ITERATES OF THE SUSPENSION MAP AND MITCHELL'S
FINITE SPECTRA WITH Ak-FREE COHOMOLOGY
GREG ARONE
June 16, 1998
Abstract. We study certain cross-effects of the unstable homotopy of sphe*
*res.
These cross-effects were constructed by Weiss, for different purposes, in*
* the con-
text of "Orthogonal calculus". We show that Mithchell's finite spectra wi*
*th Ak-
free cohomology (constructed in [Mt85]) arise naturally as stabilizations*
* of Weiss'
cross-effects. In particular, we find that after a suitable Bousfield loc*
*alization, our
cross-effects, which capture meaningful information about the unstable ho*
*motopy
of spheres, are homotopy equivalent to the infinite loop spaces associate*
*d with
Mitchell's spectra. This last result is a partial generalization of the m*
*ain result of
Mahowald and the author in [AM97 ].
0. Introduction
Let X be a topological space. We are interested in studying the difference be*
*tween
the homotopy type of X and that of S2X, the double suspension of X (for techni-
cal reasons the statements turn out to be a little cleaner if one works with do*
*uble
suspensions rather than single suspensions). One naive way to compare X and S2X
would be by means of the map X ! S2X given by taking the smash product of the
identity map on X with the inclusion S0 ! S2. Of course, this map is null-homot*
*opic
and thus is unlikely to provide useful information about the difference between*
* X
and S2X. A much better idea is to consider the Freudental (double) suspension m*
*ap
w1 : X ! 2S2X. Let F1(X) be the homotopy fiber of this map. We regard F1(X)
as measuring the difference between the homotopy type of X and that of S2X. We
would like to iterate the process and find higher analogues of the suspension m*
*ap
and higher iterated differences of the suspension map. Thus at the second stage
we would like to find a suitable way to compare F1(X) with F1(S2X). Obviously,
there exists a natural map F1(X) ! 2F1(S2X). However, this map turns out to
be null-homotopic, just as the map X ! S2X is null-homotopic. It follows from t*
*he
___________
1991 Mathematics Subject Classification. 55P40, 55P42, 55P65.
Key words and phrases. spheres, periodic homotopy, orthogonal calculus.
The author is partially supported by the NSF
1
2 GREG ARONE
remarkable analysis performed in [W95 ], that there exists a natural transforma*
*tion
w2 : F1(X) ! 4F1(S2X), extending the naive map F1(X) ! 2F1(S2X), which
deserves to be called the analogue of the map X ! 2S2X.
Remark 0.1. the map F1(X) ! 4F1(S2X) had been constructed in [C83 ], in the
case of X being an odd-dimensional sphere localized at 2. The construction in [*
*C83 ]
uses specific calculations in unstable homotopy of spheres. However, Weiss' cal*
*culus
tells us that the map exists for very general reasons.
Let F2(X) be the homotopy fiber of w2. Iterating Weiss' construction, one obt*
*ains
a sequence of functors F0; : :;:Fm ; : :(:where F0(X) = X) equipped with natural
maps
wm : Fm-1 (X) ! 2mFm-1 (S2X)
such that there are fibration sequences
F1(X) ! F0(X) w1!2F0(S2X)
F2(X) ! F1(X) w2!4F1(S2X)
..
.
Fm (X) ! Fm-1 (X) wm!2mFm-1 (S2X)
..
.
We think of Fm (-) as the m-th iterated difference, or the m-th cross-effect,*
* of the
double suspension map.
Our next step is to investigate the layers in the Goodwillie towers of the fu*
*nctors
Fm (X), first for a general space X and then for X an odd-dimensional sphere lo*
*calized
at a prime. We will assume some familiarity with Goodwillie's theory of "Taylor
towers". The references for this material are [G90 , G92 , G3 ]. See also [Jo*
*95] for
an exposition of some of the material in [G3 ]. Let Dn(X) be the n-th layer in*
* the
Goodwillie tower of the identity. By layers of the Goodwillie tower we will usu*
*ally
mean not the infinite loop space that is the actual homotopy fiber in the tower*
*, but
the associated spectrum. We recall the description of Dn(X) from [Jo95, AM97 ]
Dn(X) ' Map *(Kn; 1 X^n) hn
where Kn is (the double suspension of) the geometric realization of the poset of
(non-trivial) partitions of a set with n elements.
