from Theorem 5.11. The assembly
map for F induces an assembly map
( dF (X) K^d)h d ! dF (X K)h d,
for X 2 C and K 2 T .
If C is T , D is S, and X = S, then this reads
(CF (d) ^ K^d)h d ! dF (K)h d,
where CF (d) = dF (S). By construction, this map is the identity if K = S,
and it follows that it will be an equivalence for all finite K, or all K under
the additional hypothesis that F is finitary. A similar argument holds if both
C and D are S: here the assembly map can be constructed for all K 2 S.
When the domain category C is also stable, and D = S, there is an elegant
addendum to Theorem 5.11 essentially due to R.McCarthy [McC ].
Given Y 2 G  S, there is a natural norm map N(Y ) : YhG ! Y hG
satisfying the property that N(Y ) is an equivalence if Y is a finite free G
CW spectrum. As in [K5 ], we let the Tate spectrum of Y , TG (Y ), be the
cofiber.
Proposition 5.12. [K5 ] Let F : C ! S be any homotopy functor, with C
stable. For all d 1, there is a homotopy pullback diagram
PdF (X) ______//_( dF (X))h d
 
 
fflffl fflffl
Pd1F (X) _____//T d( dF (X)).
6. Examples
6.1. Suspension spectra of mapping spaces. Fix a finite C.W. complex
K. Let MapT (K, X) be the space of based continuous maps from K to a
space X. Similarly, given a spectrum Y , let MapS (K, Y ) be the evident
function spectrum.
In [G2 ], Goodwillie proved that the functor from spaces to spectra sending
X to 1 MapT (K, X) is analytic with radius of convergence equal to the
dimension of K.
In [A ], Arone gave a very concrete model for the associated Goodwillie
tower {P*K(X)}. The paper [AK ] includes further details about Arone's
construction while building in extra structure.
16 KUHN
Let E be the category with objects the finite sets d, d 1, and with
morphisms the epic functions. Ed will denote the full subcategory with
objects c with c d.
Given a based space X, let X^ : Eop ! T be the functor sending d to
X^d. Then Arone's model for PdK : T ! S is given by
PdK(X) = MapS Ed(K^ , 1 X^ ),
the spectrum of natural transformations between the two contravariant func
tors of Ed. The natural transformation
1 MapT (K, X) ! PdK(X)
is induced by sending f : K ! X to f^ : K^ ! X^ and then stablizing.
A by product of this construction is that there is a homotopy pullback
square of Smodules which has some of the same flavor as Proposition 5.12:
PdK(X) ______//_MapS d(K^d, 1 X^d)
 
 
fflffl fflffl
PdK1(X) ____//_MapS d(ffid(K), 1 X^d),
where ffid(K) K^d denotes the fat diagonal.
Thus the dth fiber, DKd(X), can be described as follows. Let K(d)denote
K^d=ffid(K). Then we have
DKd(X) = MapS d(K(d), 1 X^d)
' MapS (K(d), 1 X^d)h d
' (D(K(d)) ^ X^d)h d.
Here D(K(d)) denotes the equivariant Sdual of K(d), and the equivalences
follow from the fact that K(d) is both finite and dfree away from the
basepoint. It follows that the dth Taylor coefficient of the functor sending
X to 1 MapT (K, X) is D(K(d)).
Remark 6.1. In [K3 ], we observed that, when X is also a finite complex, the
tower P*K(X) also arises as by taking the Sdual of a natural filtration on
the nonunital commutative Salgebra D(X) K.
By Alexander duality, D(K(d)) can be identified an appropriate equi
variant desuspension of the suspension spectrum of a configuration space.
Specializing to the case when K = Sn, this takes the following concrete
form. Let C(n, d) denote the space of d distinct little ncubes in a big n
cube [May ]. Via a ThomPontryagin collapse, there is a very explicit duality
map of d spaces [AK ]
C(n, d)+ ^ Sn(d)! Snd
CALCULUS AND CHROMATIC HOMOTOPY 17
One proof that Arone's model works when K = Sn goes roughly as fol
lows. Suppose X = nY . One has the usual filtered configuration space
model Cn(Y ) for n nY [May ]. Thus one has maps
n
1 FdCn(Y ) ! 1 n nY ! PdS (Y ).
The nontriviality of the second map is proved by showing that the composite
is an equivalence. By induction on d, is suffices to show that crd applied to
this composite is an equivalence, and the verification of that leads back to
the above explicit duality map.
