On vanishing Tate cohomology and
decompositions in Goodwillie calculus
Kristine Bauer
Randy McCarthy *y
March 24, 2003
Abstract
Our main result is that if F is a functor from a pointed category C
to spectra, the Goodwillie tower of F evaluated at X splits rationally
when X is a coHobject of C. We show that the layers of F (X) in
this case are easy to identify. The splitting of the Goodwillie tower
gives a decomposition of F (X) into a product of its layers. We use
this to recover the rational decompositions of Hochschild and higher
Hochschild homology [P00 ], [L98 ], [GS87 ]. Finally, we extend the main
theorem to include dual calculus to recover the Poincar'eBirkhoffWitt
theorem, and improve the theorem in the special case in which the
comultiplication map is cocommutative.
1 Introduction
Let F be a functor from any pointed category C to any abelian category.
As an application of [G90 ], [G92 ] and [G02 ], JohnsonMcCarthy provide a
______________________________
*The second author was partially supported by NSF grant DMS0071482
1
Goodwillie calculus theory for such functors ([JM4 ]). There is a tower
6..6
mmmmmm.
mm
mmmmmm qn+1
mmmm pn fflffl
F (X) ___________//_RRPnF (X)
RRRRpn1RR 
RRR qn
RR((R fflffl
Pn1F (X)


fflffl
..
.
of universal degree n approximations PnF (X) to F (X). The layers DnF (X)
of F (X) are the fibers of the maps qn. This model of calculus can be extended
to functors F from C to spectra, as in [McC02 ] (by dualizing). In this paper,
we provide a criterion on X for the Goodwille tower of F to split rationally
when evaluated at X. In particular, we show that if X is an object with a
öc multiplication" map r from X to the coproduct X _ X  in other words,
if X is a coHobject in C  then rationally the first map of the fiber sequence
qn
DnF (X) _____//PnF (X)____//_Pn1F (X)
has a splitting map. This results in the main theorem of the paper:
Theorem 1.1. If F is a homotopy functor and X is a coHobject of C then
rationally
Yn
PnF (X) ' DiF (X)
i=1
Q
and consequently P1 F (X) ' n 1DnF (X).
The proof of the main theorem is constructive. We construct maps
PnF(r)
DnF (X) _____//PnF (X)____//_PnF (_nX)___//_DnF (X)
such that the composite is multiplication by n!. To obtain the splitting, we
simply invert n! (thus the result is rational). The bulk of the proof involves
2
showing that this composite map is induced by the norm map (x3), which in
turn induces the Tate map and shows that the composite is multiplication
by n!.
Since X is a coHobject, it is equipped with öc vering maps" r defined
by
__rr_//_X_+___//
X r X
where + is the fold map. An important consequence of the proof of the main
theorem is that the maps r induce multiplication by rn on DnF (X), and
hence can be used to identify the layers of F (X). We use this idea to show
that the known rational decompositions of (higher) Hochschild homology
([GS87 ], [L98 ], [P00 ], [B ]) are actually splittings of the Goodwillie tower*
* of
the forgetful functor from augmented commutative kalgebras to kmodules.
We then compute the layers of the tower associated to higher Hochschild
homology, and show that they are suspensions of the layers of the tower
associated to Hochschild homology.
We also prove a version of the main theorem for dual calculus ([McC02 ]).
In this case, we examine the Hobjects of C. Using the dual version of our
main theorem, we are able to show that rationally, the Poincar'eBirkhoff
Witt theorem is a decomposition of the universal enveloping algebra of a free
Lie algebra into the dual layers of a dual Goodwillie tower associated to it.
Finally, we show that sometimes the Goodwillie tower splits more often
than är tionally". Recall that the Tate map provides a map from the homo
topy orbits to the homotopy fixed points of a spectrum. We are particularly
interested in the Tate map applied to the spectrum D(n)1crnF (X), the multi
additivitation of the nth cross effects of F (see section 2 for the definition*
* and
close relationship to DnF ). Define the nth Tate cohomology of F evaluated
at X to be the cofiber of this map applied to crnF (X). In the final section of
the paper, we show that if the comultiplication map is cocommutative, then
the Goodwillie tower actually splits whenever the Tate cohomology vanishes.
Kuhn has recently observed that work of Greenlees, Hovey and Sadofsky
shows that the Tate cohomology of a K(n)*local spectrum vanishes  hence
the Goodwillie calculus towers of functors from cocommutative coHobjects
to spectra split upon K(n)*localization.
This paper is organized as follows. In section 2 we define and examine the
cross effects of a functor. It is important to note that in this section we do
not construct the universal degree n approximations to F . However, we do
3
provide all of the results and constructions that we will need in the context
of this paper. A more detailed account can be had from [JM4 ] and [M ] (for
the case where the target category is spectra) or [McC02 ]. In section 3 we
describe the norm map and its relationship to the Tate map, which will be
essential in constructing the splitting map to DnF (X) ! PnF (X). Here we
only provide statements of the results we will need regarding these maps; a
more detailed account of the Tate map is given in the appendix of the preprint
version [BMc ]. In section 4 we make explicit our definition of coHobjects.
We also check that the properties we require of F are preserved by crn, Dn
and Pn. Section 5 is the statement and proof of the main theorem. We
conclude by giving two extensions of the main theorem; first the extension to
dual calculus (section 6) and finally to the case of vanishing Tate cohomology
(section 7).
The authors would like to thank Nick Kuhn for pointing out an error in
an earlier draft of this paper.
2 Preliminaries in Goodwillie Calculus
Let S be any category of rational spectra, that is, modules over the Eilenberg
MacLane spectrum HQ. Let F be a functor from a pointed category C to S.
The building blocks used to construct the universal degree n approximation
to F are called the cross effects. In this paper, we will usually phrase all
important information about the Goodwillie tower of F in terms of cross
effects. For this reason, it is important for us to spend some time introducing
them.
It is easiest to think of the cross effects as the total fibers of certain
cubical diagrams. Let n = {1, . .,.n} be a finite set of n elements. Let P (n)
be the category whose objects are the subsets of n and whose morphisms are
ordered inclusions. Note that the subset ; is an initial object of P (n) while n
is the terminal object. One can visualize this category as a cubical diagram
with an object of P (n) at each corner and morphisms as edges. The objects
; and n occupy opposite corners of the cube.
An ncube in a pointed category C (i.e. a category whose initial object
is also its terminal object) with coproducts is a contravariant functor from
P (n) to C. To construct the cross effects, we will use a particular ncube.
Let _ denote the coproduct in C. Let g : n ! obj(C) be a function which
4
generates a list of objects X1, . .,.Xn of C. Define an ncube Øg to be the
contravariant functor which is given on a subset S of n by
Øg(S) = _ g(c)
c=2S
"
and which takes an inclusion SØ____//_S0to the projection
_ g(d) ! _ g(c).
d=2S c=2S0
By convention, Øg(n) = * (the base point of C). For n = 2, Øg is the square
diagram:
X1 _ X2 _____//X1
 