For a general functor F let DnF be the n-th layer in the Goodwillie tower of *
*F and
let PnF be the n-th "Taylor polynomial" of F in the sense of Goodwillie. Finall*
*y, let
U(n) be the unitary group on n letters. We think of n as a subgroup of U (n - 1)
via the reduced standard representation. The following theorem is essentially [*
*W95 ,
Example 5.7].
HIGHER ITERATES OF THE SUSPENSION MAP 3
Theorem 0.2. There is a natural equivalence
DnFm (X) ' Map *(Kn; 1 X^n) ^hn (U(n - 1)= U(n - m - 1)+ )
A couple of notational remarks are in order. First, if G is a finite group, X*
* and Y
are spaces with an action of G on the right and the left respectively then by X*
* ^hG Y
we mean X ^ EG+ ^G Y . This still makes sense if X is a spectrum with an action
of G. Second, if k < 0 then U(n)= U(k) is the empty space. In other words, DnFm*
* is
non-trivial only if n > m, and the bottom non-trivial layer of Fm is
i j
Dm+1 Fm (X) ' Map * Km+1 ; 1 X^m+1 ^m+1 (U(m)+ )
(here we may use strict orbits instead of homotopy orbits, because the action o*
*f m+1
on U (m) is free).
It is easy to see that the map wm induces an equivalence on the bottom non-tr*
*ivial
layers, and that the mapping telescope of wm consists only of the bottom layers*
*. We
can summarize this in the following diagram, where the last column lists the ma*
*pping
telescopes of the raws:
X !w1 2S2X w1! 4S4X ! . . .1 D1F0(X)
F1(X) !w2 4F1(S2X) w2! 8F1(S4X) ! . . .1 D2F1(X)
.. .
. ..
Fm-1 (X) wm! 2mFm-1 (S2X) wm! 4mFm-1 (S4X) ! . . .1 Dm Fm-1 (X)
.. . .
. .. ..
Our next goal is to analyze in further detail the layers of Fm (X) in the spe*
*cial case
when X is a sphere localized at p and to relate them to familiar objects in hom*
*otopy
theory. For the rest of the introduction, X stands for an odd-dimensional sphe*
*re,
and cohomology is always taken with mod p coefficients. The layers of F0 are t*
*he
spectra
Map *(Kn; 1 X^n) hn
and these were studied extensively in in [AM97 ] and [AD97 ] in the case of X *
*being
an odd-dimensional sphere. The plan is to extend those results to the spectra
(1) Map *(Kn; 1 X^n) ^hn (U (n - 1)= U(n - m - 1)+ )
by induction on m.
The following was proved in [AM97 ] (in a somewhat different formulation)
Theorem 0.3. Let X be an odd-dimensional sphere. Let n > 1. If n is not a power
of a prime then the spectrum
Map *(Kn; 1 X^n) hn
4 GREG ARONE
is contractible. Suppose n = pk for some prime p. Then the integral cohomology *
*of
this spectrum is all p-torsion. The mod p cohomology is freely generated over A*
*k-1
by a polynomial algebra on k-generators. More precisely, there is an isomorphis*
*m of
Ak-1-modules (up to a degree shift depending on the dimension of X)
i !
kj
H * Map * Kpk; 1 X^p h ; Fp ~=Ak-1 P
pk
Here P is a free module on one generator over the polynomial algebra
Fp[d0; : :;:dk-1]
where |dj| = 2pk - 2pj.
Here Ak-1 is the sub-algebrakof-the1mod p Steenrod algebra generated by the e*
*le- k-2
ments Sq1; Sq2; Sq4; : :;:Sq2 if p = 2 and by the elements fi; P 1; P p; : :;*
*:P p if
p is odd. The polynomial generators dj above should be thoughtiof as thejDickson
polynomials evaluated at the polynomial generators of H * B (Z=pZ)k; Fp (squares
i k j
of the polynomial generators of H * B (Z=pZ) ; Fp if p = 2). By the Dickson pol*
*y-
nomials we mean the polynomials that give the generators of the Dickson algebra
of invariants of Fp[y1; : :;:yk]GLk(Fp)(see [Wil83 ]). The generators dj can a*
*lso be
thought of as the Chern classes of the reduced regular representation of (Z=pZ)*
*k and
thus dj = cpk-pj.