A bonus corollary of this proof is that one also establishes a rather nice
version of `Snaith splitting': the tower strongly splits when X = nY .
Example 6.2. One application of this comes from applying mod p coho
mology to the tower. One obtains a spectral sequence of differential graded
algebras {Es,tr(Sn, X)} with
Ed,*1(Sn, X) = H*((C(n, d)+ ^ ( dX)^d)h d; Z=p))
and converging strongly to H*( nX; Z=p) if X is nconnected. This E1
term is a known functor of H*(X; Z=p). The differentials have not been
fully explored, but seem to be partly determined by derived functors of
destablization of unstable modules over the Steenrod algebra, as applied to
the Amodule n H*(X; Z=p).
6.2. Suspension spectra of infinite loopspaces.1The previous example
can be used to determine the tower {PdS } for the functor from spectra to
spectra sending a spectrum X to 1 1 X.
Let Xn denote the nth space of the spectrum X. Then we have that
nXn ' 1 X for all n, and the natural map
hocolimn!1 n 1 Xn ! X
is an equivalence. From this and the last example, one can deduce that the
tower converges for 0connected spectra X and that
n S1
hocolimn!1 n PdS (Xn) ' Pd (X).
As hocolimn!1C(n, d)+ is a model for E d+, and this is weakly equivalent
to S0, it follows that the formula for the dth fiber is
1 ^d
DSd (X) ' Xh d,
and thus the dth Taylor coefficient of the functor sending a spectrum X to
1 1 X is the sphere spectrum S for all d > 0.
18 KUHN
Finally, Proposition 5.12 specializes to say that for each d > 0 there is a
pullback square
PdS1(X) ______//(X^d)h d
 
 
1fflffl fflffl
PdS1(X)_____//T d(X^d).
Example 6.3. The tower begins
P2S1(X)88
qqq
e2qqqq p1
qqqqe1 fflffl
1 1 X ________//X,
where e1 : 1 1 X ! X is adjoint to the identity on 1 X. A formal
consequence is that
1 1
1 p1 : 1 P2S (X) ! X
admits a natural section.
The map p1 fits into a natural cofibration sequence
1 p1
P2S (X) ! X ! (X ^ X)h 2.
Specializing to the case when X = S1 , this can be identified [K5 , Appendix]
with the cofibration sequence
1RP01 t!S1 ! RP11,
where t is one desuspension of the KahnPriddy transfer, and RPk1 denotes
the Thom spectrum of k copies of the canonical line bundle over RP 1.
Letting QZ denote 1 1 Z, we conclude that
1 tr : QRP+1 ! QS0
admits a section: a result `one loop' away from the full strength of the Kahn
Priddy Theorem [KaP ] at the prime 2. The odd prime version1admits a
similar proof using that, localized at a prime p, PdS (S1 ) ' * for 1 < d < p.
6.3. The identity functor for Alg. Let Alg be the category of commuta
tive augmented Salgebras. This is a model category in which weak equiva
lences and fibrations are determined by forgetting down to Smodules. More
curious is that the coproduct of A and B is A ^ B.
Let {Pdalg} denote the tower associated to the identity I : Alg ! Alg .
Given B 2 Alg , it is not too hard to deduce that the tower {Pdalg(B)}
will strongly converge to B if I(B) is 0connected, where I(B) denotes the
`augmentation ideal': the homotopy fiber of the augmentation B ! S.
Let Dalgd(B) be the fiber of Pdalg(B) ! Pdalg1(B). As already discussed
in Example 5.4, Dalg1(B) can be identified with T AQ(B), the Topological
Andr'eQuillen Homology of B with coefficients in the Bbimodule S.
CALCULUS AND CHROMATIC HOMOTOPY 19
The fact that coproducts in Alg correspond to smash products of S
modules leads to a simple calculation of the dth cross effect of I:
crd(I)(B1, . .,.Bd) ' I(B1) ^ . .^.I(Bd).
From this, one gets a formula for Dalgd(B):
Theorem 6.4.
Dalgd(B) ' T AQ(B)^dh.d
A proof of this in the spirit of this paper appears in [K4 ]. See also [Min ].
A nice corollary of this formula says the following.
Corollary 6.5. If A and B in Alg have 0connected augmentation ideals,
then an algebra map f : A ! B is an equivalence if T AQ(f) is.
