MAHOWALDEAN FAMILIES OF ELEMENTS IN STABLE
HOMOTOPY GROUPS REVISITED
DAVID J. HUNTER AND NICHOLAS J. KUHN
May 30, 1997
1.Introduction
In the mid 1970's Mark Mahowald constructed a new infinite family of elements
in the 2-component of the stable homotopy groups of spheres, jj 2 ssS2j(S0)(2)[*
*M ].
Using standard Adams spectral sequence terminology (which will be recalled in x3
below), jj is detected by h1hj 2 Ext2;*A(Z=2; Z=2). Thus he had found an infi-
nite family of elements all having the same Adams filtration (in this case, 2),*
* thus
dooming the so-called Doomsday Conjecture. His constructions were ingenious: his
elements were constructed as composites of pairs of maps, with the intermediate
spaces having, on one hand, a geometric origin coming from double loopspace the-
ory, and, on the other hand, mod 2 cohomology making them amenable to Adams
Spectral Sequence analysis and suggesting that they were related to the new dis-
covered Brown-Gitler spectra [BG ].
In the years that followed, various other related 2-primary infinite families*
* were
constructed, perhaps most notably (and correctly) R.Bruner's family detected by
h2hj22 Ext3;*A(Z=2; Z=2) [B ]. An odd prime version was studied by R.Cohen [C ],
leading to a family in ssS*(S0)(p)detected by h0bj 2 Ext3;*A(Z=p; Z=p), and a f*
*iltra-
tion 2 family in the stable homotopy groups of the odd prime Moore space. Cohen
also initiated the development of odd primary Brown-Gitler spectra, completed in
the mid 1980's, using a different approach, by P.Goerss [G ], and given the ult*
*imate
"modern" treatment by Goerss, J.Lannes, and F.Morel in the 1993 paper [GLM ].
Various papers in the late 1970's and early 1980's, e.g. [BP , C, BC ], related*
* some
of these to loopspace constructions.
Our project originated with two goals. One was to see if any of the later wor*
*k on
Brown-Gitler spectra led to clarification of the original constructions. The ot*
*her
was to see if taking advantage of post Segal Conjecture knowledge of the stable*
* co-
homotopy of the classifying space BZ=p would help in constructing new families *
*at
odd primes, in particular a conjectural family detected by h0hj 2 Ext2;*A(Z=p; *
*Z=p).
(This followed a paper [K1 ] by one of us on 2 primary families from this point*
* of
view.)
What resulted, and what we do here, is the following. We isolate the two cruc*
*ial
results from the older literature (Proposition 2.1 and Proposition 2.2 below), *
*and
present these stripped of extraneous detours. We then reorganize how these resu*
*lts
____________
1991 Mathematics Subject Classification. Primary 55Q45; Secondary 55T15.
Research by the second author was partially supported by the N.S.F.
1
2 D.J.HUNTER AND N.J.KUHN
are used, together with the new idea that BZ=p should be an intermediate space
in the constructions. This leads to streamlined proofs of the main theorems of
[M , B, C]. In the odd prime case, improvements are most dramatic: generously
counting, we need only 30 of the 75 pages making up Cohen's original proof.
With respect to our original goals, we can report the following.
Regarding Brown-Gitler spectra, in some logical sense, nothing subsequently
learned about them helps simplify the construction of these sorts of homotopy
classes. Indeed, one of our highlighted older results, Proposition 2.2, is pre*
*cisely
what is also essential when one reviews the literature relating Brown-Gitler sp*
*ectra
to pieces of double loop spaces. However, our proof of this key proposition do*
*es
take advantage of some later observations: in particular, we use Carlsson modul*
*es
[Ca ], and the fact that these modules can be realized as the cohomology of cer-
tain mapping telescopes. In a companion paper [HK ], we will discuss this more
thoroughly.
Regarding odd primary classes, to recover the main theorems of [C ], we find *
*we
need to construct one element in the cohomotopy of BZ=p (see Proposition 2.3).