The following theorem will be proved as part (b) of theorem 2.2
Theorem 0.4. Let X be an odd-dimensional sphere. The spectrum
Map *(Kn; 1 X^n) ^hn (U (n - 1)= U(n - m - 1)+ )
is contractible if n is not a power of a prime. Suppose n = pk, then there is *
*an
isomorphism (up to a dimension shift) of Ak-1-modules
i i kj i j j
H* Map * Kpk; 1 X^p ^hpk U (pk - 1)= U(pk - m - 1)+; Fp ~=
~= Ak-1 E P
where E is a free module on one generator over the exterior algebra
< __ci| pk - m i pk - 1 and i is not of the formpk - pj >
(here |__ci| = 2i - 1), and P is a free module on one generator over the polyno*
*mial
algebra
Fp[cpk-pj| pj > m]
Thus the exterior generators are precisely the Chern classes in U(pk-1)= U(pk-m*
*-1)
that are not Dickson classes, and the polynomial generators are the Dickson cla*
*sses
in B U(pk - m - 1).
HIGHER ITERATES OF THE SUSPENSION MAP 5
Consider the bottom layer of Fm (X) that is non-trivial for odd-dimensional s*
*pheres.
It follows from theorem 0.4 that it is the pk-th layer, where k is the smallest*
* integer
such that m pk - 1. Moreover, by theorem 0.4, the cohomology of this layer, as*
* an
Ak-1-module, is given by
i i kj i j j
H* Map * Kpk; 1 X^p ^hpk U (pk - 1)= U(pk - m - 1)+; Fp ~=
~= Ak-1 E
where E is a free module on one genrator over the exterior algebra
< __ci| pk - m i pk - 1 and i 6= pk - pj >
in other words, for the cohomology of the bottom non-trivial layer, the polynom*
*ial
part is trivial. In particular, the mod p cohomology of the bottom layer is a *
*finite
Ak-1-free module. It is not clear if for a general m the bottom layer is in fa*
*ct
homotopy equivalent to a finite spectrum. However, if we take m = pk - 1, then *
*the
bottom layer is homotopy equivalent to
i kj
Map * Kpk; 1 X^p ^pk (U (pk - 1)+ )
(the point is that we can use strict orbits instead of homotopy orbits, as was *
*noticed
earlier in the paper). Clearly, this is a finite spectrum, whose mod p cohomol*
*ogy
is an Ak-1 free module (if X is an odd-dimensional sphere). The existence of su*
*ch
spectra was at one time an important question in homotopy theory. The question
was motivated by the "chromatic philosophy" in homotopy theory, as such spectra
are natural candidates to be spectra of type k. The first one to construct such*
* spectra
was S. Mitchell in [Mt85 ]. It turns out that our finite spectrum above and Mit*
*chell's
Ak-1 free spectrum are "essentially the same". We formulate this imprecise stat*
*ement
as a "pre-theorem"
Pre-Theorem 0.5. Let X be an odd-dimensional sphere. The spectrum
i kj
Map * Kpk; 1 X^p ^pk (U (pk - 1)+ )
is very closely related to the Ak-1-free spectrum constructed in [Mt85 ]. In fa*
*ct, our
spectra are more or less Thom spectra over Mitchell's spectra.
We will give more precise statements in section 2. For the time being, we pre*
*tend
that instead of "very closely related" we have "equivalent". Thus Mitchell's sp*
*ectra
play a role in unstable homotopy theory. The infinite loop space associated wi*
*th
Mitchell's Ak-1-free spectrum can be thought of as the "principal part" (i.e., *
*the
bottom layer in the Goodwillie tower) of Fpk-1(X), when X is an odd-dimensional
sphere.