The converse of this corollary  that T AQ(f) is an equivalence if f is 
is true even without connectivity hypotheses: see, e.g. [K3 ]. Without any
hypotheses implying convergence, one has that if T AQ(f) is an equivalence,
so is ^f: ^A! ^B, where A^denotes the homotopy inverse limit of the tower
for A [K4 ].
Example 6.6. This tower overlaps in an interesting way with the one for
1 1 X discussed above, and the corollary leads to a simple proof of a
highly stuctured version of the classical stable splitting [Ka ] of QZ, for a
connected space Z.
It is well known that 1 ( 1 X)+ is an E1 ring spectrum. Otherwise put,
we can regard 1 ( 1 X)+ as an object in Alg. It is not hard to see that
T AQ( 1 ( 1 X)+ ) is equivalent to the connective cover of X, and there is
an equivalence
1 alg 1 1
PdS (X) ' Pd ( ( X)+ )
for connective spectra X.
Another object in Alg is P(X), the free commutative Salgebra generated
by X. As an Smodule,
1`
P(X) ' X^dh,d
d=0
and it is not hard to compute that T AQ(P(X)) ' X.
The stable splitting of QZ gets proved as follows. The inclusion
j(Z) : Z ! QZ
induces a natural map in Alg
s(Z) : P( 1 Z) ! 1 (QZ)+ .
The construction of s makes it quite easy to verify that T AQ(s(Z)) : 1 Z !
1 Z is the identity. The above corollary then implies that s(Z) is an
equivalence in Alg, and thus in S, for connected spaces Z.
A more detailed discussion of this appears in [K4 ].
20 KUHN
6.4. The identity functor for T . Let {Pd} denote the tower of the iden
tity functor on T . Given a space Z, let Dd(Z) denote the fiber of Pd(Z) !
Pd1(Z), and then let Dstd(Z) 2 S be the infinite delooping provided by
Theorem 2.3.
Goodwillie's estimates [G2 ] show that the tower {Pd(Z)} will strongly con
verge to Z when Z is connected. Thus applying ss*, one gets a strongly con
vergent 2nd quadrant spectral sequence converging to ss*(Z), with E1d,*=
ss*(Dstd(Z)).
Johnson [J], and Arone with collaborators Mahowald and Dwyer [AM ,
AD ], have identified the spectra Dstd(Z):
Dstd(Z) ' (D( Kd) ^ Z^d)h d,
where Kd is the unreduced suspension of the classifying space of the poset of
nontrivial partitions of d, and D denotes the equivariant Sdual, as before.
Using this model, the discovery of Arone and Mahowald [AM ] is that
when Z is an odd dimensional sphere, these spectra are very special spectra
that were known previously. To state the theorem, we need some notation.
Let p be a prime. 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 (BVk)maek. 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 (BVk)maek:
L(k, m) = ek(BVk)maek.
The spectra L(k, 0) and L(k, 1) agree with spectra called M(k) and L(k) in
the literature: see e.g. [MP ], [K1 , KuP ].
Collecting results from [AM ] and [AD , Thm.1.9, Cor.9.6], one has the
following theorem.
Theorem 6.7. Let m be an odd natural number.
(1) Dstd(Sm ) ' * if d is not a power of a prime.
(2) Let p be a prime. Dstpk(Sm ) ' mk L(k, m), and thus has ptorsion
homotopy if k > 0.
(3) H*(L(k, m); Z=p) is free over the subalgebra A(k  1) of the Steenrod
algebra. As a function of k, the connectivity of L(k, m) has a growth rate
like pk.
Thus the associated spectral sequences for computing the unstable ho
motopy groups of odd spheres coverges exponentially quickly, and begins
from stable information about spectra of roughly the same complexity as
the suspension spectra of classifying spaces of elementary abelian pgroups.
Remark 6.8. When m = 1, one gets a spectral sequence converging to the
known graded group ss*(S1), with E1k,*= ss*(L(k)). Comparison with my
CALCULUS AND CHROMATIC HOMOTOPY 21
work on the Whitehead Conjecture [K1 , KuP ] suggests that E2 = E1 . Greg
Arone and I certainly believe this, but a rigorous proof has yet to be nailed
down.
As discussed near the end of the next section, the properties listed in the
theorem have particularly beautiful consequences for computing the periodic
unstable homotopy groups of odd dimensional spheres.
7. Interactions with periodic homotopy
For topologists who study classical unstable and stable homotopy theory,
a major development of the past two decades has been the organization
of these subjects via the chromatic filtration associated to the Morava K
theories.