 
fflffl fflffl
X2 ________//*
Let F be a covariant functor from C to S. Let P0(n) be the subcategory of
P (n) consisting of nonempty subsets and let Xi = g(i).
Definition 2.1. The nth cross effect, crnF (X1, . .,.Xn), is the fiber of the
map
holimS2P(n)F (Øg(S)) ! holimS2P0(n)F (Øg(S)).
Note that holimS2P(n)F (Øg(S)) = F (_ni=1Xi).
The cross effect crnF (X1, . .,.Xn) is an n multivariable functor from Cxn
to S. Let crnF (X) = crnF (X, . .,.X). Notice that there is a convenient map
æ : crnF (X) ! F (_nX) arising from the definition of the cross effect.
Definition 2.2. A functor F : C ! S is a degree n functor if crkF ' * for
every k > n.
If F is degree n, then the Goodwillie tower of F is truncated  i.e. DnF
is the largest nontrivial layer of F and PkF is equivalent to F for all k n.
This is sometimes taken as the definition of degree n. Furthermore, if F is
degree n, then crnF is linear in each variable. That is, e.g.
crnF (X _ Y, X2, . .,.Xn)
' crnF (X, X2, . .,.Xn) x crnF (Y, X2, . .,.Xn). (1)
We enumerate here some properties of the cross effects and Goodwillie
towers of degree n functors which we will frequently use.
5
Lemma 2.3. If F is a degree n functor, then there exists a natural equiva
lence
m Y
crnF ( _ Xi) ' crnF (Xff(1), . .,.Xff(n)) (2)
i=1
ff2Hom (n,m)
Proof.This is a consequence of the fact that for degree n functors, crnF (X)
__
is linear in each variable. __
Remark 2.4. Let uk : _mi=1Xi ! Xk (1 k m) be the map which is the
identity on the Xk component of _mi=1Xiand which sends Xj to the basepoint
for j 6= k. The weak equivalenceQof Lemma 2.3 can be realized by the map
! : crnF (_mi=1Xi) ! ff2Hom (n,m)crnF (Xff(1), . .,.Xff(n)) defined by
Y
! = crnF (uff(1), . .,.uff(n)) O
ff2Hom (n,m)
Q
where is the diagonal map crnF (_mi=1Xi) ! Hom(n,m)crnF (_mi=1Xi).
Let ck : Xk ! _mi=1Xi be the dual map to uk which includes the kth
summand. Keeping in mind that coproducts and products in S are weakly
equivalent, we construct a homotopy inverse to ! given by the composition
of
`
` crnF(cf(1),...,cf(n))`
crnF (X) f2Hom(n,m)___________//_ crnF (_ X)
f2Hom (n,m) f2Hom (n,m) m
with the fold map.
Lemma 2.5. If F is a degree n functor, then there exists a natural equiva
lence
Y
crncrnF (X1, . .,.Xn) ' crnF (Xff(1), . .,.Xff(n)). (3)
ff2 n
Proof.Let g(i) = Xi and let cS denote the complement in n of a subset S of
n. Using the definition of the cross effect and lemma 2.3, we compute that
crncrnF (X1, . .,.Xn) = fiber{holim crnF ( _ Xc) ! holim crnF ( _ Xc)}
S2P(n) c2cS S2P0(n) c2cS
6
is equivalent to the fiber of
Y Y
holim crnF (Xff(1), . .,.Xff(n)) ! holim crnF (Xff(1), . .,.Xff(*
*n)).
S2P(n) ff2 S2P0(n) ff2
Hom(n,cS) Hom(n,cS)
Q
The first homotopy limit is just ff2Hom (n,n)crnF (Xff(1), . .,.Xff(n)) (since
thisQlimit diagram has an initial object) and the second homotopy limit is
ff2Hom 0(n,n)crnF (Xff(1), . .,.Xff(n)) where Hom 0(n, n) are the nonbiject*
*ive
set maps from n to itself. So this is equivalent to the fiber of
Y Y
crnF (Xff(1), . .,.Xff(n)) ! crnF (Xff(1), . .,.Xff(n))
ff2Hom (n,n) ff2Hom 0(n,n)
which is Y
crnF (Xff(1), . .,.Xff(n))
ff2Bij(n,n)
__
where Bij(n, n) := n is the group of bijective maps. __
Remark 2.6. Using remark 2.4,Qone can realize the weak equivalence of 2.5
by a map !0: crncrnF (X) ! Bij(n,n)crnF (X) defined by
Y
!0= crnF (uff(1), . .,.uff(n)O O æ
ff2Bij(n,n)
where æ : crncrnF (X) ! crnF (_nX) is the natural map. This is accom
plished by applying the map ! to crnF (_c2cSXc) for each S 2 P (n) (or
P0(n)).
Proposition 2.7 (Lemma 3.9 [JM4 ]). If F is a degree n functor from C
to S then DnF ' (crnF )h n.
Proposition 2.8 (Remark 2.8 [JM4 ]). If F is a degree n functor from C
to S then F ' PnF .
By using fibers instead of cofibers in the definition of the cross effect, we
obtain another cross effect, called the cocross effect, ecrn. Let P1(n) denote
the full subcategory of P (n) consisting of proper subsets of n. Let eØ be
the covariant functor from P (n) to C with eØ(S) = _ g(c) and which takes
c2S
inclusions in P (n) to inclusions in C.
7
Definition 2.9. We define the cocross effect ecrnF (X1, . .,.Xn), to be the
cofiber of the map
holimS2P1(n)F (eØg(S)) ! holimS2P(n)F (eØg(S)).
In the category of spectra, since cofibration and fibration sequences are
equivalent, we have that crnF ' ecrnF . There is a convenient map
F (_nX) ! ecrnF (X)
(dual to the map æ) which we will need.
Lemma 2.10. If F is a degree n functor, then there exist natural equivalences
a
cernF (_ X) ' ecrnF (X) (4)
n
Hom (n,n)
a
ecrnecrnF (X)' ecrnF (X). (5)
n
Proof.This is the analogue of lemmas 2.3 and 2.5, and the proof is straight
__
forward. __
3 The Tate Map
If a finite group G acts on a kmodule M, there is a natural map t : MG !
MG induced by the norm map. The norm map is constructed using the
diagonal , the action of g for each g 2 G, and the addition map + as
follows:
______t0__________________________________________*
*_________________________________________________________________@
__________________________________________________________*
*_________________________________________________________________@
____//_ ____//_ +___//_
M GM g2Gg GM M
We can extend this to a map t : MG ! MG by making two observations.
First, t0extends in the following diagram (where p is the projection map onto
the orbits)
__t0_//_
M zM==
p z z
fflfflz
MG
8
since for each g 2 G, t0(m) = t0(gm).P Second, note that the image of t0
actually lands in MG since t0(m) = g2G gm so that for any h 2 G,
X X X
ht0(m) = h gm = hgm = gm = t0(m)
g2G g2G g2G
since G is finite. Thus we have a map t which factors t0 as:
__t0__//
M MOO
p i
fflfflt ?Ø
MG ____//_MG
where i is the inclusion of the fixed points. The map t is an equivalence
whenever the order of G is invertible. In fact, the inverse is given by p O i
and we have p O i O t[m] = G[m].
We wish to extend this map to the category S of HQmodules. Suppose
that E is an HQmodule with an action of a finite group G. We can extend
the norm map T 0. Again, we do this by using the diagonal map and the fold
map. This time, we must use the weak equivalence between the product and
the coproduct. The following diagram defines T 0:
Q
Q gx() Q `
_____// E _g2G______//_E o'o_?_` E _+___//
E g2G g2G g2G E
Motivated by this diagram, we make the following definition
Definition 3.1. Let I be an indexing set and {Xi}i2I be a collection of
objects of S. Let fi,j: Xi ! Xj be maps in S for each i, j 2 I and
gi,j: Xio'o__Xj be weak equivalences in S for each i, j 2 I. A weak map
in S from Xim ! Xin is a collection of objects Xim, Xim+1, . .,.Xim+k = Xin
and arrows fi,jand gi,jsuch that for any adjacent pair Xim+j, Xim+j+1of ob
jects in the collection, there is either a map fim+j,im+j+1or a weak equivalence
gim+j,im+j+1between them.
A typical weak map might look like a zigzag
fi,j gj,k
Xi _____//Xjo'o_Xk.
9
g1j,kOfi,j
This weak map is denoted Xi ` ` ` ` ` `//Xj, where the dashed line denotes
that it's a weak map.
Note that each weak map has associated to it a map in the homotopy
category. We say that a diagram of weak maps commutes if the corresponding
diagram of maps in the homotopy category commutes. There is a weak map
T which extends the weak map T 0to homotopy orbits and homotopy fixed
points just as t extended t0 in the case of modules. This weak map is called
the Tate (weak) map. A construction of t can be found in e. g. [McC ].
We will need the following properties of the Tate map. The proofs of
these can be found in [McC ] or [BMc ].
Lemma 3.2. The Tate map is the restriction of the Norm map to homotopy
orbits. That is, the following diagram of weak maps commutes:
Q
______//Q ____g2Gg__//_Q ` '`//`` __+___//
E G E G E G E EOO
 