These results have an interpretation in terms of the chromatic filtration of *
*homo-
topy theory. As explained in [AM97 ], the cohomological properties of the lay*
*ers of
6 GREG ARONE
the functors Fm , together with the fact that when evaluated at spheres the tow*
*er
converges exponentially faster than in general, imply the following theorem
Theorem 0.6. Let X be an odd-dimensional sphere localized at p. The map
Fm (X) ! PpkFm (X)
induces an equivalence in "vi-periodic homotopy" for i k.
Taking k to be the smallest integer such that PpkFm (X) in non-trivial (for X*
* an
odd-sphere localized at p) we obtain the following special case as a corollary:
Corollary 0.7. Let X be an odd-dimensional sphere localized at a prime p. Let m
be a non-negative integer an let k be the smallest integer such that m pk-1. T*
*hen
Fm (X) is trivial in vi-periodic homotopy for i < k and the composed (weak) map
Fm (X) ! PpkFm (X) ' 1 DpkFm (X)
induces an equivalence in vk-periodic homotopy.
The corollary says that a certain unstable object, namely Fm (X) where X is an
odd-dimensional sphere, is equivalent in vk periodic homotopy (where k is the s*
*mallest
integer such that m pk- 1) to the infinite loop space of a spectrum of type k.*
* This
spectrum is always finite in mod p cohomology, and if m = pk - 1 then it actual*
*ly is
a finite spectrum (the Mitchell spectrum).
It is easy to see, for instance, that for m = 1 and any p, DpF1(X) is the spe*
*ctrum
realizing one copy of A0, and in fact it is the mod p Moore spectrum. By coinci*
*dence,
if m = 2, p = 2, D4F2(X) turns out to be the spectrum whose cohomology realizes
one copy of A1. However, for all other values of m and p, the cohomology of the
bottom non-trivial layer has more than one copy of Ak-1.
Remark 0.8. For m = 0, corollary 0.7 is essentially Serre's theorem to the effe*
*ct
that if X is an odd-dimensional sphere then the map X ! 1 1 X is a rational
homotopy equivalence (v0-periodic homotopy is, essentially, rational homotopy).*
* For
m = 1 the corollary is due to Mahowald ([M82 ]). For p = 2 and m = 2 the coroll*
*ary
is due to Mahowald and Thompson ([MT94 , Theorem 1.5]).
As a concluding remark, notice that since the map wm induces an equivalence on
the m-th layers, we obtain the following corollary
Corollary 0.9. Let Xkbe an odd-dimensional sphere localized at p. Let m = pk. T*
*he
map wpk : Fpk-1! 2p Fpk-1(S2X) induces an equivalence in vk-periodic homotopy.
On the other hand, if X is an odd sphere localized at p and m is not a power
of p, then wm seems to be far from being an equivalence. Preliminary calculati*
*ons
suggest that in this case wm induces the zero map in homology and in all the Mo*
*rava
K-theories. In fact, preliminary calculations suggest the following conjecture
HIGHER ITERATES OF THE SUSPENSION MAP 7
Conjecture 0.10. If X is an odd-dimensional sphere localized at p, and m is no*
*t a
power of p then the map
wm : Fm-1 (X) ! 2mFm-1 (S2X)
is zero on homotopy groups.
In any case, it seems that for X an odd sphere, the values of m for which Fm-1
and wm are most interesting are powers of primes.
The rest of the paper is organized as follows: In section 1 we review the re*
*le-
vant points in Weiss' orthogonal calculus and explain why theorem 0.2 is implic*
*it in
[W95 ]. In section 2 we discuss the relationship with Mitchell's spectra and ma*
*ke the
"pretheorem" precise.
Acknowledgements: It is obvious that I owe a lot to Weiss' paper [W95 ]. I wo*
*uld
like to thank Goodwillie for suggesting that Weiss' ideas should help my effort*
*s on
Mahowald's program to construct higher iterates of the suspension map.
It is a pleasure to acknowledge my immeasurable debt to M. Mahowald. His
suggestions were a crucial motivating force for this work.