This we do using "elementary" methods and maps which would have all been avail-
able in the mid 1970's. We are not able to determine whether the elements h0hj
are permanent cycles at odd primes. However our parallel development of the
p = 2 and the odd prime cases suggest that they are not, and that a commonly
held view, that the odd prime elements h0hj 2 Ext2;*A(Z=p; Z=p) are analogous
to h1hj 2 Ext2;*A(Z=2; Z=2), is perhaps misguided. We find that the elements
h2hj 2 Ext2;*A(Z=2; Z=2), which are not permanent cycles, behave more similarly.
In x2, we quickly state three key propositions. Assuming these, the main theo-
rems of [M , B, C], and related results, are then formally deduced in x3. The p*
*roof
of each key proposition is then discussed in its own section. In a final short *
*section,
we deduce some related mod 2 results using elementary means (and note that odd
prime analogues don't exist).
In the rest of the paper, we are working in the stable homotopy category. Spe*
*ctra
are completed at p, and H*(X) means cohomology with Z=p coefficients, where the
prime p will be clear from the context.
Some of the results in this paper appeared in the first author's thesis [H ].
2.Three key propositions
To state our first two propositions, we need to define certain finite spectra*
* T (n).
Recall that, if X is a path connected space, there is a stable decomposition *
*[Sn]
_
22X ' D2;mX:
m>0
Here D2;mX = F (R2; m)+ ^m X[m], where F (R2; m) is the configuration space
of ordered m-tuples of distinct points in R2, Y+ denotes a space Y with a disjo*
*int
basepoint, and the symmetric group m acts in the obvious way on both F (R2; m)
and X[m], the m-fold smash product of X with itself. (We also let D2;0X = S0.)
Fixing a prime p, we then define T (n) for n 0 by S-duality:
T (2r + ffl) = 2pr+2fflDual(D2;pr+fflS1)
for all r 0, ffl = 0; 1.
MAHOWALDEAN CONSTRUCTIONS IN STABLE HOMOTOPY 3
Viewed as a module over the mod p Steenrod algebra A, H*(T (n); Z=p) is dual
to an appropriate Brown-Gitler module. When p = 2, this is [M , Thm.2.6]; when
p is odd, this is the content of the 10 page second chapter of [C ]. Our index*
*ing
has been chosen to be consistant with the modern literature (for example [Mi , *
*x7]),
so that H*(T (n); Z=p) is the injective envelope of H*(Sn; Z=p) in the category
of unstable A-modules1. In particular, H*(T (2pj); Z=p) has bottom classes in
dimensions 1 and 2 linked by the Bockstein, and top class in dimension 2pj.
Let C(g) denote the cofiber of a map g.
Proposition 2.1.
(1) (p = 2) There exist maps gj : S2j -!T (2j-3) with Sq2j acting nonzero
on H1(C(gj); Z=2).
(2) (p odd) There exist maps gj : S2(p-1)pj-! T (2pj-1) with Ppj acting
nonzero on H1(C(gj); Z=p).
When p = 2, this is [M , Thm.2(b)]. When p is odd, this is [C , Thm.IV.2.1].
(The maps in these references are S-dual to ours.)
Proposition 2.2.
(1) (p = 2) There exist maps fj : T (2j-3) -!BZ=2 with H1(fj; Z=2) 6= 0.
(2) (p odd) There exist maps fj : T (2pj-1) -!BZ=p with H1(fj; Z=p) 6= 0.
When p = 2, this was proved by Brown and Peterson in [BP , Lemma 4.1]. When
p is odd, this is [C , Thm.III.2.2]. (Starting from these references, readers w*
*ill have
to use properties of S-duality of finite complexes and manifolds to read off the
proposition as stated.)
Proposition 2.3.
(1) (p = 2) Let M = -3RP 2. There exists a map e : BZ=2 -!M with Sq4
acting nonzero on Hi(C(e); Z=2), i = -2; -1.