fflfflfflffl ?Ø
EhG ` ` ` ` `` ` ` ` ` ``T` ` ` ` ` `` ` ` ` ` `//EhG
where T refers to the composite that defines the Tate map and the composite
of the top (weak) maps is the norm map.
Proposition 3.3. Let n be the nth symmetric group. The composition
Th n __t_//_T h_n___//T____//Th n
induces multiplication by n! on homotopy groups.
4 The category of homotopy comonoids
Let C be a pointed model category.
Definition 4.1. Let X be a cofibrant object of C. We say that X is a coH
object of C if there exists a map r : X ! X _ X which is coassociative up
to homotopy, and counital up to homotopy with counit c : X ! *.
10
The coHobjects are the comonoids of C (up to homotopy). They form
a subcategory of C whose morphisms from X to Y are the maps of f 2 C
which make the following diagrams commute:
rX
X _____//X _ X .
f  f_f
fflffl fflffl
Y __rY_//Y _ Y
Since the fold map is unital with respect to the unit u : * ! X, we can think
of the fold map as a unital multiplication with which C is already equipped.
The coHobjects of C are exactly those whose comultiplication maps r act
like algebra maps with respect to the multiplication map defined by the fold
map. In other words, for a coHobject X the following diagram commutes
up to homotopy:
_nrn fi
_nX _____//_n(_n)____//__n(_nX). (6)
+n _n+n
fflffl rn fflffl
X ____________________//__nX
If one arranges _n _n X in an n x narray, one can think of the map ø as the
transpose map. It is a reordering of the copies of X.
The first examples of coHobjects are coHspaces in the category of
pointed topological spaces. In particular, the basepointed circle S1 is a co
Hspace. The map r in this case is the pinch map which identifies the
basepoint with its antipodal point. Since the circle is a coHspace, so are all
suspensions X = S1 ^ X using the map r ^ 1.
For the main theorem, we will be considering a coHobject X and a
functor F : C ! S which preserves weak equivalences. If F preserves weak
equivalences, then when F is applied to diagram 6 the resulting diagram
still commutes up to homotopy. The following lemma will allow us to use F ,
crnF , Pn and DnF interchangeably with respect to diagrams which commute
up to homotopy.
Lemma 4.2. Let F : C ! S be a functor which preserves weak equivalences.
Then crn+1F , PnF and DnF preserve weak equivalences between cofibrant
objects.
11
Proof.Recall (Definition 2.1) that crn+1F (X) is the fiber of
holimP(n+1)F (Øg) ! holimP0(n+1)F (Øg).
We claim that if ff : A ! B is a weak equivalence then crn+1F (ff) :
crn+1F (A) ! crn+1F (B) is also a weak equivalence, as long as A and B
are both cofibrant. This is true because if A and B are both cofibrant, then
_kff : _kA ! _kB
is again a weak equivalence for all k n + 1. Thus ff induces a weak equiv
alence on each term F (Øg) of the cube defining crn+1F , and hence induces
weak equivalences on holim P(n+1)F (Øg) and holim P0(n+1)F (Øg). Then the
map induced on the fiber is also a weak equivalence.
Each of the remaining constructions defining PnF and DnF is a homotopy
construction involving only crn+1F and F .
__
__
Remark 4.3. We only require C to be a model category in order to provide
us with the correct notion of "weak equivalences" in C, and coHobjects are
only cofibrant to insure that diagrams involving coHobjects which commute
up to homotopy will still commute up to homotopy after PnF or other related
functors have been applied. We can also define coHobjects in categories C
which are not model categories by requiring that all of the maps involved
in the definition commute up to isomorphism. Since all functors preserve
isomorphisms, lemma 4.2 is then unnecessary. For example, in the category of
commutative algebras over k, the coHobjects are almost the Hopf algebras.
Notice that the coHobjects only differ from Hopf algebras because they lack
the antipodal map with which Hopf algebras are equipped.
5 The Splitting Theorem
We can now state the main Theorem.
Theorem 5.1. If F is a functor from C to S and X is a coHobject of C,
then rationally the fiber sequence
j qn
DnF (X) _____//PnF (X)____//_Pn1F (X)
12
splits on the homotopy category. Consequently,
Y
P1 F (X) ' DnF (X)
n 0
is a rational equivalence.
Without loss of generality, we may assume F is a degree n functor by
replacing F with PnF . We remark that this replacement is why Lemma 4.2
is needed.
The map j exists because DnF is defined to be the fiber of the map qn.
We can reformulate j ([JM4 ]) in terms of cross effects. Since F is degree
n, there are equivalences DnF (X) ' (crnF (X))h n and PnF (X) ' F (X).
Thus, the above fiber sequence becomes
j
(crnF (X))h n ____//_F (X)___//_Pn1F (X)
for each n. Recall that there is a map æ : crnF (X) ! F (_ X). The map j is
n
induced by æ and the fold map as follows:
j
crnF (X) _______________________//F (_nX) (7)
 ss 
 ssss 
 sss 
 yyyys 
 F (_ X) F(+)
 n h n 
 o77o LLL 
 jhonoooo LLLL 
fflfflfflfflooooo LL&&Lfflffl
j
(crnF (X))h n ____________________//_F (X)
The n action on F (_nX) permutes copies of X. However, the fold map
F (+) : F (_nX) ____//_F (X)has F (+)(F (oe._nX)) = F (+)(F (_nX)). Hence
F (+) factors through homotopy orbits. Since the map æ is nequivariant, æ
extends to a map æh n on orbits. The resulting map j is the map for which
we seek to provide a splitting.
13
When X is a coHobject, we may extend this diagram to:
"j fiOF(_nrn)
crnF (X) Ø_____//_F (_nX)________//_F (_n(_nX)) (8)

 F(+n) F(_+n)
   n
fflfflj" fflffl F(rn) fflffl
crnF (X)h nØ_____//_F (X)___________//F (_nX)
ffi
fflffl
ecrnF (X)
where ffi is the map dual to æ (see definition 2.9).
Recall that for functors whose target category is S, crnF (X) ' ecrnF (X).
We wish to show that the composite ffi O F (rn) O F (+n) O æ is the norm map
T 0of section 3. If this is the case, then the following diagram commutes (as
a diagram of weak maps)
j
crnF (X) ________//_F (_nX) (9)

q F(+)
fflffl j fflfflF(rn)
crnF (X)h n _______//_F (X)_______//_F (_ X)
OO n
OOOOTOO 
OO ffi
OO''O fflffl
ecrnF (X)h n_i__//_ecrnF (X)
where T is the Tate map. We note that the lower trapezoid of this diagram
also commutes. The key point is that the Tate map, extended to a map
from homotopy orbits to homotopy orbits (by including the fixed points into
the orbits) induces multiplication by n! on ß*, which is rationally invertible.
Thus, if f and g are two maps from crnF (X)h n to ecrnF (X) such that f Oq '
g O q then since q O i O T is rationally invertible, we actually have that f ' *
*g.
Furthermore, the fact that q O i O T ' n! provides the splitting.
Using the square diagram 6 from section 4 as a central square, we can
14
further extend the diagram to
crncrnF7(X)N7
''ooooo NNN
oooo crn(p)NNNNN
oooo fflffl NNNN
cr F (X) ______//_crnF (_ X) NN''øNNN
pp7n7 n NNN
= ppp   NNN
ppp j A j C NNN
ppp   NNN
pp j fflffl fflffl crn(fNN&&fi)
crnF (X) ________//_F (_nX)fiOF(_rn)//_F (_n_nX)ffi//_ecrnF_(_nX)//_ecrnecrnF*
* (X)
 n  qqqq
 F(+n) F(_+n) B  qqq
  n  qqq''+q
fflffl j fflfflF(rn) fflffl fflfflxxqq
crnF (X)h n ________//F (X)_________//F (_ X)___ffi//_ecrnF (X)
OO n pp
__ OOOTOOO  pppp
________OO______________________________________________________________*
*__________ffi=ppp
_______OO''O_________________________________________________________*
*______________________________fflfflwwppp
_____________________________________//_ h n
______________________________________ecrnFe(X)crnF (X)
________________________________________
_________________________
xn! ________________________________________________________*
*_____________
______________________________________________fflffl*
*fflffl
___))___________________________________________*
*____ecrnF (X)h n
(10)
The squares A and B commute by the functoriality of crn (resp. ecrn). Let
jfibe the composition of maps which makes the triangle C commute. We
will show that the lifts j and j+ exist. Then, by the commutativity of the
diagram, it will suffice to show that j = j+ O jfiO j is the norm map.
Lemma 5.2. There exists a lift j such that the diagram
crncrnF"(X)`55
ll
''lllll 
llll 
llllcr fflffl
nF(rn)
crnF"(X)`__________//crnF"(_nX)
`
 