1. Weiss' calculus
Let F be the category whose objects are finite-dimensional complex vector spa*
*ces
with positive-definite inner-product and morphisms are linear maps respecting t*
*he
inner product. Weiss' calculus [W95 ] is concerned with continuous functors fro*
*m F
to spaces (=compactly generated topological spaces with non-degenerate basepoin*
*t).
In fact [W95 ] works with real rather than complex vector spaces, but since we *
*want to
work with double suspensions we will use complex vector spaces. Typical example*
*s of
functors that we will be interested in are V 7! U(V ), V 7! B U(V ) and V 7! V *
*SV X.
In the last example, X is a fixed based space, SV is the one-point compactifica*
*tion
of V and V stands for continuous maps from SV .
Let G : F ! Spaces* be a functor. Define G1(V ) to be the homotopy fiber of
the map G(V ) ! G(V C). Clearly, G1 is again a functor of F. An important
insight of [W95 ] is that the natural map G1(V ) ! G1(V C) lifts to a natural *
*map
G1(V ) ! 2G1(V C), where the map 2G1(V C) ! G1(V C)) is given by
evaluation at zero. In fact, a little more is true:
Lemma 1.1. Let G1 be as above. Let W be an object of F. There exists a functor
F ! Spaces* given on objects by V 7! V G1(W V ) such that the natural map
G1(W ) ! V G1(W V ) lifts the map G1(W ) ! G1(W V ).
Now let X be a space and take G(V ) = V SV X. Let F1(X) be the homo-
topy fiber of the map X ! 2S2X. In the language of the previous paragraph,
F1(X) = G1(C0). Similarly, 2F1(S2X) is identified with G1(C). By lemma 1.1,
there is a natural map G1(C0) ! 2G1(C). Rewriting this map in terms of F1, we
obtain that F1 comes equipped with a natural map F1(X) ! 4F1(S2X) (lifting
8 GREG ARONE
the obvious map F1(X) ! 2F1(S2X)). Let F2(X) be the homotopy fiber of this
map. Repeating the argument above, one easily finds that F2(X) comes equipped
with a natural map F2(X) ! 6F2(S2X). Letting F3(X) be the homotopy fiber
of this map and continuing inductively, one obtains the sequence of functors Fm*
* (X)
together with fibration sequences Fm (X) ! Fm-1 (X) ! 2mFm-1 (S2X) promised
in the introduction.
Let Cn be the n-th Goodwillie derivative of F0(X) = X. Thus Cn is a spectrum
with an action of the symmetric group n, and the n-th layer of the Goodwillie t*
*ower
of F0(X) is the (infinite loop space associated with) the spectrum
(Cn ^ X^n) hn
Our next task is to describe the layers of the Goodwillie tower of the functor *
*Fm in
term of the layers of the Goodwillie tower of the functor F0. Of course the des*
*cription
that we are looking for is of the form (Cmn^ X^n)hn for some n-spectrum Cmn.
Moreover, the natural map Fm-1 (X) ! 2mFm-1 (S2X) induces a map on the layers
(Cm-1n^ X^n)hn ! 2m(Cm-1n^ X^n ^ S2n)hn
which is determined by some n-equivariant map
Cm-1n^ X^n ! 2mCm-1n^ X^n ^ S2n
(where the action of n is trivial on the 2m, and is given by the standard compl*
*ex
representation on S2n). We will describe this n-equivariant map.
Lemma 1.2. The n-th layer of the Goodwillie tower of Fm (X) is contractible f*
*or
n m. Assume that n > m. Then
DnFm (X) ' (Cn ^ X^n) ^hn (U (n - 1)= U(n - m - 1)+ )
Moreover, on the level of the n-th layers, the fibration sequence Fm (X) ! Fm-1*
* (X) !
2mFm-1 (S2X) is induced by the following n-equivariant fibration/cofibration se-
quence of spectra:
U(n - 1)= U(n - m - 1)+ ! U(n - 1)= U(n - m)+ ! U(n - 1)= U(n - m) n S2(n-m)
where U(n - 1)= U(n - m) n S2(n-m)is the Thom complex of the tautological n - m-
dimensional complex bundle over U (n - 1)= U(n - m).