(2) (p odd) Let M = S4-2p[p D5-2p. There exists a map e : BZ=p -! M
with P1 acting nonzero on Hi(C(e); Z=p), i = (4 - 2p); (5 - 2p).
When p = 2, this was implicitly stated in [K1 , paragraph before Thm.4.4].
3.The main theorems
Let [X; Y ] denote the stable homotopy classes of maps between two connective,
p-complete, spectra X and Y . Recall (see, e.g. [R2 ]) that the classic Adams
spectral sequence arises from a filtration of [X; Y ], where a map f : X -! Y h*
*as
filtration at least s if it can be written as a composite
X -f1!Y1 f2-!: :-:!Y(s-1)fs-!Y
in which each H*(fi; Z=p) = 0. Intuitively, as the filtration increases, maps *
*are
harder to understand. The spectral sequence takes the form
Exts;tA(H*(Y ; Z=p); H*(X; Z=p)) = Es;t2) [t-sX; Y ]:
____________
1It is miraculous that it is unstable.
4 D.J.HUNTER AND N.J.KUHN
Using standard notation, Ext1;*A(Z=2; Z=2) is spanned by elements
j
hj 2 Ext1;2A(Z=2; Z=2); j 0;
with hj corresponding to the indecomposable Sq2j 2 A. Adams [A ] showed that
d2(hj) = h0h2j-1and deduced that the permanent cyles in Ext1;*A(Z=2; Z=2) are
spanned by h0, h1, h2, and h3. These correspond to the classic Hopf maps 2 2
ssS0(S0), j 2 ssS1(S0), 2 ssS3(S0), and oe 2 ssS7(S0), respectively.
For p odd, Ext1;*A(Z=p; Z=p) is spanned by a0 2 Ext1;1A(Z=p; Z=p) and elements
j
hj 2 Ext1;2(p-1)pA(Z=p; Z=p); j 0;
with a0 and hj respectively corresponding to fi and Ppj 2 A. Then d2(hj) = a0bj*
*-1,
j
where bj-1 2 Ext2;2(p-1)pA(Z=p; Z=p) is the p-fold Massey product
(see [Liu]). The filtration 1 permanent cycles are spanned by a0 and h0, which
represent the elements p 2 ssS0(S0) and ff 2 ssS2p-3(S0).
Theorem 3.1.
(1) (p = 2)
(a)The map gj is represented by i1*(hj) 2 Ext1;*A(H*(T (2j-3)); Z=2).
(b)The composite S2j gj-!T (2j-3) -ss!T (2j-3)=S1 is represented by
i2*(h2j-1) 2 Ext2;*A(H*(T (2j-3)=S1); Z=2).
(2) (p odd)
(a)The map gj is represented by i1*(hj) 2 Ext1;*A(H*(T (2pj-1)); Z=2).
(b)The composite S2(p-1)pjgj-!T (2pj-1) -ss!T (2pj-1)=S1 is repre-
sented by i2*(bj-1) 2 Ext2;*A(H*(T (2pj-1)=S1); Z=p).
Here i1 : S1 -!T (2pj) and i2 : S2 -!T (2pj)=S1 are inclusions of bottom cell*
*s.
Proof.Both parts (a) are just reformulations of Proposition 2.1, noting that, e*
*.g.
when p is odd, dimensions considerations show thatjthe only primary operation
that can connect H*(T (2pj-1); Z=p) to H*(S2(p-1)p; Z=p) in H*(C(gj); Z=p) is
Ppj. Then both parts (b) follow from this, using the factorizations of Sq2j and
Ppj into a sum of primary operations composed with secondary operations (see
[A , Liu]). Once again, dimension considerations show that the only secondary
operations in this decomposition that can act nontrivially on H*(C(gj); Z=p) are
the ones associated to h2j-1(when p = 2) and bj-1 (when p is odd). __|_|
Theorem 3.2.
