THE GOODWILLIE TOWER OF THE IDENTITY FUNCTOR AND
THE UNSTABLE PERIODIC HOMOTOPY OF SPHERES
GREG ARONE AND MARK MAHOWALD
June 9, 1998
Abstract. We investigate Goodwillie's "Taylor tower" of the identity func*
*tor
from spaces to spaces. More specifically, we reformulate Johnson's descri*
*ption of
the Goodwillie derivatives of the identity, and prove that in the case of*
* an odd
dimensional sphere the only layers in the tower that are not contractible*
* are those
indexed by powers of a prime. Moreover, in the case of a sphere the tower*
* is finite
in vkperiodic homotopy.
Contents
0. Introduction 2
1. The poset of partitions of a finite set 5
2. The layers of the Goodwillie tower of the identity 12
3. Odd sphere case  the cohomology of the layers 14
3.1. The homology groups
3.2. Action of the Steenrod algebra
4. The vkperiodic homotopy of the tower 30
4.1. The case of an odddimensional sphere
4.2. The case of an evendimensional sphere
Appendix A. Background on vkperiodic homotopy 36
References 39
___________
1991 Mathematics Subject Classification. Primary 05A18, 55P47, 55Q40, 55S12; *
*Secondary
55P42, 55R12, 55S10.
Key words and phrases. poset of partitions, DyerLashof operations, periodic *
*homotopy, Good
willie calculus.
The first author was partially supported by the Alexander von Humboldt founda*
*tion.
1
2 GREG ARONE AND MARK MAHOWALD
0. Introduction
In this paper we analyze the Goodwillie tower of the identity functor evaluat*
*ed at
spheres. We find that in the case of spheres the tower exhibits a pleasant and *
*sur
prising behavior. Broadly speaking, we find two new facts that are not conseque*
*nces
of the general theory of calculus. First, in the case of an odddimensional sp*
*here
localized at a prime p, the only "layers" (homotopy fibers) in the Goodwillie t*
*ower
of the identity that are not contractible are the ones that are indexed by powe*
*rs of p.
Thus the tower "converges exponentially faster" in this case than it does in ge*
*neral.
Second, the stable cohomology of the pkth layer is free over A[k  1], where A*
*[k  1]
is a certain finite subHopf algebra of the Steenrod algebra (to be defined in *
*section
3.2). This implies, in particular, that in our case all the layers beyond the *
*pkth
one are trivial in vkperiodic homotopy (for any reasonable definition of the l*
*atter).
Thus, in vkperiodic homotopy the tower has only k + 1 nontrivial layers, name*
*ly
p0; p1; : :;:pk.
The two facts imply that the unstable vkperiodic homotopy of of an odd dimen
sional sphere can be resolved into a tower of fibrations with k + 1 stages, wit*
*h infinite
loop spaces as fibers. As indicated above, the fibers are analyzed here to a co*
*nsider
able extent. For instance, their stable cohomology is completely calculated.
In the body of the paper we will assume basic familiarity with the notion of *
*"vk
periodic" homotopy of spaces and spectra. For an informal discussion of the con*
*cept,
together with references to a more complete discussion, see appendix A. We will*
* also
assume familiarity with the basic ideas of Goodwillie's "Calculus of Functors".*
* The
basic references for this material are [G90 , G92 , G3 ].
We now proceed with a more detailed overview of the paper, its genesis and its
goals. The simplest example of periodic homotopy is v0periodic homotopy, which
is essentially the same as rational homotopy. There is an old theorem of Serre*
* on
rational homotopy of spheres, which implies that if X is an odddimensional sph*
*ere,
then the map X ! 1 1 X induces an equivalence in v0periodic homotopy. Thus
in the v0periodic world the unstable homotopy of an odd sphere is the same as *
*its
stable homotopy.
In [MT92 ] Mahowald and Thompson found an analogue of Serre's theorem for v1
periodic homotopy. Roughly speaking, v1periodic homotopy is the homotopy theory
one obtains by inverting the maps that induce an isomorphism in Ktheory. For a
based topological space X, let P2(X) be the homotopy fiber of the wellknown na*
*tural
map 1 1 (X) ! 1 1 (X ^ X ^2 E2 +) which may be defined, at least up to
homotopy, as the adjoint of the composed map
1_
1 1 1 (X) '! 1 (X^i^i Ei + ) ! 1 (X ^ X ^2 E2 +)
i=1
where the first map is given by the Snaith splitting and the second map is coll*
*apsing
THE GOODWILLIE TOWER OF THE IDENTITY 3
on the factor corresponding to i = 2. MahowaldThompson's work implies that if
X is an odd sphere (localized at 2) then the natural map X ! P2(X) induces an
equivalence in v1periodic homotopy. Using this result, the v1periodic homotop*
*y of
spheres has actually been computed in [M82 ] at the prime 2 and in [T90 ] at odd
primes. From this information, it is also possible to recover the "integral" v1*
*periodic
homotopy. This is done in several places [MT92 , T90 ].
From one point of view, our goal here is to extend the work of Mahowald and
Thompson cited above to higher order periodicity. The first technical difficult*
*y with
it seemed to be that this work used the existence of maps connected with the Sn*
*aith
splitting. Such maps are constructed by means of configuration space methods. T*
*he
homotopy fibers of these maps do not have nice configuration space models and t*
*hus
do not allow new maps to be constructed in the same way. Instead, we use the
Goodwillie tower of the identity functor, which turned out to be the perfect to*
*ol for
attacking this problem. We will analyze this tower of fibrations in the general*
* case
to some extent and apply this understanding to spheres.
The Goodwillie tower of the identity ("the Taylor tower of the identity" in G*
*ood
willie's terminology) is a sequence of functors (from pointed spaces to pointed*
* spaces)
Pn(X) and a tower of natural transformations
..
.



?
X _________Pn(X) _______Dn(X)oe
Q Q
QQ fn
QQs ?
Pn1(X) _____Dn1(X)oe

fn1

?
..
.