Proof.This is the content of [W95 , Example 5.7], except that we use complex ra*
*ther
than real vector spaces. __|_ |
In particular, the bottom non-trivial layer of Fm is the m + 1-th layer, and *
*it is
given by Cm+1 ^m+1 ((U (m)+ ) ^ X^m+1.) (here we may use strict orbits instead
of homotopy orbits, because the action of m+1 on U (m) is free). Recalling from
[Jo95, AM97 ] that Cm+1 (the m + 1-th Goodwillie derivative of the identity) is*
* given
HIGHER ITERATES OF THE SUSPENSION MAP 9
by Cm+1 = Map *(Km+1 ; 1 S0), we obtain that the bottom layer of Fm (X) is given
by the infinite loop space of
Map *(Km+1 ; 1 X^m+1 ) ^m+1 U(m)+
as stated in the introduction.
2.Cohomology and relation with Mitchell's spectra
Throughout this section, X stands for an odd-dimensional sphere. Our goal is *
*to
further analyze the layers of Fm (X) in this case. Thus we want to study the sp*
*ectra
Map *(Kn; 1 X^n)^hn (U (n-1)= U(n-m-1)+ ). In particular, we will be interested
in their cohomology. We recall the following facts from [AM97 , AD97 ]:
Theorem 2.1. Let n > 1, X and odd sphere. The spectrum
Map *(Kn; 1 X^n)hn
is contractible rationally. Moreover, it is contractible mod p unless n is a po*
*wer of p.
It follows immediately from theorem 2.1, lemma 1.2 and induction on m that the
same statement holds for all the layers of Fm for all m. Thus, if X is an odd s*
*phere
then the spectrum
DnFm (X) ~=Map *(Kn; 1 X^n) ^hn (U (n - 1)= U(n - m - 1)+ )
is contractible rationally for all n > 1 and is contractible mod p unless n is *
*a power
of p. We may, therefore, take n = pk and concentrate on the mod p homotopy type
of
k k k
Map *(Kpk; 1 X^p ) ^hpk (U (p - 1)= U(p - m - 1)+ )
Start with the casekm = 0. In this case, what one gets is the familiar spectrum
Map *(Kpk; 1 X^p )hpk . We recall from [AD97 ] that there is a smaller model f*
*or
this spectrum. To describe this model, let Tk be (the double suspension of) the
geometric realization of the category of (strict, non-zero) vector subspaces of*
* Fkp,
(which is of course the same as the category of strict non-zero subgroups of (Z*
*=pZ)k).
Extend the action of GL k(Fp) on Tk to an action of the affine group Affk(Fp) :=
GL k(Fp) n (Z=pZ)k by letting (Z=pZ)k act trivially. There is a map Tk ! Kpk
determined by sending a subgroup P of (Z=pZ)k to the partition determined by the
quotient map (Z=pZ)k i (Z=pZ)k=P . It is easily checked that this map is equiva*
*riant
with respect to the group inclusion Affk(Fp) ,! pk. The main result of [AD97 ]*
* is
that this map induces a mod p equivalence (only if X is an odd-dimensional sphe*
*re,
of course)
k 1 ^pk
(2) Map *(Kpk; 1 X^p )hpk ! Map *(Tk; X )h Affk(Fp)
10 GREG ARONE
Recall that the subgroup (Z=pZ)k of Affk(Fp) acts trivially on Tk. It follows*
* that
k i 1 ^pk j
Map *(Tk; 1 X^p )h Affk(Fp)' Map * Tk; (X )h(Z=pZ)kh GL
k(Fp)
k k
Moreover, since X is a sphere, X^ph(Z=pZ)kis a Thom space over B (Z=pZ) . Let *
*us
denote it (B (Z=pZ)k)fl. Thus, there is an equivalence
i kj i i jflj
Map * Tk; 1 X^p h Aff ' Map Tk; 1 B (Z=pZ)k
k(Fp) h GLk(Fp)
Now recall that Tk is (non-equivariantly) homotopy equivalent to a wedge of sph*
*eres,
and its only non-trivial homology group realizes the Steinberg representation of
GL k(Fp). Since the Steinberg representation is projective and self-dual (in c*
*har-
acteristic p),ithe cohomologyjof this spectrum is the image of the cohomology o*
*f the
fl
Thom space B(Z=pZ)k under the action of the Steinberg idempotent. The image
i j
of H * B(Z=pZ)k itself under the action of the Steinberg idempotent is well kn*
*own.