One of the most unexpected aspects of Goodwillie towers is that they
interact with the chromatic aspects of homotopy in striking ways. In this
section, I survey, in inverse order of when they were proved, three different
theorems of this sort.
7.1. Goodwillie towers and homology isomorphisms. There are a
couple of useful general facts about how Bousfield localization relates to
Goodwillie towers.
Let E* be a generalized homology theory. A map f : X ! Y of spaces
or spectra is called an E*isomorphism if E*(f) is an isomorphism. A nat
ural transformation f : F ! G between functors F, G : C ! S is an E*
isomorphism if f(X) is for all X 2 C. Then we have
Proposition 7.1. [K6 , Cor. 2.4] If F : C ! S is finitary and f : X !
Y is an E*isomorphism then so are DdF (f) : DdF (X) ! DdF (Y ) and
PdF (f) : DdF (X) ! PdF (Y ) for all d.
Proposition 7.2. [K5 , Lemma 6.1] If a natural transformation f : F ! G
between functors F, G : C ! S is an E*isomorphism then so are Ddf :
DdF ! DdG and Pdf : PdF ! PdG for all d.
Both of these follow by observing that the various constructions defining
Pd and Dd preserve E*isomorphisms.
The next example illustrates that the finitary hypothesis in Proposi
tion 7.1 is needed.
Example 7.3. Consider LHZ=p : S ! S, a homogeneous functor of degree
1. Then HZ=p1 is HQ*acyclic (i.e. HZ=p1 ! * is an HQ*equivalence),
but LHZ=p(HZ=p1 ) = HZp is not.
For an application of Proposition 7.1 to the homology of mapping spaces,
see [K6 ]. Proposition 7.2 is crucially used in proofs of the two theorems
discussed in the next two subsections.
We end this subsection with some observations related to the phenomenon
illustrated in the last example.
22 KUHN
If F : C ! S is homogeneous of degree d, the functor LE F : C ! S will
always again be dexcisive, but need no longer be homogeneous.
Example 7.4. Let F : S ! S be defined by F (X) = (X ^ X)h 2. The
composite functor LE F will be 2excisive, but need no be longer homoge
nous, even when restricted to finite spectra. Indeed, a simple calculation
shows that
P1(LE F )(S) = hocofib{LE S ^ RP 1 ! LE RP 1}.
This can easily be nonzero. For example, when E is mod 2 Ktheory, one has
that LE RP 1 = LE S, as the transfer RP 1 ! S is an KZ=2*isomorphism.
It follows that P1(LE F )(S) has nonzero rational homology.
As a fix for this problem, we have the next lemma, which follows from
Proposition 7.2.
Lemma 7.5. If F : C ! S is homogeneous of degree d, then Pd1(LE F ) is
E*acyclic. Otherwise said, Dd(LE F ) ! LE F is an E*isomorphism.
7.2. Goodwillie towers and periodic localization. We will consider two
families of periodic homology theories.
Fixing a prime p, K(n)* is the nth Morava Ktheory.
To define the second family, recall that a plocal finite complex M is of
type n if M is K(m)*acyclic for m < n, but is not K(n)*acyclic. If M is
of type n, then M admits a vnself map, a K(n)*isomorphism
v : dM ! M.
We let T (n) denote the telescope of v. A consequence of the Nilpotence
and Periodicity Theorems of Devanitz, Hopkins, and J.Smith [DHS , HS ,
R] is that the associated Bousfield localization functor LT (n) : S ! S
is independent of the choice of both the complex and self map. Also, we
recall that T (n)*acyclics are K(n)*acyclic; thus the associated localization
functors are related by LK(n) ' LK(n)LT(n).
The main theorem of [K5 ] says that Goodwillie towers of functors from
spectra to spectra always split after applying LT(n).
Theorem 7.6. [K5 , Thm.1.1] Let F : S ! S be any homotopy functor. For
all primes p, n 1, and d 1, the natural map
pd(X) : PdF (X) ! Pd1F (X)
admits a natural homotopy section after applying LT(n).
Corollary 7.7. Let F : S ! S be polynomial of degree less than d and
G : S ! S homogeneous of degree d. Then any natural transformation
f : F ! LT(n)G will be null.
The corollary follows formally from the theorem using Lemma 7.5: we
leave verifying this as an exercise for the reader. The theorem is proved by
CALCULUS AND CHROMATIC HOMOTOPY 23
combining Proposition 5.12 and Proposition 7.2 with the following vanishing
theorem about Tate homology.