?
P1(X) ' Q(X) := 1 1 (X)
The functor Dn is the homotopy fiber of the natural transformation Pn ! Pn1 and
should be thought of as the nth homogeneous layer, or the nth differential of*
* the
identity. It follows from the general theory of calculus [G3 ] that for every *
*n there
exists a spectrum Cn, endowed with an action of the symmetric group n, such that
Dn(X) ' 1 ((C n^ X^n)hn ) := 1 ((C n^ X^n ^ En +)n ) :
4 GREG ARONE AND MARK MAHOWALD
Here, as well as everywhere else in the paper, ' stands for "weakly homotopy eq*
*uiv
alent". The spectrum Cn, considered as a spectrum with an action of n, is the n*
*th
derivative of the identity.
We need to investigate this tower, whose existence derives from the general t*
*heory.
Some information about it had been available before. As indicated in the diagr*
*am
above, it is immediate from the definitions that P1(X) ' Q(X), i.e., the linear*
* part
of homotopy theory is stable homotopy theory. The description of the second sta*
*ge
is still rather classical: as was indicated above, the second quadratic approxi*
*mation,
P2(X), is the homotopy fiber of the "stable JamesHopf" map Q(X) ! Q(X^2h2).
The second layer of the tower is D2(X) ' Q(X^2h2).
For a general n, B. Johnson was the first one to provide an explicit closed d*
*e
scription of Dn(X) in terms of standard constructions of homotopy theory. In [J*
*o95]
certain spaces n are constructed, which have the following properties:
(i) the group n acts on n,
(n1)!_
(ii) nonequivariantly, n ' Sn1,
i=1
(iii) the nth derivative of the identity is Map *(n; 1 S0), the SpanierWhiteh*
*ead
dual of n, considered as a spectrum with an action of n. Equivalently,
Dn(X) ' 1 (Map *(n; 1 X^n)hn ) :
The description of the space n is a geometric one, it is defined as a quotient *
*of the
n(n  1)dimensional unit cube by a certain subcomplex. In section 1 we reformu*
*late
the description of n. Thus we construct a certain combinatorially defined compl*
*ex
Kn. Kn has a natural action of n, and we show that for our purposes the suspens*
*ion
of Kn is equivalent to n. Thus we may write
Dn(X) ' 1 (Map *(SKn; 1 X^n)hn ) :
By the spectral sequence for the homology of Borel construction, the stable h*
*o
mology of Dn(X), i.e. the homology of the spectrum Map *(SKn; 1 X^n)hn is
essentially given by the homology of the symmetric group with coefficients in t*
*he
homology module of the (dual of) Kn tensored with the homology of X^n. Thus,
the simpler the homology of X^n is, the simpler one may expect the layers to be.
This, of course, suggests the spheres as candidates for investigation. In the c*
*ase of
an evendimensional sphere, one is led to investigating H *(n; H*(fKn)), where *
*fKn
is the dual of Kn. In the case of an odddimensional sphere, one is led to stu*
*dy
H*(n; H*(fKn) Z[1]), where Z[1] is the sign representation. Not surprisingl*
*y,
odddimensional spheres turn out to be the more basic case. In section 3, we ca*
*rry
out the homology calculations for the odd sphere case. The following theorem su*
*m
marizes some of the results in section 3.
THE GOODWILLIE TOWER OF THE IDENTITY 5
Theorem 0.1. Let X be an odddimensional sphere. If n is not a power of a prim*
*e,
then
Dn(X) ' *:
If n = pk, then Dn(X) has only pprimary torsion.
For a spectrum E, let Hs*(1 E) =iH*(E) be the stablejhomology of E. In sectio*
*n 3
we write an explicit basis for Hs* Dpk(S2s+1); Z=pZ and investigate the action*
* of the
i j
Steenrod algebra on the stable cohomology H *sDpk(S2s+1); Z=pZ . We prove that
the stable cohomology of Dpk(S2s+1) is A[k  1] free, where A[k  1] is a certa*
*in finite
subalgebra of the Steenrod algebra. In section 4 we feed this result into the v*
*anishing
line theorems of AndersonDavis [AD73 ] and MillerWilkerson [MW81 ] to conclu*
*de
that the vk1 periodic homotopy of Dpj(S2s+1) is zero for j k and moreover tha*
*t the
Goodwillie tower converges in vk periodic homotopy. This implies the main theor*
*em
of the paper which is the following:
Theorem 4.1 Let X be an odddimensional sphere localized at a prime p. The map
X ! Ppk(X)
is a vjperiodic equivalence for all k 0 and for all 0 j k.
In the last subsection we formulate and prove the analogue of theorem 4.1 for*
* even
dimensional spheres. Basically, the tower is still finite, but it is "twice as *
*long".
Acknowledgements. The first named author would like to offer a special than*
*ks
to T. Kashiwabara for many helpful conversations on this material. We also are
grateful to N. Kuhn for many indepth comments on some of the early versions of
the manuscript.
1. The poset of partitions of a finite set
Let n be an integer, n > 1. Let n_= {1; : :;:n}. A partition of n_is an equi*
*valence
relation on n_(similarly, one defines partitions of an arbitrary finite set). P*
*artitions are
ordered by refinements, and may be considered as a category. Let kn be the cate*
*gory
of partitions of n_. Thus, for two partitions 1, 2, there is a morphism 1 ! 2 i*
*ff
1 is a refinement of 2. It is clear that kn has an initial and a final object. *
*Denote
these ^0and ^1respectively. Let N okn be the simplicial nerve of kn. Since kn h*
*as an
initial and a final object, the geometric realization of N okn is contractible.*
* Let Kn
denote the subcomplex of the realization of N okn, whose simplices are those wh*
*ich
do not contain the morphism ^0! ^1as a face. Thus the zerosimplices of Kn are
partitions of n_and isimplices are increasing chains of partitions
(^0= 1 0 < 1 < . .<.i i+1= ^1)
such that not both inequalities ^0 0 and i ^1are equalities. Let eknbe the full
subcategory obtained from kn by deleting ^0and ^1. Let No fknbe the simplicial *
*nerve
6 GREG ARONE AND MARK MAHOWALD
of eknand let fKnbe its realization. It is easy to see that Kn is homeomorphic *
*to the
unreduced suspension of fKn. Equivalently, there is a cofibration sequence
gKn+! S0 ! Kn:
Here by cofibration sequence we mean that Kn is homeomorphic to the homotopy
cofiber of the map gKn+! S0. The + subscript stands for an added disjoint basep*
*oint.
For a partition let r() be the number of its components. Let S = (S0; S1; : *
*:;:Si)
be a sequence of integers such that n S0 S1 . . .Si 1, let KSn Nikn be
defined as follows
KSn= {(^0 0 . . .i ^1) 2 Nikn  r(j) = Sj forj = 0; : :;:i}
Let Kinbe the set of nondegenerate isimplices of fkn. Thus
G
N ikn = KSn:
nS0S1:::Si1
G
N ifkn= KSn:
n>S0S1:::Si>1
G
Kin= KSn:
n>S0>S1>:::>Si>1
Notice that if i > n  3 then Kin= ;. Therefore, fKn is n  3dimensional. Noti*
*ce
also that KSnis defined even if S is empty, and therefore the sets N1 kn and K*
*1nare
defined and have one element each. By our convention, N 2kn = K2n= ;.
Definition 1.1.Tn is the following based simplicial set: The set of isimplices*
*, Tni, is
Tni = Ni2kn+ 8i 0:
in particular, Tn0= ;+ = * and Tn1= S0. The face maps are defined as follows:
If 0 < j < i then dj : Tni! Tni1is given by:
dj(0; : :;:i2) = (0; : :;:^j1; : :;:i2):
For j = 0; i the formulas are
(
^0
d0(0; : :;:i2) = (1; : :;:i2) if0 =
* otherwise
(
^1
di(0; : :;:i2) = (0; : :;:i3) ifi2=
* otherwise
The degeneracy maps are defined similarly. If 0 j i then sj : Tni ! Tni+1 is
determined by
sj(^0= 1; 0; : :;:i2; i1= ^1) = (^0= 1; 0; : :;:j1; j1: :;:i2; i1= ^1):
THE GOODWILLIE TOWER OF THE IDENTITY 7
It is easy to check that Tn is indeed a simplicial set and that its realizati*
*on is
SKn = SfKn, where S and denote reduced and unreduced suspension respectively.
(This is, essentially, Milnor's suspension construction [Mi72 , page 120], appl*
*ied to gKn
twice).
The symmetric group n acts on kn, and therefore on KSn, Kin, Kn etc. The acti*
*on
of n on KSnis not, in general, transitive. We need to write KSnas a union of n
orbits. The orbits of zerosimplices (partitions of n_) are, simply, partitions*
* of positive
integers. A partition P of a positivePinteger n is a collection n1; : :;:nk of*
* positive
integers such that n1 . . .nk and ni = n. We call {ni} the components of P .
Such a partition of n is not trivial if 1 < k < n. We denote the set of partiti*
*ons of n
by Q(n).
Proposition 1.2. The quotient set K0n=n is naturally isomorphic to the set of n*
*on
trivial partitions of n.
Proof.Every partition of n_induces a partition P of the integer n: the compone*
*nts
of P are cardinalities of the components of . It is elementary to show that th*
*is
assignment is surjective and that 1 and 2 induce the same partition of n if and
only if they are in the same orbit of n. ___ 
IfPP is as above, we will call P the type of . We will sometimes use formal su*
*ms
lnil. il_to describePpartitions of integers. A formal sum as above stands for*
* a
partition of n = lnilil with nilcomponents of cardinality il.
P
Proposition 1.3. Let be a partition of type P = lnil. il_. The set of partit*
*ions
of type P is nequivariantly isomorphic to the set of cosets n= , where is the
stabilizer group of . There is an isomorphism
Y
~= nilo il:
l
Proof.Easy. ___ 
We need to classify orbits of Kin, i > 0, in a manner similar to the one we h*
*ave for
the orbits of K0n. It is sometimes convenient to represent orbits with certain *
*labeled
trees. A tree will always have a root r. Distance will mean the number of edges*
* in
the unique path between two nodes. Let v be a node.
Definition 1.4.A tree is balanced if all its leaves have the same distance from*
* the
root.
Given a tree, we define a height function on its nodes, which we denote h(v),*
* by
letting h(v) be the minimal distance from v to a leaf. Define the height of a t*
*ree to
be h(r)  1.
Definition 1.5.A tree is labeled if to every node v there is assigned a positiv*
*e integer
l(v).
8 GREG ARONE AND MARK MAHOWALD
We will make free use of such expressions as sibling nodes, a single child, t*
*he
subtree spanned by a node, etc. We say that a balanced tree has no forking on l*
*evel
j if all the nodes of height j have only one child. We say that two labeled tre*
*es T
and T 0are isomorphic as labeled trees (or just isomorphic) if there is an isom*
*orphism
of unlabeled trees : T ! T 0such that for any node v of T except possibly the*
* root,
l(v) = l( (v)).
Definition 1.6.A labeled tree is standard if
1) it is balanced,
2) l(r) = 1,
3) no two sibling nodes span isomorphic labeled subtrees.
Condition (3) implies that every node of height 1 has exactly one child. In o*
*ther
words, in a standard tree there is no forking on level 1.
Given a standard tree, we define the degree function of its nodes as follows:*
* If
h(v) = 0 then deg(v) = l(v). If h(v) > 0, let u1; : :;:uk be the children of v*
*, then
deg(v) = l(v)(deg(u1) + . .+.deg(uk)). The degree of a tree is the degree of it*
*s root.
Proposition 1.7. There is a bijective correspondence between orbits of Kinand s*
*tan
dard labeled trees of height i + 1 and degree n.
XL
Proof.For i = 0, let P = njl.jl_be an orbit. Then P is represented by the fol*
*lowing
l=1
tree:
1
R
nj1 . . .njL
? ?
j1 jL
For i > 0, the assignment of trees to orbits is constructed inductively. But be*
*fore we
describe it, we need some more definitions. For a finite set S_, let kS_be the *