(1) (p = 2)
(a)The composite fj O gj : S2j -! BZ=2 is represented by i1*(hj) 2
Ext1;*A(H*(BZ=2); Z=2).
(b)The composite S2j fjOgj---!BZ=2 -ss!(BZ=2)=S1 is represented by
i2*(h2j-1) 2 Ext2;*A(H*((BZ=2)=S1); Z=2).
(2) (p odd)
(a)The composite fjOgj : S2(p-1)pj-!BZ=p is represented by i1*(hj) 2
Ext1;*A(H*(BZ=p); Z=p).
MAHOWALDEAN CONSTRUCTIONS IN STABLE HOMOTOPY 5
(b)The composite S2(p-1)pjfjOgj---!BZ=p ss-!(BZ=p)=S1 is represented
by i2*(bj-1) 2 Ext2;*A(H*((BZ=p)=S1); Z=p).
As before, i1 : S1 -! BZ=p and i2 : S2 -! (BZ=p)=S1 are inclusions of bottom
cells.
Proof.Using naturality properties of the Adams spectral sequence with respect
to composition, this is an immediate consequence of Theorem 3.1 and Proposi-
tion 2.2. __|_|
Remarks 3.3.
(1) When p = 2, these last two theorems appear in work by Cohen, J.Jones, and
Mahowald [CJM , Cor.4.7 and the argument on p.118], and they then go on to easi*
*ly
conclude that the composite
j fjOgj
S2 - --! BZ=2 -!BSO(2)
is represented by i2*(h2j-1) 2 Ext2;*A(H*((BSO(2)); Z=2), where i2 : S2 -!BSO(2)
is the inclusion of the bottom cell. (Since 2MSO(2) ' BSO(2), this is then in-
terpreted as a result about the Kervaire invariant of 2j - 2 dimensional orient*
*ed
manifolds immersed in Euclidean space with codimension 2.) Though they pro-
ceed roughly as we have here, we have a couple of quibbles about their argument*
*s.
Firstly, they don't deduce part (1)(b) of Theorem 3.1 from part (1)(a) as we do.
They use instead compatibility of the maps gj with respect to various pairings,*
* a
method that fails at odd primes. Secondly, they use essentially circular reason*
*ing in
their argument for the existence of fj: they argue that fj as in Proposition 2.*
*2 exists
by using the fact that T (2j) was shown to be an appropriate S-dual of a Brown-
Gitler spectrum by Brown and Peterson in [BP ]. But this theorem of Brown and
Peterson was itself proved by using the existence of maps fj.
(2) When p is odd, Theorem 3.1(2)(b) essentially appears as [C , Cor.III.3.6],
with an argument like ours. However, Cohen never combines Theorem 3.1(2) with
Proposition 2.2(2) to deduce Theorem 3.2(2). Indeed, Proposition 2.2(2) is only
used by him as a technical lemma enroute to proving that the family of spectra
T (2pr + 2); r 0; are S-dual to odd primary Brown-Gitler spectra (that he has
constructed).
(3) When p is odd, the action of Z=(p - 1) ' (Z=p)x on Z=p induces the "eigen-
spectra" stable decomposition
BZ=p ' Z(1) _ . ._.Z(p - 1);
indexed so that Z(j) has bottom cell in dimension 2j - 1. (Z(p - 1) is the p-
localization of Bp.) Obviously, both Proposition 2.2(2) and Theorem 3.2(2) can
be refined by replacing BZ=p by Z(1).
If one is interested in constructing families of elements in the stable homot*
*opy
groups of spheres, Theorem 3.2 suggests hunting for elements in the cohomotopy *
*of
BZ=p and (BZ=p)=S1 that are tractable and nontrivial when respectively restrict*
*ed
to their bottom cells S1 and S2.
6 D.J.HUNTER AND N.J.KUHN
Certainly the simplest and best known element in ss*S(BZ=p) is the Kahn-Priddy
tZ=p
map t : BZ=p -! S0 defined as the composite BZ=p -! BZ=p+ ---! S0, where
tZ=p is the Z=p-transfer.