Theorem 7.8. [K5 , Thm.1.5] For all finite groups G, primes p, and n 1,
LT(n)TG (LT(n)S) ' *.
In [K5 ], I manage to first reduce the proof of the theorem to the case
when G = p. There are familiar `inverse limits of Thom spectra' models
for LT(n)T p(LT(n)S). Using these, the equivalence LT(n)T p(LT(n)S) ' *
can be shown to be equivalent to the case when X = S of the following
statement about the Goodwillie tower of 1 1 X.
1 k
Theorem 7.9. [K5 , Thm.3.7] holim kLT(n)PpS ( X) ! LT(n)X admits
k
a homotopy section.
This theorem follows immediately from the existence of the natural section
jn(X) of LT(n)e1(X) : LT(n) 1 1 X ! LT(n)X to be discussed in the next
subsection.
Remark 7.10. A weaker version of Theorem 7.8 with K(n) replacing T (n)
appears in work by Greenlees, Hovey, and Sadofsky [GS , HSa ], and certainly
inspired my thinking, if not my proof. Theorem 7.8 when G = Z=2 is
equivalent to the main theorem of [MS ].
7.3. The periodic homology of infinite loopspaces. Using the full
strength of the Periodicity Theorem, Bousfield and I have constructed `tele
scopic functors' as in the next theorem.
Theorem 7.11. [B1 , K2 , B3 ] For all p and n > 0, there exists a functor
n : T ! S such that n( 1 X) ' LT(n)X. Furthermore n(Z) is always
T (n)local.
Some further nice properties of n will be discussed in the next subsection:
see Proposition 7.16. Here we note the following corollary.
Corollary 7.12. After applying LT(n), the natural transformation
e1(X) : 1 1 X ! X
admits a section
jn(X) : LT(n)X ! LT(n) 1 1 X.
The section is defined by applying n to the natural map
j( 1 X) : 1 X ! Q 1 X.
Remark 7.13. jn is unique up to `tower phantom' behavior in the following
sense: for all d, the composite
jn(X) 1 1 LT(n)ed(X) S1
LT(n)X ! LT(n) X ! LT(n)Pd (X)
LT(n)pd(X)
is the unique natural section of LT(n)PdS1(X) ! LT(n)X. Here
uniqueness is a consequence of Corollary 7.7.
24 KUHN
In [K4 ], I use jn to prove a splitting result in a manner similar to Exam
ple 6.6. The natural transformation
jn(X) : X ! LT(n) 1 1 X
induces a map of commutative augmented LT(n)Salgebras
sn(X) : LT(n)P(X) ! LT(n) 1 ( 1 X)+ .
As Example 6.6, sn(X) has been constructed so that it is easy to see that
T AQ(sn(X)) : LT(n)X ! LT(n)X is homotopic to the identity, and one
learns that sn(X) induces an equivalence of localized Goodwillie towers.
Because the towers have been localized with respect to a nonconnected ho
mology theory, the convergence of localized towers is problematic. However,
one can easily deduce the first statement of the next theorem, and starting
from this, I was able to establish the rest.
Theorem 7.14. [K4 ] For all X 2 S,
sn(X)* : K(n)*(P(X)) ! K(n)*( 1 X)
is monic, and fits into a chain complex of commutative K(n)*Hopf algebras
n+1O
K(n)*(P(X)) sn*!K(n)*( 1 X) ! K(n)*(K(ssj(X), j)).
j=0
This sequence of Hopf algebras is exact if X is T (m)*acyclic for all 0 <
m < n, and only if X is K(m)*acyclic for all 0 < m < n.
Note that the two acyclicity conditions on X are empty if n = 1. They
agree if n = 2, by the truth of the Telescope Conjecture when n = 1.
Recall that the Telescope Conjecture asserts that a K(n)*acyclic spec
trum will always be T (n)*acyclic, and is believed to be not true for n 2.
Peter May has remarked that maybe the theorem could be used to disprove
it. I end this subsection by describing one way this might go.
If Z is a connected space, let jn(Z) : LT(n)P(Z) ! LT(n)P(Z) be the
composite
sn( 1 Z) 1 LT(n)s(Z)1
LT(n)P(Z) ! LT(n) QZ ! LT(n)P(Z).
Here we have written P(Z) for P( 1 Z).