*category
of partitions of S_. Let S_1and S_2be two finite sets. Let 1 = (10 . . .1i) and
2 = (20 . . .2i) be isimplices of N okS_1and N okS_2respectively. A morphism
ae : 1 ! 2 is a map of sets S_1 ! S_2 which for every 0 j i maps every
component of 1jinto a component of 2j. ae is an isomorphism if it has a twosid*
*ed
inverse. 1 and 2 are isomorphic if there exists an isomorphism 1 ! 2.
Definition 1.8.Let = (0 . . . j) be a chain of partitions. Let S_ be a
component of j. Then 0; : :;:j1 determine a (j  1)simplex of No kS_. We call*
* it
the restriction of to S_and denote it S_.
Let = (0 . . .i) be an isimplex of No kn. Thus iis the coarsest partition
in the chain. Let S1_, S2_be two components of i. We say that S1_and S2_induce
THE GOODWILLIE TOWER OF THE IDENTITY 9
isomorphic blocks if S1_and S2_are isomorphic. Of course, a necessary conditi*
*on
for S1_and S2_to induce isomorphic blocks is that S1_and S2_are isomorphic sets.
The property of inducing isomorphic blocks defines an equivalence relation on t*
*he
components of i. We consider S1_as an element of N i1KS1_and the orbit of S1_
under the action of S1 as an element of (Ni1KS1 )S1. Two isimplices 1 and 2
are in the same orbit of n if and only if they have the same isomorphism classes
of blocks, counting with multiplicities. Thus, every orbit B of Kincan be writ*
*ten
uniquely as a formal sum X
B = nl. Bl_
i j l
where Bl_are elements of Ki1kl for some k1; : :;:km such that
kl
X
nlkl= n
l
and B1_; B2_; : :;:Bm__are pairwise distinct. Assume by induction that we have *
*assigned
to Blpairwise nonisomorphic standard labeled trees Tlof height i. Now for ever*
*y Tl
replace the label 1 at the root with nland join all roots to a common new root.*
* Thus
we have constructed a standard tree of height i + 1, which is the tree assigned*
* to B.
It's easy to check that the construction is welldefined, i.e., that two isimp*
*lices are
in the same orbit if and only if the above procedure associates to them isomorp*
*hic
trees. ___ 
To describe the orbit of a given isimplex of type B as above, we notice, as w*
*e did in
the case of 0simplices, that the stabilizer group of has, up to an isomorph*
*ism,
the following form:
~=(n1o B1) x (n2o B2) x . .x.(nm o Bm )
where Bl are stabilizers of representatives of Bl. Notice that all groups in s*
*ight
are naturally subgroups of n, and the set of partitions of type B can be identi*
*fied
equivariantly with the cosets n= . The stabilizer groups of two representatives*
* of
a given orbit are conjugate. In the course of the paper we will sometimes conf*
*use
between the set of orbits and a set of arbitrarily chosen representatives of or*
*bits.
The group is isomorphic to a semidirect product
i j
~=(n1x n2x . .x.nm ) n xn1B1x xn2B2x . .x.xnmBm
where there is an obvious action of n1x. .x.nm on xn1B1x. .x.xnmBm. Inductively,
one may write in the form
Gi+1n (Gin (. .n.G0))
where each Glis a product of symmetric groups and there is a "wreath product ty*
*pe"
action of Gl on Gl1n (Gl2n . .n.G0) . As a matter of fact, Gl is isomorphic *
*to
10 GREG ARONE AND MARK MAHOWALD
the product of powers of symmetric groups indexed by nodes on level l in the tr*
*ee
corresponding to the type B of . The size of each symmetric group is given by t*
*he
corresponding label and the power to which it is raised is given by the label o*
*f the
father node.
From here until the end of the section, let p be a fixed prime number.
Definition 1.9.Let = (^0= 1 0 . . .j) be a chain of partitions. We
say that j is a pcoarsening of ^0= 1 0 . . .j1 if for every component S_
of j the following holds:
1) The number of components of j1 contained in S_is a power of p.
2) Any two components of j1 contained is S_induce isomorphic blocks.
We will say that is a ppartition if it is a pcoarsening of ^0. Obviously,*
* the
property of being a ppartition is invariant under the action of n and therefor*
*e we
may speak about ppartitions of numbers. A ppartition is simply a partition wh*
*ose
components all have cardinality which is a power of p. We let Pe(n_) denote the*
* set
of ordered ppartitions of n_and P (n) denote the set of unordered ppartitions*
* of n_,
which is the same as the set of ppartitions of n. We use the following "logari*
*thmic"
notation for elements of P (n): a sequence (n0; n1; :P:):denotes the partition *
*with nj
components of cardinality pj for all j 0. Thus n = jnjpj.
Definition 1.10. Let = (0 . . . i) be an isimplex of N okn. An ordered
pramification of is a chain of partitions
(^0= 1 ffi0 0 ffi1 1 . . .i ffii+1 i+1= ^1)
such that for all j = 0; 1; : :;:i + 1, ffij is a pcoarsening of (^0= 1 ffi0*
* 0 ffi1
1 . . .j1).
Recall that we denote by the subgroup of n which stabilizes . acts on
the set of ordered pramifications of . We define an unordered pramification of
to be an orbit of an ordered pramification under the action of . It is clear *
*that
if 1 and 2 are two isimplices in the same orbit of n then the set of unordered
pramifications of 1 is isomorphic to the set of unordered pramifications of 2.
Let be an unordered pramification of . Consider as a 2i+2simplex of No kn
and consider the orbit of under the action of n. This orbit is represented by*
* a
standard tree of height 2i + 3. It follows easily from the definitions that all*
* the nodes
of even height in this tree are labeled by powers of p and that there is no for*
*king
on odd levels. It is also easy to see that the set of orbits of ordered pramif*
*ications
of under the action of n is isomorphic to the set of orbits under the action of
, which is the set of unordered pramifications of . We denote the set of orde*
*red
pramifications of by eP() and the set of unordered pramifications of by P (*
*).
Thus P () ~=Pe() . We denote by Pe(Bl)kP(Bl)the fibered product of k copies of
eP(Bl) over P (Bl). Thus a point in eP(Bl)k e
P(Bl)is a ktuple of elements of P(Bl*
*) which
THE GOODWILLIE TOWER OF THE IDENTITY 11
are all in the same orbit of . We will need inductive formulae for eP() and P *
*().
Suppose that i, the coarsest partition in , has nlblocks of type Blfor l = 1; :*
* :;:L
YL
where Bl are pairwise nonisomorphic. We denote a generic element of P (nl)by
l=1
(n01; : :;:nj11); (n02; : :;:nj22); : :;:(n0L; : :;:njLL)
Proposition 1.11. There is an isomorphism of equivariant sets
0 1
BB Q C
i j jnjCC
eP() ~= a BB________lnl________01x Y Y Pe(Bl)p lC
Y BBQ Y l j P(Bl) CC
P (nl)@ l@ njo pjA A
l j=1:::jll
Proof.Fix a pcoarsening ffii+1of . By definition, every component of ffii+1con*
*tains
a power of p of components of i+1 of type Bl for some l. Let us say that ffii+1*
* has
njlcomponents containing pj components of type Bl. A pramification of whose
coarsest partition is ffii+1is determined by a collection of pramifications of*
* the blocks
Blsuch that any two blocks which are in the same component of ffii+1have isomor*
*phic
pramifications. This set is isomorphic to
Y Y ie pj jnjl
P(Bl)P(Bl) :
l j
On the other hand, the set of pcoarsenings of which have njlcomponents contai*
*ning
pj components of type Bl is clearly isomorphic to
Q
________lnl________01
Q Y
l@ njlo pjA
j=1:::jl
The proposition follows byYtaking union over the set of types of pcoarsenings *
*which
is isomorphic to the set P (nl). ___ 
l
Corollary 1.12. There is an isomorphism of sets
0 1
a Y Y nj
P () ~= @ (P (Bl))lA
Y l j njl
P (nl)
l
Proof.Recall that
P () ~=Pe() ~=Pe()(n1oB1x...xnLoBl):
12 GREG ARONE AND MARK MAHOWALD
Applying proposition 1.11 one readily sees that
0 1
! nj
a BY Y i pj j lC
P () ~= B@ P (Bl)P(B ) CA:
Y l j l pj
P (nl) njl
l
i j j
But P (Bl)pP(Bl)~=P (Bl) where the right hand side can be considered as a set *
*with
a trivial action of pj. The corollary follows. ___ 
2. The layers of the Goodwillie tower of the identity
In this section we will describe Dn(X), the nth layer of the Goodwillie towe*
*r of
the identity in terms of the complexes Kn of the previous section. This amount*
*s,
basically, to a reformulation of the main result of Johnson in [Jo95]. In [AK9*
*7 ] a
different way to derive our description of Dn(X) is presented.
Theorem 2.1.
Dn(X) ' 1 Map *(SKn; 1 X^n)hn :
Proof.By [Jo95, corollary 2.3]
Dn(X) ' 1 Map *(n; 1 X^n)hn
where n is defined in [Jo95, definition 4.7]. We recall the definition. Let
2 n2
In = {t = (t11; t12; : :;:t1n; t21; : :;:tnn) 2 R  0 tij 1}
2
be an n2dimensional cube. Let In(n1)be the subspace of In defined by tii= 0 *
*for
i = 1; : :;:n. Thus In(n1)is an n(n  1)dimensional cube. For 1 i < j n def*
*ine
Wij= {t 2 In(n1) tik= tjk for1 k n}:
Define also
Z = {t 2 In(n1) tij= 1 for some1 i; j n}:
Then [
n = In(n1)={Z [ Wij}:
i 1. Rationally
Dn(X) = Map *(SKn; 1 X^n) hn ' *
Proof.We saw in the previous section that there exists a tower of fibrations wi*
*th n
stages converging to Dn(X), in which the fibers are of the form
Map *(SiKi2n+; 1 X^n)hn
where Ki2nis the set of nondegenerate i  2chains of partitions. Thus
_
Map *(Ki2n+; 1 X^n)hn ' 1 X^nh
THE GOODWILLIE TOWER OF THE IDENTITY 15
where the wedge sum on the right hand side in indexed by representatives of orb*
*its
of Ki2n+. Thus each stabilizer group can be written as a semidirect product
Gi1n (Gi2n (. .n.G0))
where each Gl is a product of symmetric groups. Since we consider only non
degenerate orbits, none of the Gls is trivial. In particular, G0 is not trivia*
*l. The
proposition follows since for k > 1 and X an odddimensional sphere, X^khkis ra*
*tio
nally trivial. ___ 
Notice that proposition 3.1 implies the theorem of Serre that the map X ! Q(X) *
*is
a rational equivalence for an odd sphere X.
From now on all spaces considered will be localized at a fixed prime p. All h*
*omology
groups are taken with Z=pZ coefficients. We will calculate H*(D n(X)) explicitl*
*y. The
case n = 1 is trivial. Assume, till the end of the section, that n > 1.
The plan is to use the homology spectral sequence associated with the tower of
fibrations Totin(Don(X))(as defined on page 14). To see that such a spectral se*
*quence
exists, note that since we are dealing with spectra, smashing with a fixed spec*
*trum
preserves fibration sequences and finite towers of fibrations. The homology spe*
*ctral
sequence is obtained by smashing our tower of fibrations with the EilenbergMac*
*Lane
spectrum HZ=pZ and considering the homotopy spectral sequence of the resulting
(finite) tower (see [BK72 , page 259] for a reference on the spectral sequence *
*of a tower
of fibrations). The first term of the spectral sequence has the following form:
i j i j
Ei;t1= Hti Map *(SiKi2n+; 1 X^n)hn = Ht Map *(Ki2n+; 1 X^n)hn
with a differential
d1 : Ei;t1! Ei+1;t1:
We may view this E1 term as a cochain complex Co of graded Z=pZmodules. The
module of icochains is
i j
Ci = H* Map *(Ki2n+; 1 X^n)hn :
P j o
The differential @i : Ci ! Ci+1 is given by the alternating sum j(1) dj, whe*
*re
dojis induced by the face map dj in N okn. If we write, as we did in the proof*
* of
proposition 3.1, _
Map *(Ki2n+; 1 X^n)hn ' 1 X^nh;
then for j = 0 and j = i, dijis the zero homomorphism, and for 1 j i  1,