When p = 2, t restricted to S1 is j, which is detected by h1 2 Ext1;*A(Z=2; Z*
*=2).
One recovers Mahowald's original jj family.
Theorem 3.4. The composite (t O fj O gj) 2 ssS2j(S0) is represented by h1hj 2
Ext2;*A(Z=2; Z=2).
Proof.By dimension reasons, t has Adams filtration 1, and not 0. Thus t will
be represented by an element "h12 Ext1;*A(Z=2; H*(BZ=2)) such that i*("h1) =
h1 2 Ext2;*A(Z=2; Z=2). By Theorem 3.2(1)(a), (fj O gj) is represented by i*(hj*
*) 2
Ext1;*A(H*(BZ=2); Z=2). Thus (t O fj O gj) will be represented by "h1i*(hj) =
i*("h1)hj = h1hj. __|_|
Now consider the same construction when p is odd. In contrast to the even
case, t restricted to S1 is null. Indeed, under the decomposition of BZ=p given*
* in
Remarks 3.3(3), t factors through the summand Z(p-1) ' Bp. Thus, if the map
fj in Proposition 2.2 has been chosen to land in the Z(1) summand, the composite
(t O fjO gj) will be 0.
The maps e of Proposition 2.3 amount to next simplest maps out of BZ=p. The
next proposition is simply a convenient reformulation of Proposition 2.3.
Proposition 3.5.
(1) (p = 2) Let M = -3RP 2. There exists a diagram
i1 _____- _______oei2
S1 _________-BZ=2 (BZ=2)=S1 S2
| | | j j
| |e |e0 j
| | | j
|? i |? ss |? j+
S-2 __________M- __________-S-1
in which the left square commutes, e0 is induced by e so that the middle
square commutes, and the right triangle commutes up to multiplication by
an odd integer.
(2) (p odd) Let M = S4-2p[p D5-2p. There exists a diagram
i1 _____- _______oei2
S1 _________BZ=p- (BZ=p)=S1 S2
| | | j j
|ff |e |e0 j
| | | j ff
|? i |? ss |?j+
S4-2p _________-M _________S5-2p-
in which the left square commutes, e0 is induced by e so that the middle
square commutes, and the right triangle commutes up to multiplication by
an integer prime to p.
MAHOWALDEAN CONSTRUCTIONS IN STABLE HOMOTOPY 7
In each part of this proposition, the lower horizontal maps form the obvious
cofibration sequence, and we note that, for all primes p, the projection map ss*
* has
order p.
Once again, dimension reasons imply that the maps e and e0have Adams filtra-
tion 1, and not 0. Thus combined with Theorem 3.2, Proposition 3.5 implies the
following theorem.
Theorem 3.6.
(1) (p = 2) Let M = -3RP 2.
(a)The composite (e O fj O gj) : S2j -!M is represented by i*(h2hj) 2
Ext2;*A(H*(M); Z=2).
(b)The composite (ss O e O fjO gj) 2 ssS2j+1(S0) is an element of ord*
*er 2
represented by h2h2j-12 Ext3;*A(Z=2; Z=2).
(2) (p odd) Let M = S4-2p[p D5-2p.
(a)The composite (eOfjOgj) : S2(p-1)pj-!M is represented by i*(h0hj) 2
Ext2;*A(H*(M); Z=p).
(b)The composite (ss O e O fj O gj) 2 ssS2(p-1)pj+2p-5(S0) is an elem*
*ent
of order p represented by h0bj-1 2 Ext3;*A(Z=p; Z=p), up to multipl*
*i-
cation by an element in (Z=p)x .
Part (1)(b) of this theorem says that, at the prime 2, h2h2j-1is a permanent
cycle representing an element of order 2 in ssS*(S0). This is the main theorem *
*of
[B ]. This proof of Bruner's theorem is roughly that of [CJM , Thm.4.12] and [K*
*1 ,
Thm.5.1], except that [CJM ] don't recover that the elements have order 2, and *
*[K1 ]
more awkwardly shows this.