 
fflfflF(_rn) fflffl
F (_ X) _____n_____//_F (_ _ X)
n n n
commutes. Moreover, j is homotopic to a diagonal map.
Proof.First, notice that by Remark 2.6 the following diagram commutes up
15
to homotopy:
0 Q cr F (X)
crncrnF (X) _!'__//_Bij(n,n)n
 OO
j u
fflffl Q 
crnF (_ X)_____//_ crnF (_ X)
n Bij(n,n) n
Q
where is the diagonal map and u = ff2Bij(n,n)crnF (uff(1), . .,.uff(n)) is
built of the maps ui of Remark 2.4.
If we rearrange this diagram, one sees that this provides a lift:
Q
crnF (X) oo!0_
Bij(n,n)88OO ' crncrnFp(X)p
''qqqqq  pppp
qqq uO ppppjp
qqqcr  wwpp
nF(r)
crnF (X)_______//crnF (_nX)
In fact, since X is a coHobject, the map r is counital. Therefor uiO r is
homotopic to the identity on X for each 1 i n. It follows that j is
homotopic to the diagonal map. We will not distinguish between the lift j
and the associated map (!0)1 O j .
__
__
Lemma 5.3. There exists a lift j+ such that the diagram
F (_ _ X) _____//ecrnF (__X)__//ecrecrF (X)
n n n nq n
qqq
F(+) crnF(+) qq''qqq
fflffl fflfflxx+qqqq
F (_ X) ______//_ecrF (X)
n n
commutes. Moreover, j+ is homotopic to a fold map.
Proof.This is formally dual to Lemma 5.2, using Lemma 2.10. The details
__
of the proof are left as an exercise to the reader. __
The map ø with
Y Y
ø : crnF (X) ! crnF (X)
f2Bij(n,n) f2Bij(n,n)
16
is given on the factor indexed by f 2 Bij(n, n) by shuffling each of the n
variables of crnF (X) by f. Call ø the twist map.
Lemma 5.4. The map jfi, which is the composite
0 Q cr F (X)
crncrnF (X) __!'_//_Bij(n,n)n
 " ` PPPP
  PPPP
  PPP
fflffl Q fflffl PPPP
! crnF (X) PPP''ø
crnF (_nX)____'//_Hom(n,n) PPP
PP PPPP
 PPP''0øP PPPP
j PPPP PPPP
fflffl PP''Q PPP''Q
F (_ _ X) _______//_ ecrnF (X)___////_ ecrnF (X)
n n  Hom(n,n) Bij(n,n)
d
'  '
fflffl fflffl
cernF (_ X)________//_ecrecrF (X)
n n n
is the twist map.
Proof.Note that here we have used Lemma 2.10 and the fact that coproducts
are weakly equivalent to products in the lower corner of the diagram.
We will begin by examining the map j0fi. Recall that æ : crnF (X) !
FQ(_nX) and ffi : F (_nX) ! ecrnF (X) are the usual maps. On the factor of
1
Hom(n,n)crnF (X) indexed by f, notice that ! : crnF (X) ! crnF (_nX) is
given by crnF (cf(1), . .,.cf(n)) where ci : X ! _nX is the inclusion into the

ith summand. Let cf := cf(1)_ . ._.cf(n). Then the map æ is given on the
fth factor by either of the composites
1
crnF (X)__cr___!________//crnF (_ X)
nF(cf(1),...,cf(n)) n
j  j
fflffl fflffl
F (_ X) _______F(cf)____//_F (_ _ X)
n n n

Similar computations provide a factorization of d, projected onto the gth
factor, using the map ui : _nX ! X which is the identity on the ith
17
summand and 0 elsewhere (this is dual to ci). Let ug := ug(1)_ . ._.ug(n).
We obtain a commuting diagram
1
crnF (X) __cr___!________//crnF (_ X)
nF(cf(1),...,cf(n)) n
j j
fflffl fflffl
F (_ X) ______F(cf)_____//_F (_ _ X)______d_______//_ecrnF (_ X)
n TT T n n n
TT T F(ug) cernF(ug(1),...,ug(n))
TT T fflffl 
TT** fflffl
F (_ X)_________d________//ecrF (X)
n n
(11)
The key point now is to understand the composition represented by the
dashed arrow. By functoriality, we need only understand the composition
(ug) O (cf(1)_ . ._.cf). To understand this composition, it is easiest to
introduce the labels
0 1 0 1
X1,* X1,1 _ . . ._ X1,n
B C `` '' B C
B _ C c B _ _ C
B . C f B . . C
B .. C __________//_B.. ..C
B C B C
@ _ A @ _ _ A
Xn,* Xn,1 _ . . ._ Xn,n
`` ''
 ug




fflffl
X*,1 _ . . ._ X*,n
keeping in mind that X*,*= X for all choices of *'s. On any summand, Xi,*,
we have
cf(i) ug(f(i))
Xi,*_____//Xi,1_ . ._.Xi,n___//_Xg(f(i)),f(i)
which is nonzero if and only if g(f(i)) = i. Since this is true for all 1 i *
* n,
this implies that f has an inverse and that g = f1 . That is, f 2 Bij(n, n).
18
Furthermore, the composition of uf1 with cf yields
0 1
X1,*
B C
B _ C
B . C
B ..C ____//_X1,f(1)_ . . ._ Xn,f(n)
B C
@ _ A
Xn,*
which is exactly the twist by the action of f.
We have shown the composition is exactly nonzero when f 2 Bijso that
diagram 11 actually defines
Y Y
jfi: crnF (X) ! ecrnF (X).
Bij(n,n) Bij(n,n)
By the equivariance of æ and d, we see that this extends to show that jfiis
__
the twist map. __
Now, looking at the composites of the maps of lemmas 5.2, 5.3 and 5.4 we
find by lemma 3.2 that we have constructed the norm map. Thus the induced
map T is indeed the Tate map which gives us the rational equivalence we
sought. This concludes the proof of the Theorem 5.1.
Remark 5.5. In fact, the proof of the theorem only requires the condition
that there exists a maps
F (X) ! F (_nX)
which are compatible with F (+) : F (_nX) ! F (X) in the sense of definition
4.1. Since the structure maps r are actually induced as maps in C in most
of our examples, we choose to state the theorem in terms of coHobjects.
However, one should note that the theorem could be stated more generally
by adjusting definition 4.1 to accommodate this situation.
Example 5.1. The following example is due to Tom Goodwillie.
Let F : T op ! S be the functor F (X) = C*(X ^ X= ) where is the
diagonal map : X ! X ^ X. We show that rationally the Goodwillie
tower of F splits when X is a coHspace, but not in general.
First note that the cofiber sequence
_____// ____//_X^X_
X X ^ X
19
induces a short exact sequence
C*(X) _____//C*(X ^ X)_____//C*(X^X_.)
The functor X 7! C*(X) is a homogeneous functor of degree one and the
functor X 7! C*(X ^ X) is homogeneous of degree two. Since Dn (as well
as Pn) preserve short exact sequences, we have that F (X) must also be a
functor of degree at most two.
We want to examine the fiber sequence
D2F (X) ! P2F (X) ! P1F (X)
The following 3 by 3 diagram with exact rows and columns captures all of
the essential information for the Goodwillie tower of F (X):
D2C*(X) ____//_D2C*(X ^ X) '___//_D2F (X)
  
 ' 
fflffl fflffl fflffl
P2C*(X) _____//P2C*(X ^ X) _____//P2F (X)
  