dijis a direct sum of the transfer maps associated with the inclusion of stabil*
*izers
of representatives of orbits of chains of the form (0; : :;:i2) into stabilize*
*rs of
representatives of orbits of chains of the form (0; : :;:^j1; : :;:i2). The i*
*nclusions
are welldefined up to conjugation, and therefore the transfer maps are wellde*
*fined.
16 GREG ARONE AND MARK MAHOWALD
To study this spectral sequence we will need to study the homology of (reduce*
*d)
Borel constructions on X with respect to certain subgroups of n. We recall a few
standard facts about the homology of X^nhn= X^n ^n En + as described in terms
of DyerLashof operations. Let H* be a graded Z=pZ module. Let l(H*) be the free
graded Z=pZ module generated by allowable DyerLashof words of length l over H*
(see [CLM76 , I.2] and [BMMS86 , page 298] for details). Thus, if l > 0, the*
*n l(H*)
is generated by the following set
{fiffl1Qs1. .f.iffllQslu 
u 2 H*;Pffli2 {0; 1}; si> 0; psi ffli si1ifp > 2
2s1  li=2[2si(p  1)  ffli] u}
{Qs1. .Q.slu 
u 2 H*;P si> 0; 2si si1 ifp = 2
s1  li=2si u}
where u 2 H*. By convention, 0(H*) = H*. The operations Qs come from elements
in the mod p homology of symmetric groups. Qs raises degree by 2(p  1)s if p >*
* 2
and by s if p = 2 (the fis in the oddprimary case are homology B"ocksteins, an*
*d thus
lower the degree by 1). Qsu = 0 if s < u_2for p > 2 and u even (if s < u *
*for p = 2)
u_ p u
and Q 2u = u for p > 2 (Q u = u u for p = 2). Thus we include powers of
elements of H* in l. The convenience of this will become clear later  its purp*
*ose is
to make the "negligible" summands in the proof of lemma 3.11 negligible.
We will sometimes abbreviate l(H*) as l when H* is clear from the context.
Given a graded vector space D, let V (D) be the augmentation idealLof the free
symmetric algebra generated by D. Thus V (D) is the quotient of 1k=1Dk by the
ideal generated by the relations a b  (1)abb a where a; b are homogeneo*
*us
elements. Let E(D) be the quotient of V (D) by theLideal generated by ap . Sinc*
*e the
relations are homogeneous, we may write E(D) ~= Ek(D). By abuse of notation,
we will write Ek(D) as Dkk. From now on, whenever we write Dkk, where D is a
graded Z=pZ module, we mean Ek(D).
The mod p homology of X^nhnis described in terms of the DyerLashof operation*
*s.
For any based space X, the following is true (the homology is taken with Z=pZ
coefficients)
0 1
M O n
(1) H *(X^nhn) = @ (l(H *(X)) l )nlA
(n0:::)2P(n)l0
Because of our choice to suppress the pth powers of elements of H*(X), the s*
*plitting
in (1) is not natural, but depends on a choice of basis of H *(X). The point is*
* that
the projection map V (D) i E(D) splits, but not naturally. However, it is easy *
*to
THE GOODWILLIE TOWER OF THE IDENTITY 17
see that if we filter P (n) by the number of components, then the splitting is *
*natural
up to elements of lower filtration. Thus we get a (more or less natural) expans*
*ion of
H*(X^nhn) into a direct sum indexed by ppartitions of n. We will call the term*
*s in
the expansion the standard summands (or just summands) of H *(X^nhn).
It is well known that the standard summands are detected by certain elementary
Abelian subgroups of n. We proceed to recall the basic facts about this. Let us*
* begin
k
with k as a summand of H *(pk), or more generally of H *(X^ppk). For k = 0; 1; *
*: : :
let Ak ~=(Z=p)k. Ak = pk, therefore the action of Ak on itself defines an inc*
*lusion
(up to conjugation) of Ak ,! pk. We will consider Ak as a subgroup of pk via th*
*is
inclusion (this is the subgroup that is defined as k in [KaP78 , page 95]). Thu*
*s Ak
acts transitively on pk_. Ak is very useful for detecting elements in the homol*
*ogy of
pk: the pure part of H *(pk) is detected on Ak. This was probably first proved *
*by
Kahn and Priddy in [KaP78 ]. For a more detailed account we recommend [AdMi95 *
*].
We need a slightly more general version of this:
Proposition 3.2. Let X be a based space. Let H* = H *(X). Recall that k is a
k ^pk
summand of H *(X^phpk). Write H *(Xhpk ) ~=k A. Consider the homomorphism
k
H*(X ^ BAk +) ! H*(X^phpk)
induced by inclusion of subgroups and the diagonal map X ! X^n. This map is
onto the summand k and zero on the summand A.
Proof.For X = S0 (a zerodimensional sphere) this is precisely [KaP78 , Proposi*
*tion
3.4]. The proof generalizes straightforwardly. The idea is to reduce the questi*
*on from
pk to its pSylow subgroup and then proceed by direct calculation. ___ 
Now consider a summand on the right hand side of (1) corresponding to a p
partition (no: : :; nl; : :):2 P (n).QThis summand is detected, in a suitable s*
*ense, by
the elementary Abelian group A = lAxnll. Consider the space X^lnl as a space *
*with
a trivial action of A. It is easy to see that there is a diagonal map X^lnl ! X*
*^n
that is equivariant with respect to the subgroup inclusion A ! n. We have the
following proposition
Proposition 3.3. With notation as above, consider the homomorphism
H *(X^lnl ^ BA+ ) ! H*(X^nhn)
induced by group inclusion A ! n and the diagonal map X^lnl ! X^n. This ho
momorphism is onto the summand corresponding to to the ppartition (no: : :; nl*
*; : :):
and zero on the other summands of the same filtration and the summands of higher
filtration.
18 GREG ARONE AND MARK MAHOWALD
Proof.The case X = S0 is wellknown. It is largely proved in [KaP78 ] and in mo*
*re
detail in [AdMi95 ]. It is a longish, but straightforward, exercise to extend *
*the result
to a general X. ___ 
The reason that we need proposition 3.3 is that we have to study the transfer*
* map
in the homology of Borel construction. Let n0p0_+ n1p1_+ . .n.kpk_be a pparti*
*tion
of n. Let P = n0 o p0x . .x.nk o pk and 0P= n0p0x . .x.nkpk. Let X
be any based space. The homology groups H *(X^nhn), H *(X^nhP) and H *(X^nh0P) *
*each
have a summand isomorphic to n00 n0 . . .nkknk which we denote simply .
Write H*(X^nhn) = A, H*(X^nhP) = B and H*(X^nh0P) = B0. Consider the
homomorphisms A ! B and A ! B0 induced by the appropriate
transfers. These homomorphisms can be represented as two by two matrices of maps
! !
! ! B ! ! B0
A ! A ! B and A ! A ! B0
Proposition 3.4. The map ! in both matrices is an isomorphism.
Proof.The proof is similar to the proof of the main theorem of [KaP78 ]. To pro*
*ve
that the homomorphism ! is an isomorphism it is enough to prove that it is
surjective. To do this for the case of the first matrix, it is enough to show t*
*hat the
composite homomorphism
H*(X^lnl ^ BAn00x . .x.Ankk+) i*!H*(X^nhn) tr!H*(X^nhP)
is surjective onto . This composed homomorphism can be analyzed by means of a
suitable version of the double coset formula. It is not hard to show that the c*
*omposed
map above is the same as the homomorphism induced by the group inclusion An00x
. .x.Ankk! P , essentially because of two reasons: the normalizer of An00x . .x*
*.Ankk
in n is the same as in P and the transfer from an elementary Abelian group to
a proper subgroup is zero (see [KaP78 ]). The argument for the second matrix is
similar. ___ 
We will need to consider a slightly more general situation. Let n = i0p0+i1p1+.*
* .i.kpk
as before. Let K1; K2; : :;:Kj be disjointPsubsets of {0; 1; : :;:k} whose uni*
*on is
{0; 1; : :;:k}. for 1 l j, let ml= t2Klitpt. Consider the group m1 x . .m.j*
*as
a subgroup of n. It is easy to see that is a summand of H *(X^nhm1x...mj). Wri*
*te
H*(X^nhm1x...mj) = C and consider the matrix
!
! ! C
A ! A ! C
describing the transfer map. We have the following proposition
Proposition 3.5. In the matrix above the map ! is an isomorphism.
THE GOODWILLIE TOWER OF THE IDENTITY 19
Proof.Similar to the proof of the previous proposition. ___ 
Next we need to generalize the formula (1) and proposition 3.3 to H*(X^nh), w*
*here
is the stabilizer of a chain of partitions of n. More precisely, we will show*
* that
H*(X^nh) splits as a certain direct sum indexed by unordered pramifications of*
* .
We first show how to associate a graded Z=pZvector space to a pramification o*
*f .
Indeed, let = (^0= 1 0 . . .i i+1 = ^1) be an ichain of partitions
of n_and let ffl be an unordered pramification of . Recall that we associate *
*with
ffl a standard tree of height 2i + 3 in which all the nodes of even height are *
*labeled
by powers of p and there is no forking on odd levels. Given such a tree, a nod*
*e v
in the tree and a graded Z=pZvector space H* = H *(X), we construct a graded
Z=pZvector space Hv*and, for future use, a detecting elementary abelian group *
*Av
as follows: if the height of v is zero then it is labeled by pk for some k (sin*
*ce 0 is
even) and we define Hv*= k(H*) and Av = Ak. Assume now that we defined Hv*
and Av for all v of height j  1 or less. Let v be a node of height j. Let l be*
* the label
of v. Assume, first, that j is even. Then l = pk for some k. Let u1; : :;:um be*
* the
children of v. We define
Hv*= k(Hu1* . . .Hum*)
and
Av = Ak x (Au1x . .x.Auk)
(Av should be thought of as the diagonal subgroup of pko (Au1x . .x.Auk)). Now
assume j is odd. Then v has only one child u and we define
Hv*= (Hu*)ll
and
Av = (Au)xl:
Let Hffl*be the module associated with the root of the tree and similarly let A*
*fflbe
the elementary abelian group associated with the root of the tree. Afflis in f*
*act a
subgroup of n (determined up to conjugation).
Lemma 3.6. Let = (^0= 1 0 . . . i i+1 = ^1) be an ichain of
partitions of n_. Recall that P () is the set of unordered pramifications of .*
* There
is an isomorphism M
H*(X^nh) ~= Hffl*:
ffl2P()
Proof.We will prove it by induction on i. The induction starts with i = 1. In *
*this
case = (^0; ^1) and the lemma is given precisely by (1) and proposition 3.3. A*
*ssume
the lemma holds for i  1. Let = (^0= 1 0 . . .i i+1 = ^1) be an
ichain of partitions. Consider i, the coarsest partition in the chain. The rel*
*ation
of inducing isomorphic blocks is an equivalence relation on the components of i.
Let i have nl components of type Bl, where the Bls are pairwise distinct orbits*
* of
20 GREG ARONE AND MARK MAHOWALD
i  1 chains of partitions of a set with kl elements and l = 1; : :;:L. Thus, *
*in the
tree corresponding to the orbit of , the root has L children labeled n1; : :;:n*
*L. The
stabilizer group of has the following form
~=n1o B1 x . .x.nL o BL:
Thus 0
0 1 1
OL M JlOi i jjnl
H *(X^nh) = B@ @ j H *(X^klhB) j A CA
l=1 (nl0;nl1;:::;nlJl)2P(nl)j=0 l nlj
which implies
0 0 11
M LO OJli i ^k jj nlj
H*(X^nh) = @ @ j H *(XhBll) AA :
YL l=1 j=0 nlj
P (nl)
l=1
YL
We see that H *(X^nh) splits as a direct sum indexed by P (nl). By the induc*
*tion
l=1
assumption, M
H*(X^klhBl) ~= Hffl*
ffl2P(Bl)
Obviously, if H1*and H2*are two graded modules then l(H1* H2*) ~= l(H1*)
l(H2*), therefore
M
jH *(X^klhBl) ~= jHffl*
ffl2P(Bl)
Thus 0 0 1 1
0 1 nl
M BBLO BBJlO M j CCCC
H*(X^nh) = B@ B@ @ jHffl*A CACA:
YL l=1 j=0 ffl2P(Bl) l
P (nl) nj
l=1
By multiplying out, we see that H*(X^nh) splits as a direct sum indexed by the *
*set
0 1
a Y Y nj
@ (P (Bl)) lA
Y l j njl
P (nl)
l