Part (2) of Theorem 3.6 says that, at an odd prime p, h0bj-1 is a permanent c*
*ycle
representing an element of order p in ssS*(S0) and that i*(h0hj) is a permanent*
* cycle
in the Adams spectral computing the stable homotopy groups of a mod p Moore
space. These are the two main theorems of [C ], but our proof here is not Cohen*
*'s:
as mentioned in the introduction, our argument is roughly 45 pages shorter than
his.
Remark 3.7.It has been widely thought that the family of elements h0hj is the a*
*na-
logue, at odd primes, of the 2 primary family h1hj 2 Ext2;*A(Z=2; Z=2). Thus, s*
*ince
h1hj is a permanent cycle when p = 2, it has been conjectured that h0hj should *
*be
a permanent cycle at odd primes2. Our constructions indicate that the h0hj fam-
ily would be better deemed analogous to the family h2hj 2 Ext2;*A(Z=2; Z=2).
Since these are not permanent cycles, we conjecture that neither are h0hj 2
Ext2;*A(Z=p; Z=p).
There is an easy way to use parts (a) of Theorem 3.6 to construct more infini*
*te
families of permanent cycles in Ext*;*A(Z=p; Z=p). Suppose given ffi 2 ssSd(S0*
*) of
____________
2Indeed the main theorem of [CG ] asserts precisely this. Unfortunately, N.M*
*inami has pointed
out that the first sentence on p.186 of [CG ] is incorrect in an essential way,*
* and we ultimately
learn nothing about the existance of odd primary jj from these authors' efforts.
8 D.J.HUNTER AND N.J.KUHN
order p. Then, when p is odd, ffi factors:
ffi
S4-2p _____-S4-2p-d
| j jj3
|i j ffi0
|?j
M;
and there is an analogous diagram if p = 2.
If ffi is represented by d 2 Exts;d+sA(Z=p; Z=p), we might hope that the comp*
*osite
(ffi0OeOfjOgj) 2 ssS2(p-1)pj+2p+d-4(S0) will be represented by dh0hj through Ad*
*ams
filtration s + 2, so that dh0hj is a permanent cycle. (This would be dh2hj if p*
* = 2.)
This will be the case if the lift ffi0can be chosen so that it also has Adams f*
*iltration s,
and is represented by an element d02 Exts;d-sA(Z=p; H*(M)) satisfying i*(d0) = *
*d.
The following corollary, a strengthening and generalization to all primes of [K*
*1 ,
Thm.4.4], contains an easy to verify condition ensuring that this will be the c*
*ase.
Corollary 3.8.Suppose that ffi 20ssSd(S0)0is represented by d 2 Exts;d+sA(Z=p; *
*Z=p)
and pffi = 0. Suppose that ExtsA;d+s +1(Z=p; Z=p) contains only permanent cycles
for s0 s - 1.
(1) (p = 2) Then dh2hj 2 Exts+2;*A(Z=2; Z=2) will be a permanent cycle, for
all j.
(2) (p odd) Then dh0hj 2 Exts+2;*A(Z=p; Z=p) will be a permanent cycle, for
all j.
The proof in [K1 ] works for all primes without change.
Example 3.9. Let p be odd. The element fi2 2 ssS4p2-2p-4(S0) has order p and
is represented by k0 = 2 Ext2;2(p-1)(2p+1)A(Z=p; Z=p) [R2 , p.205]*
*. We
conclude that k0h0hj 2 Ext4;*A(Z=p; Z=p) will be a permanent cycle, for all j.
Remark 3.10.By construction, in the situation above, the composite (ffi0OeOfjOg*
*j)
will be an element in the Toda bracket . Thus, even when *
*the Ext
condition of Corollary 3.8 fails, one still might be able to deduce the Adams s*
*pectral
sequence name for this composite using work of Moss [Mo ] (though checking his
Ext conditions will require having some control over certain infinite families *
*of Ext
groups3).