It was first calculated by Mitchell and Priddy in [MtP83 ], and then again in a*
* more
"algebraic" fashion (at theiprime 2) byjCarlise and Kuhn in [CK89 ]. In particu*
*lar,
it is well-known that St H* B (Z=pZ)k is free over Ak-1. Same holds if one rep*
*laces
B(Z=pZ)k with the Thom complex of any multiple of the regular representation. T*
*he
calculation is a modification of the well-known calculations of Mitchell and Pr*
*iddy
cited above. A detailed account will appear either in the final version of [AD9*
*7 ] or
in a separate note. In fact, as stated in theorem 0.3, there is an isomorphism,*
* up to
a dimension shift, of Ak-1-modules
i i jflj
H * Map * Tk; 1 B (Z=pZ)k h GL ~=Ak-1 P
k(Fp)
where P is a free module on one generator over the (doubled, if p = 2) Dickson
algebra Fp[d0; : :;:dk-1].
This finishes the discussion of the case m = 0. To extend it to all values of*
* m, i.e.,
to analyze the spectra
i kj i j
Map * Kpk; 1 X^p ^hpk U (pk - 1)= U(pk - m - 1)+
where X is an odd-dimensional sphere, we use lemma 1.2 and induction on m.
Theorem 2.2. Let X be an odd-dimensional sphere. The spectrum
Map *(Kn; 1 X^n) ^hn (U (n - 1)= U(n - m - 1)+ )
is contractible if n is not a power of a prime. Suppose n = pk, then:
(a) The map Tk ! Kpk induces an equivalence
i kj
Map * Kpk; 1 X^p ^hpk (U (pk - 1)= U(pk - m - 1)+ ) !
HIGHER ITERATES OF THE SUSPENSION MAP 11
i kj
! Map * Tk; 1 X^p ^h Affk(Fp)(U (pk - 1)= U(pk - m - 1)+ )
(b) There is an isomorphism (up to a dimension shift) of Ak-1-modules
i i kj i j j
H* Map * Kpk; 1 X^p ^hpk U (pk - 1)= U(pk - m - 1)+; Fp ~=
~= Ak-1 E P
where E is a free module on one generator on the exterior algebra
< __ci| pk - m i pk - 1 and i is not of the formpk - pj >
with |__ci| = 2i-1 and P is a free module on one generator over the polynomial *
*algebra
Fp[cpk-pj| pj > m].
(c) If m is not a power of p then the map wm induces the zero map on the mod
p cohomology of the layers. If m = pj then wm induces the inclusion of the id*
*eal
generated by dj.
Proof.The proof is by induction on m, using lemma 1.2 for the induction step. F*
*or
m = 0 part (a) is the equivalence (2) and parts (b) and (c) are given by theore*
*m 0.3.
Part (a) for a general m follows from part (a) for m - 1. Part (c) for a gener*
*al m
follows from part (b) for m - 1 and elementary considerations about characteris*
*tic
classes. Part (b) for m follows from part (b) for m - 1 and part (c) for m. __*
*|_ |
Taking m = pk-1 in part (a) of theorem 2.2, we find that the spectrum DpkFpk-1(*
*X)
is equivalent to the image under the Steinberg idempotent of the suspension spe*
*ctrum
of a certain Thom space over the homogeneous space (Z=pZ)k\ U(pk - 1). This is
almost precisely Mitchell's construction in [Mt85 ] of a finite Ak-1-free spect*
*rum,
except he does not take a Thom space and that he uses the special orthogonal ra*
*ther
than the unitary group, but this makes very little difference. This is what we *
*meant
in the "pretheorem" in the introduction.
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(G. Arone) Department of Mathematics, University of Chicago, 5734 University
Avenue, Chicago, Il 60637, USA
E-mail address: arone@math.uchicago.edu