which, by corollary 1.12 is isomorphic to P (). It is tedious, but entirely str*
*aight
forward to verify that the summand corresponding to ffl 2 P () is indeed Hffl*.*
* ___ 
THE GOODWILLIE TOWER OF THE IDENTITY 21
Remark 3.7. Recall that given an unordered pramification ffl of we constructe*
*d an
elementary abelian group Affl(just before lemma 3.6). It is not hard to show th*
*at Affl
detects the summand Hffl*of the homology of X^nh in the sense of proposition 3.*
*3. If
ffl1 and ffl2 are different pramifications of then Affl1and Affl2are nonconj*
*ugate in .
The map on the homology of Borel constructions induced by the inclusion (defined
up to conjugation) Affl,! n is nonzero only on the summand Hffl*and summands
corresponding to elementary Abelian groups with strictly fewer components (the
number of components of a subgroup G of n is the number of components of the
induced partition of n).
Definition 3.8.An integer n is pure if n = pk for some integer k. If n is pure,*
* an
ichain of partitions of n_, = (^0= 1 0 . . .i i+1= ^1), is pure if j is
a pcoarsening of 1 0 . . .j1 for all j = 0; : :;:i + 1.
Clearly, purity is preserved by the action of n, hence we may speak about pure
orbits of chains of partitions. It is easy to see that an orbit is pure iff the*
* corresponding
tree has one branch and has all labels powers of p. Given a pure , the correspo*
*nding
stabilizer group has the form ~= pk0o pk1o . .o.pkifor some (k0; : :;:ki) such
P
that kj = k. Also, consider the chain of partitions (^0= 1 ffi0 0 . . .
ffii i ffii+1 i+1= ^1) in which ffij = j for all j = 0; : :;:i + 1. Since is p*
*ure,
it is easy to see from definition that it is a pramification of . The correspo*
*nding
summand of H*(X^nh) is of the form k0(k1: :(:ki)) . We call it the pure summand
associated with . All other summands are impure.
Let Ci be the graded Z=pZmodule of icochains in the complex Co defined abov*
*e.
Cican be written as a direct sum Ci ~=P iIiwhere P iand Iiare the pure and impu*
*re
summands of Ci (P iis often trivial). The coboundary map @i : P iIi ! P i+1Ii+1
can be represented by a matrix of matrices as follows
!
P i! P i+1 P i! Ii+1
Ii ! P i+1 Ii ! Ii+1 :
Proposition 3.9. P i! Ii+1 are zero matrices for all i
Proof.It is not hard to show (similar to proposition 3.2) that the pure summands
are detected by the elementary Abelian group Ak (the transitive elementary Abel*
*ian
subgroup of pk). More precisely, the homomorphism
k
H*(X ^ BAk +) ! H*(X^ph o o...o)
pk0 pk1 pki
is onto the pure summand and zero on the impure summands. The proposition
follows, using the double coset formula. ___ 
22 GREG ARONE AND MARK MAHOWALD
Corollary 3.10. The pure summands span a subcomplex of Co, which we denote
P o. P ois nontrivial only if n is a power of p. There is a short exact sequ*
*ence of
cochain complexes
0 ! P o! Co ! Io ! 0
where Io is the complex of impure summands.
The following lemma is important:
Lemma 3.11. Io is acyclic.
Proof.We will use the following evident proposition
Proposition 3.12. Let
C10 C20! C11 C21! C12 C22! . .!.C1j C2j! . . .
be a cochain complex of graded Z=pZvector spaces, where C10is the trivial modu*
*le.
Suppose that for all j 0 the differential C1jC2j! C1j+1C2j+1is given by a matr*
*ix
!
C1j! C1j+1 C1j! C2j+1
C2j! C1j+1 C2j! C2j+1
where C2j! C1j+1is an isomorphism. Then the complex is acyclic.
Consider now the cochain complex Io. Obviously, I0 is the trivial module (sin*
*ce
n > 1). For j 1 we will write Ij as a direct sum of two modules Ij1 Ij2, which
we nowMproceed to define. Recall that Ij is the direct sum of the impure summan*
*ds
of H *(X^nh) where ranges through a set of representatives of orbits of non
degenerate (j  2)chains of partitions (^0= 1 < 0 < . . .< j2 < j1 =
^1). Moreover by lemma 3.6 we know that given = (^0= 1 < 0 < . . .<
j2 < j1 = ^1), the impure summands of H *(X^nh) are indexed by unordered
pramifications
ffl = (^0= 1 ffi0 0 < . . .ffij2 j2 ffij1 j1 = ^1)
such that not for all i ffii = i. We say that an unordered pramification ffl *
*of is
admissible if there exists 0 l j  2 such that ffim = m for all 0 m l and
m = ffim+1 . If ffl is not admissible than we say it is unadmissible. Let Pa*
*() and
Pu() be the set of admissible, impure and unadmissible, impure pramifications *
*of
. We define 0 1
M M
Ij1= @ Hffl*A
2(Kj2n)n ffl2Pa()
THE GOODWILLIE TOWER OF THE IDENTITY 23
and 0 1
M M
Ij2= @ Hffl*A:
2(Kj2n)n ffl2Pu()
By lemma 3.6, Ij ~=Ij1 Ij2for all j. Clearly, I11is the trivial module. It rema*
*ins to
prove that the map Ij2! Ij+11induced by the coboundary map in Io is an isomorph*
*ism
for all j. Then the lemma will follow from proposition 3.12.
First we establish that Ij2and Ij+11are abstractly isomorphic. Let
= (^0= 1 < 0 < . .<.j2 < j1 = ^1)
be a (j  2)chain and let
ffl = (^0= 1 ffi0 0 < . . .ffij2 j2 ffij1 j1 = ^1)
be an unadmissible pramification of . Let l be the smallest index such that ff*
*il+16=
l+1. Such an l exists because otherwise ffl would be pure. Call l the level o*
*f ffl.
Suppose first that l = 1. We claim that if 1 = ffi0 then Hffl*is the trivial *
*module.
Indeed, in this case it is easy to see that the tree corresponding to ffl has a*
*ll the nodes
on level 0 labeled by 1, but not all the nodes on level 1 labeled by 1, since *
*1 6= 0.
It follows that Hffl*has a tensor factor of the form (0(H*))kk, where k > 1. Bu*
*t it
is easy to see that (0(H*))kk is the trivial module if H* has exactly one gener*
*ator
of odd degree, which it does if X is an odddimensional sphere. Thus if l = 1 *
*and
1 = ffl0 we say that ffl is a negligible pramification and Hffl*is a negligib*
*le summand of
I2j. We denote the set of nonnegligible unadmissible impure pramifications of*
* by
Pun(). We proceed to establish an isomorphism between the sum of nonnegligible
summands of I2jand I1j+1. We may assume now that l6= ffil+1because otherwise ff*
*l is
either negligible (if l = 1) or admissible (if l > 1). It follows that the se*
*quence
0= (^0= 1 < 0 < . .<.l< ffil+1< l+1< . .<.j2 < j1 = ^1)
is a nondegenerate j  1chain of partitions. It is obvious by inspection that
ffl0= (^0= 1; ffi0; 0; ffi1; : :;:ffil; l; ffil+1; ffil+1; ffil+1; l+1; : :*
*;:j2; ffij1; j1 = ^1)
0 1
is an admissible, impure pramification of 0and thus Hffl*is a summand of Ij+1.*
* It is
0
also obvious by inspection that Hffl*is isomorphic to Hffl*and that the above p*
*rocedure
establishes an abstract isomorphism between the sum of nonnegligible summands *
*of
Ij2and Ij+11. It remains to prove that the coboundary homomorphism of Io induces
an isomorphism between the two. We now may write this map as follows:
0 1 0 1
M M M M 0
@ Hffl*A! @ Hffl*A
2(Kj2n)n ffl2Pnu() 2(Kj2n)n ffl2Pnu()
24 GREG ARONE AND MARK MAHOWALD
where ffl0is obtained from ffl by the procedure0described above. This map can b*
*e de
scribed as a matrix M of maps Hffl1*! Hffl2*. To show that this map is an isomo*
*rphism
it is enough to show that the matrix is block upper triangular with respect to *
*a cer
tain ordering of the indexing set and that all the diagonal blocks are isomorph*
*isms.
To show that all the diagonal blocks are isomorphisms we need to show0that for *
*any
(j  2)chain as above and for any ffl 2 Pun() the map Hffl*! Hffl*, induced b*
*y the
transfer map from to 0 , is an isomorphism. We may write
~=Gj1n (. .G.l+1n (Gln (. .G.0)))
where all Gi are products of symmetric groups. Now consider 0 . It is not diffi*
*cult
to see that for i = l + 1; : :;:j  1
i j
0 ~= G0j1n . .G.00l+1n G0l+1n (Gln (. .G.0))
Q 0
where G00l+1n G0l+1is a subgroup of Gl+1of the form imi o piand Giis a subgro*
*up
0
of Gi of the form required for corollary 3.5. The fact that the map Hffl*! Hffl*
**is an
isomorphism follows from propositions 3.4 and 3.5.
It remains to show that the matrix M is equivalent to a block upper triangular
one with respect to some ordering of the indexing set. It is easy to see, using*
* remark
3.7 and the double coset formula, that if is a j  2chain of partitions, and *
*ffl is an
unadmissible pramification of (so0Hffl*2 Ij2), then the only summands of Ij+1*
*1that
Hffl*maps nontrivialy on are Hffl*and summands whose detecting elementary Abel*
*ian
group has strictly fewer components than the elementary Abelian group detecting
Hffl*. This completes the proof of lemma 3.11. ___ 
An immediate consequence of lemma 3.11 is the following theorem:
Theorem 3.13. Let X be an odddimensional sphere localized at a prime p. Assu*
*me
n is not a power of p. Then
Dn(X) ' 1 Map *(SKn; 1 X^n)hn ' *:
Proof.Let E1 be the first term of the spectral sequence associated with the ske*
*letal
filtration of Kn abutting to
H *(Map *(SKn; 1 X^n)hn ) :
We saw that E1 can be identified with the cochain complex Co of graded Z=pZvec*
*tor
spaces. Moreover, there is a short exact sequence P o! Co ! Io. It is obvious t*
*hat
since n is not a power of a prime, P ois trivial. By lemma 3.11, Io is acyclic.*
* It follows
that E2 is zero. Therefore, H *(Map *(SKn; 1 X^n)hn ) is zero and the proposit*
*ion
follows. ___ 
THE GOODWILLIE TOWER OF THE IDENTITY 25
Thus if X is an odddimensional sphere localized at a prime p then the only i*
*n
teresting values of n are powers of p. If n = pk then the E2 term of the spect*
*ral
sequence computing
H* (Map *(SKn; 1 X^n)hn )
may be identified with the cohomology of the cochain complex P o. So we proceed*
* to
analyze the complex P o.
Definition 3.14. An ordered partition of a positive integer k is an ordered seq*
*uence
K = (k1; : :;:kj) of positive integers with k1 + . .+.kj = k.
For future use, we denote by 2l the ordered partition
(1; : :;:1; 2; 1; :::;:1)
_________z________"
2 at placel
Ordered partitions of k are partially ordered by refinement, we write K J if K
is a refinement of J. Moreover, ordered partitions form a lattice: any collect*
*ion of
partitions {Km }m has well behaved greatest common refinement and least common
coarsening (denoted by \m (Km ) and [m (Km ) respectively). In fact, the latti*
*ce of
ordered partitions of k is isomorphic to the Boolean lattice of subsets of k__*
*1_ordered
by inclusion. Given K = (k1; : :;:kj), let K be the group pk1o. .o.pkjand let K*
* be
k
the summand k1k2. .k.jof H*(K ) or more generally of H*(X^phK) depending on
the context. We denote by Nj(k) the set of ordered partitions of k with j compo*
*nents.
The following is obvious by inspection:
P 0~=0
For j > 0 M
P j~= K :
K2Nj(k)
In particular, if j > k then P j~=0.
In the following definition, the underlying assumption is that X is a 2s + 1
dimensional sphere and u is a generator of H 2s+1(X).
Definition 3.15. For a fixed k, let CU* be the free graded Z=pZ module on the
following generators:
if p > 2
{fiffl1Qs1. .f.ifflkQsku  sk s; si> psi+1 ffli+18i};
if p = 2 n o
Qs1. .Q.sku  sk 2s + 1; si> 2si+1:
Thus CU* is generated by the "completely unadmissible" words of length k (hen*
*ce
the notation).
26 GREG ARONE AND MARK MAHOWALD
Theorem 3.16. Let n = pk. The cohomology of P ois concentrated in degree k.
Moreover there are isomorphisms of modules over the Steenrod algebra
H k(P o) ~=CU* ~=k H*(Map *(SKn; 1 X^n)hn )