  
fflffl fflffl fflffl
D1C*(X) ____//_D1C*(X ^ X) ____//_D1F (X)
The columns are exact since we have a fiber sequence D2 ! P2 ! P1 and,
since all of the functors involved are reduced, P1 = D1. Since C*(X) is a
homogeneous functor of degree 1, D2C*(X) = 0 and similarly, since C*(X ^
X) is a homogeneous functor of degree 2, D1C*(X ^ X) = 0. That means we
have equivalences D2F (X) ' D2C*(X ^ X) = C*(X ^ X) and D1F (X) '
D1C*(X) = C*( X). Since F is of degree two, we also have that P2F (X) '
C*(X ^ X= ). That is, we have an exact sequence
C*(X ^ X) _____//C*(X ^ X= ) _____//C*( X).
inducing the long exact sequence
. . .____//H*+1( X)
@
fflffl
H*(X ^ X) _____//H*(X ^ X= ) _______//H*( X)
@
fflffl
H*1(X ^ X) ____//_. . .
20
on homology. If this map splits, then the map @ is zero. It is easy to find
spaces for which this can not be the case. One simple example is given by
S0. Also, notice that up to the suspension isomorphism the map @ can be
expressed as
*
H*(X) _____//H*(X ^ X) .
On (rational) cohomology, the map * : H*(X ^ X) ! H*(X) is the cup
product. Therefore, if this map is zero then X has trivial cup products. Since
spaces don't generally have trivial cup products, this generally doesn't split.
However, when X is a coHspace, the Hopf algebra H*(X) is exterior on the
indecomposables, the cup products are trivial.
We end this section by showing that the layers DnF (X) can be identified
by the image of a certain map. Let r : F (X) ! F (X) be the composite
r = F (+) O F (rr). Note that in the case X = S1, this induces an rfold
covering map of S1.
Remark 5.6. The map Dn( r) := P1 ( r)DnF(X) from DnF (X) ! DnF (X)
induces multiplication by rn.
To see this, recall that DnF (X) = (D(n)1crnF (X))h n by [JM4 ]. We have
Q (n) (n)
D(n)1crnF (_kX) ~= Hom (n,r)=rnD1 crnF (X) since D1 crnF is a multilinear
functor. The following diagram describes r:
D(n)1crnF (X)
RRR
F*(rr) RRRRRRR
fflffl RR))RQ
(n)
D(n)1crnF (_ X)__'_//_ D1 crnF (X)
r Hom(n,r)=rn
ll
F*(+) llll+lll
fflffluullll
D(n)1crnF (X)
Now, by arguments analogous to Lemmas 5.2 and 5.3, we have that the maps
and + of the diagram are the diagonal and fold map (recall that coproducts
and products are weakly equivalent). Hence, r is multiplications by rn.
The result follows from the fact that r is a n equivariant map, so passes
to homotopy orbits.
21
5.1 Higher Hochschild Homology
Let A be a commutative algebra over a field k of characteristic zero. Let
X. be any finite pointed simplicial set. We can form a simplicial kalgebra
by composing the functor X. with the functor  A from the category of
finite pointed sets to Aalgebras which takes the set [[n]] = {0, 1, . .n.} to
[[n]] A = A kn. The chain complex associated to the resulting simplicial
algebra computes the Hochschild homology of A when X ' S1 and computes
the nth higher Hochschild homology when X ' Sn. Denote the homology
of the chain complex associated to X. A by HHX*(A). Rational decompo
sitions of Hochschild homology have been studied extensively [L98 ], [GS87 ],
[R93 ]. A rational decomposition of higher Hochschild homology which recov
ers the known decomposition of Hochschild homology has been discovered by
1^X
Pirashvili [P00 ] and a rational decomposition for HHS* (A) was found by
the first author, recovering the decompositions of the other cases [B ].
The goal is to compute the layers of the Goodwillie tower computing
higher Hochschild homology in terms of the layers of Hochschild homology.
The chain complex (S1 ^ X) A is a commutative differential graded Hopf
algebra [B ]. Using the comultiplication map, consider (S1 ^ X) A as a co
Hobject in the category of commutative augmented Aalgebras, ACommA .
Let U be the forgetful functor from the category ACommA to the category
1^X 1
A  mod of Amodules. By Theorem 5.1, HHS* = DnU((S ^ X) A).
n
The Goodwillie tower of the functor U has been computed [KM ]. Let M be
a cofibrant object of ACommA . Then the linear part, D1U(M), is I=I2(M)
where I(M) is the augmentation ideal of M over A. The higher layers are
given by DnU(M) = (I=I2(M)) nn= Sn(I=I2(M)) where Sn denotes the n
th part of the symmetric algebra. Since A is a cofibrant object in ACommA ,
so is Sd A and we will use this to compute the Goodwille tower.
By Remark 5.6 the "rfold cover map" r induces multiplication by rn
on each layer DnU((S1 ^ X) A) and one can use this to show that the
decomposition using Theorem 5.1 agrees with the decomposition of [B ] and
hence also those of [P00 ], [L98 ] and [GS87 ].
Since D1U is a linear functor, it commutes with suspensions. Note that
the suspension in ACommA is given by S1  and the suspension in Amod
is given by ~Z[S1] A  where ~Z[S1] is the free module generated by S1.
Therefore D1U(Sd A) = eZ[Sd] A D1U(S0 A). Since DnU(Sd A) =
22
(D1U(Sd A)) nnwe have
DnU(Sd A) = (eZ[Sd] D1U(S0 A)) nn (12)
= [eZ[Sdn] (D1U(S0 A)) n ] n (13)
(14)
where the action of n on the last line is given diagonally. On the first
factor, n acts by permuting the copies of Sd. Each flip of factors Sd induces
multiplication by (1)d on homology. On the second factor, n acts by
permuting the factors D1U(S0 A). Taking the orbits of this action produces
the homogeneous degree n part of the symmetric algebra. Taking both of
these actions together, we have
(
dnSnD1U(S0 A) d is even;
DnU(Sd A) =
dn nD1U(S0 A) d isodd,
where n is the homogeneous degree n part of the exterior algebra. Finally,
if one then computes that DnU(S1 A) = n nD1U(S0 A) then we can
express the layers of higher Hochschild homology in terms of Hochschild
homology by realizing that (up to a sign) DnU(Sd A) is an dfold suspension
of DnU(S1 A). That is, the layers of higher Hochschild homology depend
only on the layers of Hochschild homology.
6 Dual Calculus
There is a version of calculus which is strictly dual to the Goodwillie calculus
tower which we have been using so far. To obtain this theory, one simply
replaces homotopy limits with homotopy colimits, coproducts with products,
fibers with cofibers, etc. For details, see [McC02 ].
If one replaces coproducts by products and fibers by cofibers in 2.1, one
obtains the dual cross effects, crnF (X). The dual cross effects can be thought
of asQthe total cofibers of cubical diagrams. There is a natural map æ :
F ( n X) ! crnF (X).
One can use the dual cross effects to define a codegree n approximation
23
to F , and these assemble into a tower
..
.OO


 pn
P nFO(X)O______//F8(X)8
rrr
qn rrn1rr
 rrr p
P n1FO(X)O



..
.
which is universal with respect to maps to F from codegree n functors. The
nth dual layer of F , DnF , is the cofiber of the map qn : P n1F ! P nF .
There is also a notion of dual cocross effects analogous to the cocross
effects and obtained by replacing coproducts with productsQand limits with
colimits. There is a natural map ~æ: ecrnF (X) ! F ( n X).
To state the dual version of the theorem, we must describe the appropriate
counterpart for coHobjects. The following is the expected definition:
Definition 6.1. A fibrant object X of C is an Hobject if X is equipped with
a map ~ : X x X ! X which is unital and associative up to homotopy.
The unit map for ~ is given by the inclusion of the basepoint. Let be
the diagonal map : X ! X x X. The following diagram commutes up to
homotopy:
Q Q n Q Q
X ___fiO____//_ X
n n n
~n Q~n 
 fflffl
fflffl n Q
X ____________//_nX
Q Q
where ø is the map which transposes the entries of n nX. The Hobjects
in the category of pointed topological spaces are precisely the Hspaces.
Theorem 6.2. If F is a functor from C to S which preserves weak equiva
lences and if X is an Hobject of C, then rationally the cofiber sequence
qn n j n
P n1F (X) ____//_P F (X)___//_D F (X)
24
splits in the homotopy category. Consequently,
Y
P 1F (X) ' DnF (X)
n 0
is a rational equivalence.
The proof of this dual version of our main theorem proceeds in essentially
the same manner as the proof of the main theorem. Here is a sketch:
We first reduce to the case where F is a codegree n functor, since if F is
not codegree n we may replace F by P nF . When F is codegree n, we have
equivalences P nF (X) ' F (X) and DnF (X) ' crnF (X)h n. We can then
express the map j as in the following diagram:
j n h
F (X)L_____________________//crF7(X)7 n
 LLLL j* ppppp 
 LLL ppp 
 LL%% ppp 
 Q h n 
F( )  F ( X) i
 t n 
 ttt 
 ttt 
Qfflfflyytt fflffl
F ( X) ___________j__________//_crnF (X)
n
where F ( ) factors through the fixed points since it is nfixed, and æ*
extends because æ is nequivariant.
Now,Qusing the fact that X is an Hobject and using the map eæ: fcrnF (X) !
F ( n X), we can expand this diagram to
ecrnF (X)

j~
Qfflffl
F ( X) _F(~)__//F (X)___j_//crnF (X)h n
n 

F() F() i
Q fflfflQ fflfflQ fflffl
F ( X)F(~Ofi)//_F ( _X)j__//_crnF (X)
n n n
which commutes. We claim that the ö utside" map  that is, the composition
æOF (~Oø)OF ( )Oæ~ is the norm map. Just as before, one can show this by
25
Q Q
usingQthe equivalences crnF ( n X) ' Hom(n,n)crnF (X) and crncrnF (X) '
n
ncr F (X) and by showing that the relevant maps are the diagonal, twist
and fold maps. Once this is done, we can further expand our diagram to
q n
ecrnF (X)_____//_crFk(X)hOnkWWW
WWWW
 WWWWWiW OOOOOT
~j WWWWWW OOOO
Qfflffl WWWWWWOOO''
F ( X) __F(~)___//F (X)____j__//crnF (X)h n
n 