The theorem says that if Z is T (m)*acyclic for all 0 < m < n, and only
if Z is K(m)*acyclic for all 0 < m < n, there is a short exact sequence of
K(n)*Hopf algebras
n+1O
K(n)*(P(Z)) jn*!K(n)*(P(Z)) ! K(n)*(K(ssSj(Z), j)).
j=0
It appears that for some Z, a calculation of both K(n)*(P(Z)) and jn*
may be accessible. If one could find a K(n  1)*acyclic space2 Z, and
____________
2If a space Z is K(n  1)*acyclic, then it is K(m)*acyclic for all m < n b*
*y [B2].
CALCULUS AND CHROMATIC HOMOTOPY 25
explicit calculation showed that the above sequence is not exact, it would
follow that Z would not be T (m)*acyclic for some 0 < m < n. The first
example to check is when p = 2, n = 3, and Z = K(Z=2, 3): it is known
that this space is K(2)*acyclic, but it is unknown whether or not it is
T (2)*acyclic.
7.4. The periodic homotopy groups of odd dimensional spheres.
Let v : dM ! M be a K(n)*isomorphism of a space M whose suspension
spectrum is a finite compex of type n. If Z is a space, one can use v to
define periodic homotopy groups by letting
v1 ss*(Z; M) = colimr[ rdM, Z]*.
It is clear that these behave well with respect to fibration sequences in the
Z variable. These can be similarly defined for spectra, and it is evident that
there is an isomorphism v1 ss*( 1 X; M) = v1 ss*(X; M).
The direct limit appearing in the definition suggests that these functors
of spaces do not necessarily commute with holimits of towers. However
Arone and Mahowald note that the properties listed in Theorem 6.7 imply
that the tower for an odd dimensional sphere leads to a convergent spectral
sequence with only a finite number of infinite loop fibers for computing
periodic homotopy. More precisely, they show prove the following.
Theorem 7.15. Let m be odd. With (M, v) as above, the natural map
v1 ss*(Sm ; M) ! v1 ss*(PpnSm ; M)
is an isomorphism.
Bousfield notes that periodic homotopy can be computed using the tele
scopic functor n (at least for the version of n defined in [B3 ]).
Proposition 7.16. [B3 , Thm.5.3(ii) and Cor.5.10(ii)] There are natural
isomorphisms
v1 ss*(Z; M) ' [M, n(Z)]*.
Furthermore, given f : Y ! Z, v1 ss*(f; M) is an isomorphism if and only
if n(f) : n(Y ) ! n(Z) is a weak equivalence.
Assembling all of the various results, one deduces the following theorems.
Theorem 7.17. When m is odd, there is a spectral sequence converging to
v1 ss*(Sm ; M) with
(
[M, LT(n)L(k, m)]*+km for0 k n
E1k,*=
0 otherwise.
Theorem 7.18. Let m be odd. The spectrum n(Sm ) admits a finite de
creasing filtration with fibers LT(n) mk L(k, m) for k = 0, . .,.n.
26 KUHN
Example 7.19. Using that L(0, m) = S0, L(1, m) = RP 1=RP m1, and
some naturality properties of Goodwillie towers, one can quite easily deduce
that there is a weak equivalence
1(S2k+1) ' 2k+1LK(1)RP 2k.
One can define periodic homotopy groups with integral coefficients. Ob
serve that v1 ss*(Z; M) is really only dependent on the spectrum 1 M. As
in [K2 ], one can construct a sequence under S of finite spectra of type n,
C1 ! C2 ! . . .
so that the induced map C(n) ! S is a K(n)*isomorphism, where C(n) =
hocolimrCr.
One defines v1 ss*(Z) by letting
v1 ss*(Z) = colimrv1 ss*(Z; D(Cr)).
The results above show that alternatively this can be computed as
v1 ss*(Z) = C(n)*( n(Z)).
Various people have observed that C(n) is independent of choices (see
e.g. [Mil]): indeed it is the fiber of the finite localization map S ! Lfn1S.
Observations of Bousfield [B3 , Thm.3.3] can be interpreted as saying that
there are isomorphisms
ss*(MfnX) = eC(n)*(X) = C(n)*(LT(n)X),
where Ce(n) is the cofiber of C(n + 1) ! C(n), and MfnX is the fiber of
LfnX ! Lfn1X.
Thus we have our last theorem.
Theorem 7.20. When m is odd, there is a spectral sequence for computing
v1 ss*(Sm ) with
(
eC(n)*+km (L(k, m)) for0 k n
E1k,*=
0 otherwise.
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Department of Mathematics, University of Virginia, Charlottesville, VA
22903
Email address: njk4x@virginia.edu