Where the action of the Steenrod algebra on CU* is given by the Nishida relatio*
*ns
([CLM76 ]).
Proof.The results about the cohomology of P oare known, and are more or less
implicit in [Ku85 ] (see also [Ku82 ] and [KuP85 ]), although the language the*
*re is
somewhat different from ours. First of all, let us see that the claim is plausi*
*ble by
counting dimensions. Consider a pure summand of a form k1k2. The module
k1+k2is a submodule of k1k2 (however, this obvious inclusion is not the same
as the transfer map in homology  if it was, the theorem would be easier to pro*
*ve).
When we consider k1+k2as a subobject of k1k2 we will denote it ak1k2  the
module generated by words which are admissible at place k1 (from the left). Let
uk1k2 be the quotient of k1k2 by ak1k2. Thus uk1k2 is generated by words
which are unadmissible at place k1. By a slight abuse of notation, we will write
k1k2= ak1k2 uk1k2:
The splitting is valid on the level of vector spaces, and is valid up to filtra*
*tion on the
level of Amodules. More generally, given an ordered partition K = (k1; : :;:kj*
*) of
k, we may write K as a direct sum of 2j1 modules. These 2j1 "subsummands"
are indexed by sequences (s1; : :;:sj1) where each si stands for either the le*
*tter a
or the letter u. The subsummand corresponding to a sequence S = (s1; : :;:sj1)
is generated by the words which are admissible (resp. unadmissible) at the pla*
*ce
k1 + . .+.ki if si is a (resp. si is u). We denote this subsummand by SK. Le*
*t u_
stand for the sequence (u; u; : :;:u). Clearly, if si is a for some i, then
SK~= (s1;:::;^si;:::;sj1)(k1;:::;ki+ki+1;:::kj):
Thus every "subsummand" is canonically isomorphic to u_Kfor some K. It is easy *
*to
see that for any K 2 Nj(k), the summand u_Koccurs in Pj+i(i 0) with multiplici*
*ty
! !
X k1  1 kj  1
. . .
(i1;:::;ij)i1 ij
where the summation is over jtuples (i1; : :;:ij) of nonnegative integers who*
*se sum
is i. It is also easy to see that
! !
X X k1  1 kj  1
(1)i . . . = (1  1)k11. .(.1  1)kj1
i (i1;:::;ij)i1 ij
where 00 = 1. Thus the alternating sum of multiplicities of all subsummands is*
* 0
except for the subsummand u_(1;1;:::;1), for which the "total multiplicity" is *
*1. Also, it
THE GOODWILLIE TOWER OF THE IDENTITY 27
is obvious that u_(1;1;:::;1)~=CU*. Thus CU* concentrated in dimension k is a "*
*lower
bound" for H *(P o). Thus we have to show that the rank of the coboundary map in
P ois as large as possible. This boils down to analyzing the effect of the tran*
*sfer map
k1+...+kj
on the pure part of the homology of spaces of the form X^phk1o...okjand showing*
* that
the intersection of the images of such various transfer maps is as small as pos*
*sible
(this, and much more, was done in [Ku85 ]). It is helpful to consider the chain*
* com
plex Po which is the "reverse" of P o. Pk ~=P kfor all k and the boundary maps *
*in Po
are induced by inclusion of groups where the coboundary maps in P oare induced *
*by
transfer maps. Indeed, given two ordered partitions K K0of k, the map K0 ! K
induced by the transfer has a retraction induced by inclusion of subgroups. It*
* is a
retraction (up to multiplication by a unit in Z=pZ because all the groups in si*
*ght
contain a common pSylow subgroup. We let eK;K0 denote the idempotent (up to a
unit in Z=pZ) homomorphism given by the composition K !i*K0 tr*!K : In the
special case K = (1; : :;:1), K0= 2l, we denote eK;K0simply el. The following c*
*rucial
properties of these idempotents are proved in [Ku85 ]:
1) el1el2= el2el1if l1  l2 2
2) elel+1el= el+1elel+1
Moreover, for any ordered partition K of k and a collection {K0i}i2Iof ordered *
*par
titions such that K K0ifor all i 2 I the folloing holds:
3) Im(eK;[i2IK0i) = \i2IIm(eK;K0i)P
4) ker(eK;[i2IK0i) = i2Iker(eK;K0i) .
The basic reason that properties (1)(4) hold is that the (dual of the) summand*
* K
of the cohomology of K is detected by the ring of invariants H *(Ak)PK , where *
*PK
is the parabolic subgroup of GL k(Fp) associated with the partition K, and thus*
* pro
preties (1)(4) can be read off the structure of the Hecke algebra of endomorph*
*isms
of Z=pZ[GL k(Fp)=B] where B is the Borel subgroup of GL k(Fp). As a matter of f*
*act,
(3) and (4) are only proved in [Ku85 , theorem 4.11 (2) and (3)] for the specia*
*l case
K0i= 2i, I = {1; : :;:k  1}, but the general case can be deduced from it quite*
* easily.
Property (3) implies, by the inclusionexclusion principle, that the rank of *
*the
coboundary maps in P ois as large as it can be. Therefore H i(P o) ~= 0 for i *
*< k
and H *(P o) is concentrated in degree * = k, moreover, H k(P o) is abstractly *
*isomor
phic to CU*, at least as a graded vector space. Property (4) implies the same *
*for
H*(Po). It remains to show that the isomorphisms are isomorphisms of Steenrod
Algeba modules, and not only of graded vector spaces.
The graded vector space Hk(P o) can be identified with the cokernel of the co*
*bound
ary homomorphism Pk1 ! Pk. The maps i* : (1;:::;1)! 2l l = 1; : :;:k  1
assemble to the boundary homomorphism Pk ! Pk1 in Po. H k(Po) it the kernel of
this map
k1"
Hk(Po) = ker{(1;1;:::;1)! 2l}:
l=1
28 GREG ARONE AND MARK MAHOWALD
L k1
Obviously, H k(Po) = \k1l=1ker(el) and H k(P o) = coker{ l=1Im (el) ! (1;:::;*
*1)}.
There is a homomorphism of Steenrod algebra modules CU* ! Hk(Po), given by the
Adem relations, which is clearly injective and thus is an isomorphism. On the o*
*ther
hand, there is a homomorphism of Steenrod algebra modules Hk(Po) ! Hk(P o) given
by the composition Hk(Po) ! (1;:::;1)! Hk(P o). We claim that this homomorphism
is surjective, and therefore is an isomorphism. To prove that the map is surjec*
*tive,
we need to show that for any element of u 2 (1;:::;1)there exists an element v 2
k1l=1Im(el) such that u + v 2 Hk(Po) (we consider H k(Po) as a subspace of (1;*
*:::1)).
To see this, let
w = (1e1)(1e2) . .(.1ek1)(1e1)(1e2) . .(.1ek2)(1e1) . .(.1ek3)(1
e1) . .(.1  e1)(1  e2)(1  e1)u:
P k1
Let v = w  u. It is easy to see that v 2 l=1Im(el). It is also easy to see t*
*hat since
the idempotents ei satisfy the braid relations, so do the idempotents 1  ei an*
*d that
as a consequence (1  el)w = w for all l = 1; : :k: 1, and thus w = u + v 2 Hk*
*(Po).
It follows that H k(P o) is isomorphic to CU* as a module of the Steenrod algeb*
*ra.
Once we know that the cohomology of P ois concentrated in degree k, it follows
that the spectral sequence collapses at E2 for dimensional reasons. Thus E2 ~=E*
*1 .
Since E1 has only one column, there is an isomorphism
i k j
E*;k1~=H*k Map *(SKpk; 1 X^p )hpk :
__
_ 
3.2. Action of the Steenrod algebra. Let n = pk. Our goal in this subsection is
to study the action of Steenrod algebra on
H*(Map *(Kn; 1 X^n)hn )
where X is an odddimensional sphere localized at a prime p.
Let A be the modp Steenrod algebra. Let A[k] be the subalgebra of A, generat*
*ed
by the Milnor basis (see [Mar , ch. 15] for notation and basic definitions) el*
*ements
P10; P11; : :;:P1kand (if p > 2) by Q0; : :;:Qk.
Theorem 3.17. Let X be a 2s + 1dimensional sphere localized at a prime p. Let
n = pk. The module
H* (Map *(SKn; 1 X^n)hn )
is free over A[k  1].
Proof.Our situation is very similar to that of [W81 , theorem 2.1]. The idea of*
* the
proof is taken from there entirely.
Theorem 3.16 gives us a basis for
H *(Map *(SKn; 1 X^n)hn ) :
THE GOODWILLIE TOWER OF THE IDENTITY 29
We dualize this basis to get a basis for the cohomology groups
H *(Map *(SKn; 1 X^n)hn ) :
The action of A is given by dualizing the homology Nishida relations as describ*
*ed in
[CLM76 , page 6]. To make the connection with [W81 ] explicit, we will rewrit*
*e the
basis in terms of Steenrod operations rather then the DyerLashof operations. *
*We
define a correspondence between the two kind of operations as follows: If p = 2*
* then
(Qi*) $ P i+1:= Sqi+1, and if p > 2 then (Qi)* ! fiP iand (fiQi)* ! P i(we
remind the reader that on the left hand side fi stands for the homology B"ockst*
*ein
and thus lowers degree by 1 while on the right hand side it stands for the coho*
*mology
B"ockstein and hence raises degree by 1.) By comparing the dualized Nishida rel*
*ations
with the Adem relations in the Steenrod algebra, it is not hard to see that this
correspondence establishes an isomorphism (up to a dimension shift) of Amodules
between H* (Map*(SKn; 1 X^n)hn ) and the module generated by admissible (in the
sense of the Steenrod algebra) words P s1. .P.sksuch that if p = 2 then sk 2s *
*+ 2
and if p > 2 then sk s + 1 (in the case p > 2 there are also B"ocksteins which*
* we
omitted). It is interesting to notice that in case X = S1, the cohomology that *
*we get
is isomorphic, as a module over the Steenrod algebra, to the cohomology of cert*
*ain
subquotients of symmetric product of the sphere spectrum which was first comput*
*ed
in [N58 ] and further studied in [W81 ]. These subquotients of the symmetric pr*
*oduct
spectra play a key role in [Ku82 , KuP85 ]. We conjecture that the spectrum
k
Map *(SKpk; 1 S^p )hpk
i.e. the pkth layer of the Goodwillie tower of the identity evaluated at S1 is*
* homotopy
equivalent (up to a suitable suspension) to the spectrum denoted L(k) in [Ku82 ,
KuP85 ]1 In any case, when X = S1, our statement is equivalent on the level of
cohomology to [W81 , theorem 2.1]. We sketch Welcher's proof, and indicate the
required very minor generalization. Given a sequence I = (s1; : :;:sk) we deno*
*te
by P Iu the element P s1. .P.sku, where P i= Sqi if p = 2. Suppose first that
p = 2. Following [W81 ] we define Bskto be the vector space generated by the s*
*et
{P I I = (2kj1; : :;:4jk; 2jk); where j1 . . . jk s + 1}: By computing the
Poincare series, one can easily show that BskA[k1] ~=H *(Map *(SKn; 1 X^n)hn )
as graded Z=pZ vector spaces. The calculation is exactly as in [W81 ] and we om*
*it
it. It follows that if the A[k  1] module generated by Bn is free, then it mu*
*st be
H*(Map *(SKn; 1 X^n)hn ) . This part of the proof again carries over from [W81*
* ].
___________
1Added in revision: since this paper was written, W. Dwyer, jointly with the *
*firstnamed author,
proved this conjecture. Details will appear in [AD97 ]. The overall connection *
*of the material in
this paper with the work of Kuhn, Mitchell and Priddy is made clear and explici*
*t in [AD97 ]. As a
byproduct, this leads to a substantial simplification of some of the proofs in *
*this paper (especially
those in section 3).
30 GREG ARONE AND MARK MAHOWALD
If p > 2, the same strategy applies with
Bsk= {P I I = (pk1j1; : :;:pjk1; jk); where j1 . . .jk s + 1}:
__
_ 
4. The vkperiodic homotopy of the tower
4.1. The case of an odddimensional sphere. Let p be a fixed prime. All spaces
in this section are automatically localized at p. In the previous section we sa*
*w that
in the Goodwillie tower of the identity evaluated at an odd dimensional sphere,*
* the
only layers that are nontrivial are those indexed by powers of p. So, there ex*
*ists a
tower of fibrations converging to the homotopy type of S2s+1
..
.