F() F() i
Q fflfflQ fflfflQ fflffl
F ( X) F(~Ofi)//_F ( X)__j___//crnF (X)
n n n
where q and i are the relevant quotient and inclusion maps, respectively. The
map T is the Tate map, and this commutes with the rest of the diagram as
before because the outside composition is the norm map. In fact, the entire
diagram commutes except possibly for the map i. We wish to show that i
provides a splitting map for j. In other words, we'll show that j O F (~) O eæO*
* i
is rationally homotopic to the identity map on crnF (X)h n. This is the same
as showing that T OqOi is rationally homotopic to the identity map. However,
recalling the definition of the map T , we have that T O q O i ~ N O i where N
is the norm map since N actually lands in the fixed points. One can easily
check that N O i is multiplication by n! which is rationally equivalent to the
identity map on crnF (X)h n. Thus i provides the splitting.
Remark 6.3. Define a map r := F (~) O F ( ). Then Dn( r) induced
multiplication by rn. The argument for this is exactly dual to Remark 5.6.
6.1 The Poincar'eBirkhoffWitt Theorem.
We seek to recover the rational version of the Poincar'eBirkhoffWitt Theo
rem as an application of Theorem 6.2. First, we recall the definitions required
to state this theorem. This background can be found in e.g. [W94 ].
Let Liek be the category of Lie algebras over a field k of characteristic 0.
If A is any algebra over k, A can be thought of as a Lie algebra by giving it
the bracket [x, y] = xy  yx, where x, y 2 A. Denote A as a Lie algebra by
A. Let G be a Lie algebra over k with bracket [ , ].
26
Definition 6.4. The universal enveloping algebra of G is an algebra over k,
U(G) with associated Lie algebra U(G) together with a morphism i : G !
U(G) which is universal with respect to algebras over k. That is, if f : G ! A
is a map of Lie algebras, then there is a map g : U(G) ! A such that the
following diagram commutes:
G ___i_//DU(G)
DD
DDD g*
f DD!!Dfflffl
A
where g* is the map induced by g.
One can construct U(G) as follows: Let i : G ! T (G) be the inclusion of
G into its tensor algebra induced by including G into the first graded piece
of T (G). Let I be the ideal of T (G) generated by the relations
i([x, y]) = i(x)i(y)  i(y)i(x)
where x, y 2 G. Then U(G) = T (G)=I.
Let ß : T (G) ! U(G) be the quotient map and let Tm (G) = mj=0G j. One
can see that U(G) inherits a grading from T (G) by setting Um (G) = ß(Tm (G)).
If G is free as a module over k, then the Poincar'eBirkhoffWitt theorem
shows that Um (G)=Um1 (G) ~=Sm as kmodules, where Sm := G m = m is the
mth homogeneous graded piece of the symmetric algebra, and that U(G) ~=S
as kmodules, where S is the whole symmetric algebra. In fact, if {ei}i2I is
a basis for G as a kmodule, then
{ei1 . . .eimij 2 I; i1 . . .im ; m 1}
is a basis for U(G) (where I is some indexing set).
From [MM65 ] we know that U(G G) ~= U(G) U(G) and that U(G)
is a cocommutative Hopf algebra. The multiplication structure map for the
Hopf algebra is induced by concatenation on T (G), and the comultiplication
is induced by the diagonal map : G ! G G. Let C be the category of
cocommutative, coaugmented coalgebras over k. In C, the product is given
by and the diagonal map for any object is given by the comultipication.
The Hopf algebra structure makes U(G) an Hobject in the category C.
27
Let F be the forgetful functor from C to the category of kmodules. We
can now apply theorem 6.2 to F (U(G)) to obtain a splitting of U(G) as a k
module. We want to show that this splitting recovers the Poincar'eBirkhoff
Witt theorem. In other words, we want to show that DnF (U(G)) ~=Sn.
Note that for g 2 U1(G)=U0(G), the image of the comultiplication map is
(g) = g 1 + 1 g. Denote an element in Un(G)=Un1(G) by (g1, . .,.gn).
By induction, we have
X
(g1, . .,.gn) = (gff(1), . .,.gff(p)) (gff(p+1), . .,.gff(p+q*
*))
p+q=n
ff2(p,q)shuffles
where oe, a (p, q)shuffle means that oe is a permutation of {1, . .,.n} with
oe(1) . . . oe(p) and oe(p + 1) . . . oe(p + q) Since multiplication is
induced by concatenation, the map 2 = ~ O is
X
~ O (g1, . .,.gn) = (gff(1), . .,.gff(p), gff(p+1), . .,.gff(p+*
*q)).
p+q=n
ff2(p,q)shuffles
However, in U(G) = T (G)=I, for any permutation oe 2 n we have
(gff(1), . .,.gff(n)) = (g1, . .,.gn).
Therefore ~ O is simply multiplication by the number of ways of shuffling
(g1, . .,.gn) into two factors. An easy inductive argument shows that the
number of (p, q)shuffles with p + q = n is 2n.
We now know that the map 2 = ~ O induces multiplication by 2n on
Un(G)=Un1(G) ~=Sn. However we also know that 2 induces multiplication
by 2n on DnF (U(G)). This shows that the two decompositions are the same
 the map 2 plays the role of a linear operator on the kvector space U(G)
whose image determines the decomposition associated to it. Hence theorem
6.2 recovers the Poincar'eBirkhoffWitt theorem.
7 Splittings and Tate Cohomology
We want to consider another extension to Theorem 5.1.
28
Definition 7.1. [McC02 ] Let F be a homotopy functor from C ! S. Define
the nth Tate cohomology of F at X to be
T aten(F ; X) := cofiber((D(n)1crnF (X))h n ! (D(n)1crnF (X))h n)
where the map from the homotopy orbits to the homotopy fixed points is the
Tate map.
Theorem 7.2. Let F be a homotopy functor from C ! S and let X be a
coHobject of C. If the map r is cocommutative, then the fiber sequence
DnF (X) ! PnF (X) ! Pn1F (X)
splits whenever the Tate cohomology vanishes for all n.
Proof.The proof only requires a small adjustment to the proof of theorem
5.1. If r is cocommutative (hence rn is cocommutative), then it is a nfixed
map and since ffi is nequivariant we have a factorization
F(rn)
F (X) ________________________//NF (_nX)
 NNNN rrr99 
 NNNN rrr 
 NNN rrr 
 && r 
ff F (_ X)h n ffi
 pp n 
 ppp 
 pppp h n 
fflfflwffiwpp fflffl
cernF (X)h n_____________________//ecrnF (X)
This extends Diagram 5 to:
j
crnF (X) ________//_F (_nX) (15)