?
_________ _______oe 2s+1
S2s+1 Rk Dpk(S )
QQ 
Q Q fk
QQs ?
Rk1 _____oDpk1(S2s+1)e

fk1

?
..
.



?
R0 = Q(S2s+1)
where Rk = Ppk(S2s+1).
Moreover, Dpk(S2s+1) is an infinite loop space, and we saw in theorem 3.17 th*
*at
the cohomology of the associated spectrum is free over Ak1. This implies that
the Dpk(S2s+1) is trivial in vk1periodic homotopy and so are all the higher l*
*ayers.
In other words, in vkperiodic homotopy, the tower has only k + 1 nontrivial l*
*ayers
(Dp0; : :;:Dpk). We would like to conclude that the map S2s+1! Rk is an equival*
*ence
in vkperiodic homotopy. Apriori, it is not clear that the tower converges in v*
*k periodic
homotopy. Consider, for instance, the Postnikov resolution of a space X. The la*
*yers
in this resolution are trivial in vkperiodic homotopy, but X need not be, beca*
*use
the tower does not converge after inverting vk. Thus our goal in this section *
*is to
study the convergence of this tower in vkperiodic homotopy. It turns out that *
*since
THE GOODWILLIE TOWER OF THE IDENTITY 31
the connectivity of the layers grows so fast, the tower converges in the sense *
*that we
need. The main theorem of this paper is the following
Theorem 4.1. The map
S2s+1! Rk
is a viperiodic equivalence for all k 0 and all i k.
Proof.Let k be fixed all along. We are going to use theorem 3.17 in conjunction*
* with
the "vanishing line" theorems in [AD73 , MW81 ]. For the rest of the proof we *
*assume,
for simplicity, that p = 2, the odd primary case is only marginally more compli*
*cated.
Recall the following theorem:
0
Theorem 4.2. [AD73 , theorem 1.1] If M is an Amodule and Pts0is the lowest de*
*gree
pstwith s < t such that H(M; Pts) 6= 0, then Exts;t(M; Z=2Z) = 0 for ds > t + c,
where d = deg(Pts00) and _c_d1is approximately t0  2.
Let i
kj
M = H* Map * SK2k; 1 (S2s+1)^2 h
2k
By theorem 3.17, M is free over A[k  1], and since Pts2 A[k  1] if s + t k it
follows that the lowest degree Ptswith s < t s.t. H(M; Pts) 6= 0 is at least Pt*
*s00, where
( k1
___ ifk  1 is even
s0 = k2_
2 ifk  1 is odd
( k1
___ + 2 ifk  1 is even
t0 = k2_
2 + 1 ifk  1 is odd.
Thus Pts00= 2s0(2t0 1) = 2k+1  2s0> 2k  1.
Corollary 4.3. The Adams Spectral Sequence converging to the homotopy of D2k+i
has an (s; t  s) vanishing line of slope which is smaller than __1__2k+i2= __*
*1___vk+i1. It also
has a vertical intercept smaller than k + i.
Since vi acts on the level of the Adams spectral sequence as multiplication b*
*y an
element on a line of slope _1_vi, it follows that D2k is vk1trivial and mor*
*e generally,
if i > 0 then D2k+iis vktrivial.
We need to prove that the Goodwillie tower converges to S2s+1 in vkperiodic
homotopy. Till the end of this section, let ss*() denote ss*(; Vk1), where V*
*k1 is
a finite space (not a spectrum) of type k with a vk self map (see appendix). S*
*ince
S2s+1= holimRj we have to show that
v1kss*(holim Rj) ~=limv1kss*(Rj):
32 GREG ARONE AND MARK MAHOWALD
Let Qj = fiber(Rk+j ! Rk). There is a tower of fibrations
..
.





?
Qj _______Dpk+joe


gj


?
Qj1 _____Dpk+j1oe


gj1


?
..
.





?
Q1 = Dpk+1
Our statement is equivalent to showing that the vkperiodic homotopy of the inv*
*erse
limit of this tower is trivial. In other words, we want to show that
v1kss*(holim Qj) ~=0
or equivalently
v1k(lim ss*(Qj)) ~=0:
Let ff = (: :;:ff2; ff1) 2 limss*(Qj). Then ffj 2 ssd(Qj), gj(ffj) = ffj1, d =*
* deg(ff). We
identify an element of ssdDpk+j with its pullback at the E1 term of the corresp*
*onding
ASS. (We will also assume that such an element has (s; t  s) bidegree (0; d).*
* It
will be clear that from our point of view it is a harmless assumption, it amoun*
*ts to
taking the worst possible case.)
Suppose that ff = (: :;:ffj+1; ffj; 0; : :):, where j > 1 and ffj 6= 0. Since*
* ffj1 = 0,
ffj can be thought of as an elementfofissdj(D2k+j).fLetikj be the maximal integ*
*er such
that vkjk(ffj) 6= 0. Let dj+1 = fifivkjk(ffj)fifi= dj + kj(2k+1  1). It follow*
*s from corollary
THE GOODWILLIE TOWER OF THE IDENTITY 33
4.3 that dj+1 is bounded by dgj+1, which is determined by the following equatio*
*ns
8
< y(k+j)_g= __1___k+j
dj+1 2 2
: __y___g= __1___k+1
dj+1dj 2 2
Here (dgj+1; y) are the coordinates of the intersection of the line passing thr*
*ough
(0; k + j) and having slope __1___2k+j2(the "vanishing line") and the line pas*
*sing through
(dj; 0) and having slope __1___2k+12(the line along which vk moves ffj). Solvi*
*ng for dgj+1
we obtain the following bound
gdj+1= _____k_+_j____ ____dj____ k+1 3_
(2) ___1__ __1___+ 2k+12_< 2 (k + j) + dj:
2k+12 2k+j2 1  2k+j2 2
Now let ff = (: :;:ff2; ff1) be any element of limss*(Qj). We may assume ff1 =f*
*0i(by fi
applying vk enough to annihilate ff1). Let dj be the maximal d such that d = fi*
*fivkjk(ffj)fifi
and vkjk(ffj) 6= 0. It follows from (2) that the sequence dj has the rate of gr*
*owth of at
i jj
most 3_2 and thus it grows slower than the connectivity of D2k+j(the connectiv*
*ity
of D2k+jhas the rate of growth 2j), which proves the theorem. ___ 
4.2. The case of an evendimensional sphere. Throughout this subsection, let
X denote an evendimensional sphere (possibly localized at a prime p). In this *
*case
the tower is still finite in vkperiodic homotopy, but it is "twice as long" as*
* in the
odd sphere case. More precisely, there is the following version of our main the*
*orems.
Theorem 4.4. If n does not equal pk or 2pk for some prime p, then
Dn(X) ' 1 Map *(SKn; 1 X^n) hn ' *:
If n = pk or n = 2pk then Dn(X) has only pprimary torsion.
Thus, if X is an even sphere localized at p, there is a regraded Goodwillie t*
*ower,
34 GREG ARONE AND MARK MAHOWALD
which, in the case p > 2, looks as follows
..
.



?
X ______R2k____oD2pke
@ 
@ 
@@R ?
R1k_____oDpke



?
..
.