 F(+)
 
fflffl j fflfflF(rn)
crnF (X)h n _______//_F (X)_______//_F (_ X)
OO n
OOOO'OO  
OO ff ffi
OO''Offlffl fflffl
ecrnF (X)h n____//_ecrnF (X)
29
We wish to show that oe O j is a weak equivalence. When the nth Tate
cohomology vanishes, the map T is an equivalence so that crnF (X)h n '
crnF (X)h n which in turn is equivalent to ecrnF (X)h n.
Let F and G be functors to spectra. From the construction of PnF , we
know that if crnF (X) ' crnG(X) via a natural transformation ! : F ! G,
then there is a pull back diagram:
Pn(!)
PnF (X) ______//_PnG(X) (16)
 
 
fflfflPn1(!) fflffl
Pn1F (X) ____//_Pn1G(X)
In this case, we wish to use G(X) = ecrnF (X)h n and ! = oe. By us
ing Lemma 5.2, weQknow that crn(ffi) O crnF (rn) is the diagonal map into
ecrnecrnF (X) ' n ecrnF (X). Here, we are making use of the equivalence
ecrnF (X) ' crnF (X) repeatedly. From there, one sees that
Y h n
cernF (X) ' ecrnF (X)
n
so that the map ecrn(oe) is an equivalence. Thus, we have a pullback diagram
Pn(ff)
PnF (X) ____________//_Pn(cernF (X)h n)
 
 
fflffl Pn1(ff) fflffl
Pn1F (X) __________//_Pn1(cernF (X)h n)
Now, since the Tate map is an equivalance, and since crnF (X)h n is a homo
geneous degree n functor, we have that crnF (X)h n is also a homogeneous
degree n functor. So, this pullback diagram is actually
F (X) ____ff//_crnF (X)h n
 
 
fflffl fflffl
Pn1F (X) __________//*
__
and the result follows by taking parallel fibers. __
30
Remark 7.3. Note that the theorem only actually requires that the map
F (X) ! F (_nX) be öc commutative", i.e. that this map is fixed under
the action of n. In [McC02 ], McCarthy proves a similar theorem for "stable
functors", that is, for functors to spectra with the property that the inclusion
map
X _ Y ! X x Y
induces an equivalence F (X _ Y ) ! F (X x Y ) for all X and Y in C. If F
is a stable functor, then every object X 2 C becomes a coHobject via the
diagonal map
F (X) ! F (X x X) ' F (X _ X).
Since the diagonal map is cocommutative, this recovers McCarthy's theorem.
In particular, functors from the category C of left (resp. right) kmodules
are stable functors since it is already the case in C that products are equival*
*ent
to coproducts. Note also in this case that every object is both a coHobject
and an Hobject, so that both the dual towers and the regular towers split
when the Tate groups vanish. McCarthy's result shows that in fact the layers
of the dual tower and the regular tower, and hence the two splittings, agree.
8 Appendix
Recall from section 3 that if a finite group G acts on a kmodule M, there is
a natural map t : MG ! MG factoring the norm map as:
__t0__//
M MOO
p i
fflfflt ?Ø
MG ____//_MG
which is an equivalence whenever the order of G is invertible.
We wish to extend this map to the category S of HQmodules. Let T be
an FSP. Let E_be it's associated spectrum
E_k = hocolimn nE(Sn+k )
and suppose E_ is an HQmodule with an action of a finite group G. We
wish to explicitly construct a weak map E_hG ! E_hG which will be the
31
correct analogue for the Tate map for modules. Such constructions can be
found in e.g. [WW ], [GM ], [DGM ], [McC ]. The construction we provide h*
*ere
is due to Tom Goodwillie and closely follows the recollection provided in the
appendix of [McC ]. We begin with some background.
8.1 Group Actions
Let T op be the category of base pointed topological spaces. In the following,
G is a finite group. The definitions and constructions which follow hold for
all groups G, but since we are only interested in the case G = n, it won't
hurt for us to assume that G is a finite group throughout.
An action of a group G on a space X is a map G+ ^ X ! X. We
denote the image of g ^ x by gx. Equivalently, a group action is a functor
X : G ! T op where G is the category with one object and Hom G(*, *) = G.
In this case, X denotes both the functor and the topological space X which
is the image of the unique object of G. We say that G acts freely on X if for
all nonbasepoint elements x 2 X, gx 6= x unless g is the identity element of
G.
The following is a small collection of constructions involving G actions.
The orbits of the G action on X are defined to be
XG = colimGX = X={x ~ gx}.
If X is a Gspace and Y is a Gopspace (the action of G is on the right), then
we may define a smash product Y ^ X with Gaction g(y ^ x) = yg1 ^ gx.
Define Y ^G X = {yg ^ x ~ y ^ gx} and notice that
(Y ^ X)G = Y ^G X.
The homotopy orbits of the G action on X are
XhG = hocolimG X = (EG+ ^ X)G
where EG is any contractibe free G space. Later, we will use a specific model
of EG given by the simplicial construction
q+1^
EG = [q] 7! G+ 
32
where + denotes a disjoint basepoint.
The fixed points and homotopy fixed points of the G action on X are
respectively
XG = limGX = {xgx = x for allg 2 G}
and
XhG = holimG X = Map (EG+ , X)G .
If X is a Gspace, there are two important free G spaces associated to
X. The first is G+ ^ X with Gaction given by h(g ^ x) = gh1 ^ hx. The
second is Map (G+ , X) with Gaction given by h(f)(g) = hf(gh). There are
equivalences
(G+ ^ X)G ! X
g ^ x 7! gx
and
X ! Map (G+ , X)G
x 7! f(g) = gx.
Since the action of G on G+ ^ X and Map(G+ , X) is free, there are G
equivariant equivalences
(G+ ^ X)hG := (EG+ ^ G+ ^ X) ! (G+ ^ X)G
and
Map (G+ , X)G ! Map (EG+ , Map (G+ , X))G =: Map (G+ , X)hG.
We conclude this section by defining an importantQrelationship between
these two free Gspaces. Using the inclusion _G X ! G X we can produce
a map Y
G+ ^ X ~=_ X ! X ~=Map (G+ , X).
G G
Let this composition be named fl and notice that fl is given by
(
x u = g
fl(g ^ x)(u) =
* otherwise.
By the BlakersMassey theorem, if X is kconnected then fl is (2k  1)
connected. We will take advantage of this in the future to produce equiva
lences using fl and stabilization.
33
8.2 Gactions on Functors with Smash Product
In this section, we assume a familiarity with the definition of functors with
smash products and we describe the group actions on these. Readers who
are not familiar with these definitions should consult e.g. [McC ].
If E is an FSP, then we denote by E_the spectrum associated to E with
E_k= E(Sk). Since every E_ is naturally equivalent to an spectrum (see
[McC ] for a construction), we may assume that E_is an spectrum.
Define the homotopy orbits of E to be the FSP EhG with EhG(X) =
1 (E(X)hG). (Technically, we have to indicate why this is again an FSP.)
Note that since
EhG(X) = 1 (E(X)hG)
= hocolimn n(E(Sn ^ X)hG)
= hocolimn n(hocolim GE(Sn ^ X)),
EhG is a homotopy colimit construction. Therefore, if E* is a simplicial F SP ,
the natural map E*hG ! (E*)hG is an equivalence since the geometric
realization is also a colimit construction and öc limits commute". Let E_hG
be the spectrum associated to EhG.
The homotopy fixed points of E is the FSP with EhG (X) = ( 1 E(X))hG.
Since homotopy fixed points are a limit construction we cannot make a similar
statement relating (E*)hG with E*hG. However, there is a natural map
(E*)hG ! E*hG.
Let E_hG be the spectrum associated to EhG .
We have FSP's (G+ ^ E)(X) = G+ ^ E(X) and Map (G+ , E)(X) =
Map (G+ , E(X)). Note that the relevant homotopy orbit and homotopy fixed
point constructions are given by
(G+ ^ E)hG(X) = 1 [(G+ ^ E(X))hG]
= hocolimn n[(G+ ^ E( nX))hG]
Map(G+ , E)hG(X) = [ 1 Map (G+ , E(X))]hG
= [hocolim n nMap (G+ , E( nX))]hG
34
8.3 The Tate Map
We would like to produce a weak map E_hG__T__//E_hGwhich factors the
(weak) norm map for spectra. That is, we'd like T to satisfy the following
diagram of weak maps
diagonalQE ff Q E _ E fold
E_______//g2G_____//g2G_` ``//g2G_______//E_OO
 
 
 
fflffl T 
E_hG ` ` ` `` ` ` ` ` `` ` ` ` ` `` ` ` `//E_hG
where the vertical maps are the natural maps and the dotted lines are the
weak maps (the dotted arrow in the top row is given by the inclusion of the
coproduct into theQproduct and the BlakersMassey theorem).QThe map oe is
the "shuffle map" g2G g which acts on the factor of g2G X indexed by g
by g.
Before we construct the map T for a general FSP, we first consider the
special case where the FSP is of the form G+ ^ E. In this case, we'll produce
an equivalence (G+ ^ E)hG ! (G+ ^ E)hG.
If X is a space, we almost have an equivalence (G+ ^ X)hG ' (G+ ^ X)hG
given by assembling the maps from subsection 8.1:
(G+ ^ X)hG (G+ ^ X)hG . (17)