?
R20_____oD2p0e



?
R10= Q(S2s+1)
where R1k= Ppkand R2k= P2pk. If p = 2 then the tower looks just as in the odds*
*phere
case.
Theorem 4.5. If p > 2 then the map
X ! R2k
is a vkperiodic equivalence for all k 0. If p = 2 then the map
X ! Rk+1
is a vkperiodic equivalence for all k 0.
Proof of theorems 4.4 and 4.5.Rather than adapting the calculations of section *
*3 to
this case, we make use of the Goodwillie calculus and the James fibration. Cons*
*ider
the sequence of natural maps
j ^2
X !s X ! X
where s and j are the suspension map and the James map respectively. If X is an
odddimensional sphere localized at a prime then this is a fibration sequence. *
*For a
THE GOODWILLIE TOWER OF THE IDENTITY 35
general X, this is a fibration sequence in the metastable range [Ja53]. Let F *
*(X) be
the homotopy fiber of j. Since the composition j O s is trivial, there is a nat*
*ural map
f : X ! F (X)
which is a homotopy equivalence for odd spheres localized at a prime. We want to
conclude that the Taylor polynomials of F (X) are the same as of the identity w*
*hen
evaluated at odd spheres.
Proposition 4.6. Let
f : G(X) ! F (X)
be a natural transformation of reduced analytic functors. Suppose there exists*
* a
space K such that f induces an equivalence
G(S2iK) ! F (S2iK)
for all i 0. Then the map
Pnf : PnG(S2iK) ! PnF (S2iK)
is an equivalence for all i and n.
Proof.By induction on n. The case n = 0 is trivial. Indeed, since we assume G a*
*nd
F are reduced
P0G(K) ' P0F (K) ' *:
Assume that the proposition is true for n  1. It is clear that it is enough to*
* show
that
Dnf : DnG(K) ! DnF (K)
is an equivalence. Recall that the maps G(S2iK) ! PnG(S2iK) and F (S2iK) !
PnF (S2iK) are (n + 1)(k + 2i) + c connected, where k is the connectivity of K.*
* It
follows that the map Pnf(S2iK) is (n + 1)(k + 2i) + c connected. Using our indu*
*ction
assumption, it follows that the map Dnf(S2iK) is (n + 1)(k + 2i) + c connected.*
* By
Goodwillie's classification of homogeneous functors, there exist spectra Gn an*
*d Fn
with an action of n which represent DnG and DnF . Thus, the map
Dnf : (Gn ^ (S2iK)^n)hn ! (Fn ^ (S2iK)^n)hn
is (n + 1)(k + 2i) + c connected. By the Thom isomorphism, this implies that t*
*he
map
Dnf : (Gn ^ K^n)hn ! (Fn ^ K^n)hn
is (n + 1)k + 2i + c connected for all i. The proposition follows. ___ 
The following proposition is an easy consequence of the general theory of cal*
*culus
36 GREG ARONE AND MARK MAHOWALD
Proposition 4.7. (1) The operator Pn commutes up to natural equivalence with
finite homotopy inverse limits of functors. In particular
Pn(F ) ' PnF:
(2) Let Sq(X) = X ^ X. Then
PnF (X ^ X) ' Pn(F O O Sq)(X):
Returning to the notation of our main text, it follows from the two propositi*
*ons
that if X is an odd sphere localized at p, then there is a fibration sequence
Pn(X) ! Pn(X) ! Pn(X ^ X)
where Pn(X) is really Pn(Id)(X). Taking X = S2k1(p), the fibration sequence be*
*comes
Pn(S2k1(p)) ! Pn(S2k(p)) ! Pn(S4k1(p)):
Thus, we have a resolution of the Goodwillie tower for an even sphere by towers*
* for
odd spheres and theorems 4.4 and 4.5 readily follow. ___ 
Appendix A. Background on vkperiodic homotopy
In this appendix we collect some material from [MS95 ] concerning the definit*
*ion
of v1khomotopy and Lfklocalization.
Let M be a finite complex, endowed with a map v : dM ! M such that MU*(v)
is not zero. This implies, in particular, that all iterates of v are essential*
*. We can
consider the homotopy theory which results from looking at the homotopy classes*
* of
maps from M to a space X. We will write ssi(X; M) = [iM; X]. We can consider
this as a Z[v] module. The periodic homotopy of X defined by v is ss*(X; M) Z[v]
Z[v; v1]. The simplest case is obtained by letting M = S1, k = 0 and v be a map
of degree two. Then the periodic theory is obtained by tensoring the homotopy w*
*ith
Z[1=2]. This is an example of a v0periodic homotopy.
Higher order periodicity is defined in terms of a family of finite complexes *
*which
are detected in BP* by some power of vn (the idea of v1periodic homotopy goes
back to Adams  it can be defined using the Adams map 13RP 2! 5RP 2). These
complexes are not unique and there does not seem to be a canonical choice, but *
*such
complexes do exist and the choices do not matter much. That's the point of the
forthcoming discussion.
Definition A.1. We take M(pi0; vi11; : :;:vikk) to be any choice of a finite sp*
*ectrum
such that
BP*(M(pi0; vi11; : :;:vikk)) = BP *=(pi0; vi11; : :;:vikk):
As shorthand, we write I for (i0; : :;:ik), and M(I)for M(pi0; : :;:vikk). W*
*e also
write I J if il jl for 0 l k, and I J if il < jl for 0 l k. Below we
collect some facts about the spectra M(I).
THE GOODWILLIE TOWER OF THE IDENTITY 37
Proposition A.2. (1)Given a multiindex I, M(I) need not exist, but M(J)
exists for some J I.
(2) There may be more than one possible choice of homotopy type for M(I), but
there are at most finitely many choices.
(3) Given M(I), there is a J I and a map
22)+...+(i j )(2pk2)
fIJ: M(I) ! (i1j1)(2p2)+(i2j2)(2p kM(kJ)
commuting with projection to the top cell. (Note that the top cell of M(I*
*) is
in dimension k + 1 + i1(2p  2) + . .+.ik(2pk  2), so the suspension is *
*just
the difference in dimension between the top cells of M(I) and M(J). To sp*
*are
notation, we will frequently omit the suspension.) The map fIJinduces the
obvious map on BP*  multiplication by pj0i0vj1i11.v.j.kikk.
(4) For each I; J there are at most finitely many choices of homotopy classes*
* for
fIJ.
(5) Given M(I); M(J); M(K) with J I and K I, and maps fIJ; fIKas above,
there exists L J; K, M(L) and fJL; fKL so that
fIJ
M(I) ! M(J)
# fIK # fJLV
fKL
M(K) ! M(K)
commutes.
(6) One can choose a sequence of spectra M(Il) and maps fIlIl+1so that given *
*any
M(I) there is an fIIlfor l sufficiently large. If F is a specific finite*
* type k
complex, then one can choose the M(Il) so that M(Il) ^ F is a wedge of 2k*
*+1
copies of F (one for each cell in M(Il)), and so that fIlIl+1factors
g h
M(Il) ^ F ! F ! M(Il+1) ^ F
where g is projection to the top cell of M(Il) smashed with F and h is in*
*clusion
of the wedge factor of F associated to the top cell of M(Il+1) (once agai*
*n we've
neglected suspensions here).
(7) The SpanierWhitehead dual of an M(I) is also an M(I). The Spanier
Whitehead dual of fIJgives the obvious projection
0 jk i0 ik
BP*=(pj ; : :;:vk ) ! BP*=(p ; : :;:vk ):
The finiteness results are consequences of the fact that a finite torsion spe*
*ctrum has
finite homotopy groups in every dimension. The existence results are all applic*
*ations
of the Nilpotence and Periodicity theorems.
38 GREG ARONE AND MARK MAHOWALD
We will make use_of_the direct system one can form by using the spectra M(I) *
*and
the maps fIJ. Let M (I) be the fiber of the projection to the top cell
21)+:::+i 2(pk1)
M(I) ss!Sk+1+i12(p1)+i22(p k:
Then there is a cofiber sequence
21)+:::+i 2(pk1)gI_
(3) Sk+i12(p1)+i22(p k! M (I) ! M(I):
Since the fIJhave been chosen to commute with the projections to the top cell*
*, we
get induced maps (of positive degree which we omit from our notation)
___ _fIJ__
M (I) ! M (J)
__I
such that fJgI = gJ.
Corresponding to the direct system of M(I)'s and fIJ's, we get a direct syste*
*m of
___ __I
M (I)'s and fJ.
Proposition A.3. The map
_k1)_
(4) S0 ! hocolim[ki12(p1):::ik2(pM(I)]
I
induced by the {gI} is Lk localization.
The next proposition gives a functorial description of vk torsion generalizin*
*g the
usual definition when X has a vk map.
Proposition A.4. Let X be a spectrum and f 2 ss*(X). The following are equiva
lent:
i)f factors as
f" g
S0 ! M ! X
where M is a complex with a vk map v such that vjf"' * for some j.
ii)f factors through a finite complex in Ck+1.
iii)f is in the kernel of
ss*X ! ss*(LfkX):
If X is a finite complex of type k, the above conditions are equivalent to
iv)If v is any vk map of X, then vjf ' * for j sufficiently large.
Here is the definition of vk periodic homotopy with integral coefficients.
Definition A.5.
v1kssk(X) = dirlim(i0; : :;:ik1)v1k[Ml(pi0; : :;:vik1k1);:X]
Here the subscript l indicates the dimension of the bottom cell of the coeffici*
*ent
spectrum.
THE GOODWILLIE TOWER OF THE IDENTITY 39
Note that this definition also makes sense unstably for l sufficiently large:*
* suppose
for some (i0; : :;:ik1), Mk(pi0; : :;:vik1k1) exists unstably, and supports *
*a vikkself map.
Then after inverting vikkwe can still form the direct limit over (j0; : :;:jk1*
*) by noting
that the stable map
0 i0k1 i ik1
Ml(pi0; : :;:vk1) ! Ml(p 0; : :;:vk1)
is the stabilization of some unstable map
0 i0k1 rvik i ik1
Ml+rvik(pi0; : :;:vk1) ! S kMl(p 0; : :;:vk1):
k 
We also need to know that a vkmap of a spectrum can be represented on the le*
*vel
of a (perhaps suitably modified) Adams spectral sequence by multiplication by an
element on the line of slope _1_vkpassing through the origin.
References
[AD73] D.W. Anderson and D.W. Davis, A vanishing theorem in homological algeb*
*ra, Comment.
Math. Helv. 48 (1973), 318327.
[AdMi95] A. Adem and J. Milgram, Cohomology of finite groups, Grundlehren der m*
*athematischen
Wissenschaften 309, SpringerVerlag (1995).
[AK97] G. Arone and M. Kankaanrinta, A functorial model for iterated Snaith s*
*plitting with
applications to calculus of functors, To appear in the the proceedings*
* of the homotopy
program at the Fields Institute.
[AD97] G. Arone and W. Dwyer, Partition posets, Tits buildings and symmetric *
*product spectra
(tentative title), in preparation.
[BK72] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localiza*
*tions, Springer
Lecture Notes, Vol. 304 (1972).
[BMMS86] P.R. Bruner, J.P. May, J.E. McClure and M. Steinberg, H1 ring spectra*
* and their
applications, Springer Lecture Notes, Vol. 1176 (1986).
[CLM76] F.R. Cohen, T.J. Lada and J.P. May, The homology of iterated loop spac*
*es, Springer
Lecture Notes, Vol. 533 (1976).
[G90] T.G. Goodwillie, Calculus I: the first derivative of pseudoisotopy the*
*ory, KTheory 4
(1990), 127.
[G92] T.G. Goodwillie, Calculus II: analytic functors, KTheory 5 (1992), 29*
*5332.
[G3] T.G. Goodwillie, Calculus III: the Taylor series of a homotopy functor*
*, in preparation.
[Ja53] I. M. James, Reduced product spaces, Annals of Math. 62 (1953), 17019*
*7.
[Jo95] B. Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc*
*. 347, Number
4 (1995), 12951321.
[KaP78] D.S. Kahn and S.B. Priddy, On the transfer in the homology of symmetri*
*c groups, Math.
Proc. Cambridge Philos. Soc. 83 (1978), 91101.
[Ku82] N.J. Kuhn, A KahnPriddy sequence and a conjecture of G.W. Whitehead, *
*Math. Proc.
Cambridge Philos. Soc. 92 (1982), 467483.
[Ku85] N.J. Kuhn, Chevalley group theory and the transfer in the homology of *
*symmetric groups,
Topology 24 (1985), 247264.
[KuP85] N.J. Kuhn and S.B. Priddy, The transfer and Whitehead conjecture, Math*
*. Proc. Cam
bridge Philos. Soc. 98 (1985), 459480.
40 GREG ARONE AND MARK MAHOWALD
[M82] M. Mahowald, The image of J in the EHP sequence, Annals of Mathematics*
* 116 (1982)
65112.
[MS95] M. Mahowald and H. Sadofsky, vn telescopes and the Adams spectral sequ*
*ence, Duke
Math J. 78 (1995) 101129.
[MT92] M. Mahowald and R. Thompson, The Ktheory localization of an unstable *
*sphere, Topol
ogy 31 (1992) 133141.
[MT94] M. Mahowald and R. Thompson, On the secondary suspension homomorphism,*
* Amer.
J. of Math. 116 (1994) 179206.
[Mar] H.R. Margolis, Spectra and the Steenrod algebra, North Holland Mathema*
*tical Library.
[MW81] H. Miller and C. Wilkerson, Vanishing lines for modules over the Steen*
*rod algebra, J.
Pure Appl. Algebra 22 (1981), 293307.
[Mi72] J. Milnor, On the construction FK, in "Algebraic Topology: A Student's*
* Guide" by J.F.
Adams, London Math. Soc. Lecture Note Series 4, Cambridge Univ. Press,*
* Cambridge
(1972), 119135.
[N58] M. Nakaoka, Cohomology mod p of symmetric products of spheres, J. Inst*
*. Poly., Osaka
City Univ. 9 (1958), 118.
[T90] R. Thompson, The v1periodic homotopy groups of an unstable sphere at *
*odd primes,
Trans. Am. Math. Soc. 319 (1990), 535559.
[W81] P.J. Welcher, Symmetric fiber spectra and K(n)homology acyclicity, In*
*diana Univ.
Math. J. 30 (1981), 801812.
(G. Arone) Department of Mathematics, University of Chicago, 5734 University
Avenue, Chicago, Il 60637, USA
Email address: arone@math.uchicago.edu
(M. Mahowald) Department of Mathematics, Northwestern University, 2033 Sheri
dan road, Evanston, IL 602882730, USA