'  
fflffl 
(G+ ^ X)G flhG

~  
=  
fflffl ~= ' fflffl
X _________//_Map(G+ , X)G____//Map(G+ , X)hG
Of course, if X is just a space, we do not know that flhG is a weak equivalence.
Before venturing into the stable world to correct this, we make the following
observation.
Remark 8.1. The weak equivalence we have constructed between (G+ ^
X)hG and Map (G+ , X)hG has a very encouraging property. Notice that the
35
following diagram commutes:
Map (G+O,OX)_______________ff_____________//_Map(G+O,OX)
 
 ' ' 
G+ ^ X _______//(G+ ^ X)hG_____//X____//_Map(G+ , X)hG
where the vertical arrow on the left hand side is the diagonal map sending
g ^ x to the constant map f(u) = gx (thisQmap is closely related toQfl). So
far, we have considered the map oe := g2GQg to be a self map of G X.
Here, we are using the identificationQof G X with Map (G+ , X). The image
of the constant map f(u) = gx under u2G u is the map h(u) = ugx, which
is exactly the image of g ^ x under the bottom row (off by a negative sign...?
change the shuffle map to twist by g1). In other words, we have shown that
this forms a piece of the Tate map.
Now, if E is an FSP then since the product and the coproduct in the
category of spectra agree, we have that the Gequivariant map fl induces a
G equivariant equivalence
hocolimn n(G+ ^ E( nX)) ! hocolimn nMap (G+ , E( nX)).
We would like to use this to produce the analogue of diagram 8.3 for FSP's.
Note that since the homotopy orbits are a homotopy colimit construction,
we have a (Gequivariant) equivalence
(G+ ^E)hG(X) := hocolimn n[(G+ ^E( nX)hG] ' [hocolim n n(G+ ^E( nX))]hG.
Using this, the equivalences
(G+ ^ E)hG(X) _______'__________// 1 E(X)
:= :=
fflffl fflffl
hocolimn n[(G+ ^ E( nX)hG] _'__//_hocolimn nE( nX)
and
hocolim n nE( nX) __'_//_Map(G+ , hocolimn nE( nX))hG
36
are induced by diagram 8.3, with 1 E(X) playing the role of X. Note that
since G is finite, we also have a Gequivariant equivalence
hocolim nMap (G+ , nE( nX)) _____//Map(G+ , hocolimn nE( nX)) .
Since G acts trivially on n, the adjunction
Map (G+ , nE( nX) ' nMap (G+ , E( nX)
is also Gequivariant. Assembling these, we have a Gequivariant equivalence
' n n hG hG
Map (G+ , 1 E(X))hG oo___[hocolimn Map (G+ , E( X))] := Map (G+ , E) (X*
*).
Now we can apply the equivalence induced by flhG to obtain
(G+ ^ E)hG(X) ` `` ` ` ` ` ` ` `` ` ` ` ` ` ` //`(G+ ^ E)hG(X)
 
 
'  'flhG
 
 
fflffl fflffl
1 E(X) _'_____//_Map(G+ , 1 E(X))hGo'o__Map (G+ , E(X))hG.
These assemble into a natural equivalence of functors with stablilization with
Gaction (G+ ^E)hG ' (G+ ^E)hG. Call the weak map representing the com
posite of weak equivalence we've just described : (G+ ^ E)hG ` ` ` ` ``//(G+ *
*^ E)hG .
Lemma 8.2.QThe equivalence (G+ ^ E)hG ' (G+ ^ E)hG factors the shuffle
map oe = g.
G
Proof.This follows from the fact that the diagram
fl ff fl
G+ ^ E ____'__//Map(G+ , E)_____//Map(G+O,OE)oo'_____G+ ^OXO
  
  
fflffl  fl 
(G+ ^ E)hG __'_____// 1 E______'_//Map(G+ , E)hGo'o__(G+ ^ X)hG
commutes. The rectangle on the left hand side commutes by remark 8.1 and
the fact that the weak equivalences indicated are induced by weak equiv
alences of spaces. The square on the right commutes because fl is a G
__
equivariant map. __
37
We now have enough information to construct the Tate (weak) map. Let
ß be the Gequivariant equivalence EG+ ^ E ! E. The Tate map is the
composite
EhGOOØ
' iØØ
V q+1
(EG+ ^ E)hG ______:=__//_[q] 7! ( G+ ^ E)hG
OOØ
' ØfØ
V q+1
[q] 7! ( Ø G+ ^ E)hG
' Ø 
V fflfflØ
[q] 7! ( q+1G+ ^ E)hG
g
V fflffl
oo__:=_____
[q] 7! ( q+1G+ ^ E)hG EG+ ^ EhG
' i
fflffl
EhG
where f and g are induced by the universal properties of the homotopy orbits
and homotopy fixed points. Note that g is not neccessarily an equivalence.
Remark 8.3. Technically, we have only defined the weak map on simplicial
level 0. We can extend to a simplicial map by using the extra degeneracy,
d0, with which EG is equipped (since it is the path space of BG). See for
example ([W94 ], section 8.3) for details.
Lemma 8.4. The Tate map is the restriction of the Norm map to homotopy
orbits. That is, the following diagram of weak maps commutes:
Q
______//Q ____g2Gg__//_Q oo'```` __+___//
E G E G E G E EOO (18)
 
fflfflfflffl ?Ø
EhG ` ` ` ` `` ` ` ` ` `` t`` ` ` ` `` ` ` ` ` `//EhG
where T refers to the composite that defines the Tate map.
38
Q
Proof.Using the equivalences G+ ^ E ' _G E ' G E ' Map(G+ , E), we
can rewrite the Norm map as
____//_ _ff_//_ +___//_
E G+ ^ E G+ ^ E E
where all of the maps are homotopic to the maps in diagram 18 above, but
composed with the appropriate weak equivalences.
Under the identification G+ ^ E ' _G E, the fold map is the same as the
projection map G+ ^ E ! E. Since the map ß : EG+ ^ E ! E is given
simplicially by the projections
q+1^
G+ ^ E ! E
ß is the fold map in simplicial degree 0. If one identifies G+ ^ E with
Map (G+ , E) instead, the diagonal map given (on the space level) by tak
ing x 2 E(X) to the constant map with value x is a homotopy inverse to
the fold map G+ ^ E ! E. Therefore, ß1 is the diagonal map in simplicial
degree 0.
For any space or FSP X, let X.denote the constant simplicial set whose
value in every simplicial dimension is X and whose face and degeneracy maps
are idX . We wish to show first that the weak map ß1 factors the diagonal
map : E. ! (G+ ^ E).Qwhich makes up the first part of the norm map
(recall that G+ ^ E ' G E). To do this, we want to show that
E.____________________________//_(G+ ^ E).
2FF2
22FFF 
22 FFF fflfflffl
22 F""F 1
22 E  ` `` ` i`` //` q+1V
22 . hG [q] 7! G+ ^ EhG
22 
22 ' ' 
22 fflffl
ßß2fflffl i1 q+1
(EhG).`` ` ` ` `//[q] 7! ( V G+ ^ E)hG
commutes. By remarkV8.3, we need only worry about simplicial map involving
EG+ := [q] 7! q+1G+  in simplicial degree 0, since such maps naturally
extend to all degrees by using d0. The map ffl is the inclusion of simplicial
q+1V
degree 0 of (G+ ^ E). into [q] 7! G+ ^ E, extended in this way, and
39
followed by the map to homotopy orbits. The square which comprises the
lower right hand corner of the diagram commutes by the equivariance of ,
and of ß. The triangle on the left hand side of the diagram commutes by
the universal properties of the colimits involved. In simplicial degree 0, the
outer diagram is
_______//_
E G+ ^ E
 
 
fflffli1 fflffl
EhG ____//_(G+ ^ E)hG
which commutes because the diagonal map is a homotopy inverse to ß, and by
the equivariance of ß. Once again, we extend this to a diagram of simplicial
maps by using d0 and this whole diagram commutes.
We already know that
G+ ^ E ___ff___//_G+O^OE
 
 
fflffl 
(G+ ^ E)hG `` `//(G+ ^ E)hG
commutes by Lemma 8.2. We can again extend this to a commuting diagram
of simplicial sets by remark 8.3.
Finally, dualize the argument used with the diagonal map to show that
(G+ ^OE).____________+_______________//_E.O<