4.An outline of the proof of Proposition 2.1
Here, for completeness, we briefly sketch the construction of the maps gj of
Proposition 2.1. In discussing this construction, and even more essentially, t*
*he
construction of the maps fj, it is useful to recall the following lemma from [C*
*MM ].
Lemma 4.1. D2;m2X ' 2mD2;mX:
____________
3Checking Moss' conditions wasn't done in [L2, L3], but perhaps could be, co*
*mpleting Lin's
arguments.
MAHOWALDEAN CONSTRUCTIONS IN STABLE HOMOTOPY 9
Recall that we defined T (2r) to be 2prDual (D2;prS1). Using the lemma, we
can write
T (2r) ' 2(p-1)prDual(D2;prS(2p-3)):
When p is odd, the construction of the maps gj then goes as follows.
Let O be the infinite orthogonal group, and let Q1S0 the component of the
identity map in QS0 = colimn-!1nSn. Viewing ff 2 ssS2p-3(S0) as an unstable
map ff : S2p-3 -!Q1S0, ff factors through the J-homomorphism:
0 J
S2p-3 ff-!O -! Q1S0:
Using the infinite loopspace structure on O, ff0has a canonical double loop ext*
*ension
ff00: 2S2p-1 -!O;
and we let ff: (2S2p-1)+ -! S0 be the stable map adjoint to the composite of
based unstable maps
00 J
(2S2p-1)+ -ff-!O+ -! QS0:
The map gj is then defined to be the 2(p - 1)pjth suspension of the S-dual of
the composite
D2;pjS2p-3 ,! (2S2p-1)+ -ff!S0:
When p = 2 there is a similar construction, starting with a map oe0 : S7 -! O
lifting oe 2 ssS7(S0).
The assertion of Proposition 2.1, that appropriate cohomology operations act
nontrivially in the cohomology of the cofibers, is proved in a couple of pages *
*in
[C , M ] using characteristic class arguments. We know of no improvement upon
these authors' arguments.
5.An outline of the proof of Proposition 2.2
In this section, we outline the proof of Proposition 2.2. Though this will ba*
*sically
be the same as in [BP , C], with the idea going back to [M ], we will take adva*
*ntage
of work in the last 15 years on U, the category of unstable A-modules. This, we
feel, greatly clarifies the presentation.
5.1. algebraic results. Let J(n) = H*(T (n); Z=p). Then J(n) is an unstable
module, and for all M 2 U, we have a natural isomorphism:
(5.1) Hom A(M; J(n)) ' (Mn)*:
By Yoneda's lemma, given a 2 Ad, the natural transformation a. : Mn -! Mn+d
will induce a map of A-modules
.a : J(n + d) -!J(n):
To give a unified discussion for all primes, we adopt the convention that if *
*p = 2,
Pn 2 A denotes Sq2n. Then one has the so-called "Mahowald exact sequences".
Lemma 5.1. [S, Prop.2.2.3] There is an exact sequence
n
0 -!J(pn - 1) -!J(pn) .P--!J(n) -!0:
10 D.J.HUNTER AND N.J.KUHN
By (5.1) above, one sees that J(n) is an injective object in U. This is no lo*
*nger
true when one regards J(n) in the category of all A-modules, but it suggests th*
*at
perhaps even in that category, one might have some control on an injective reso*
*lu-
tion of J(n). The following critical lemma is a reflection of this.
Lemma 5.2. In the category of A-modules, there exist injective resolutions,
0 -!J(n) -!I0(n) -!I1(n) -!I2(n) -!: :;:
and chain maps under .Pn : J(2pn) -!J(2n),
fln : I*(2pn) -!I*(n);
with the following property: if M is an unstable A-module,
fln : Hom A(M; tIs(2pn)) -!Hom A (M; tIs(2n))
is zero for all t - s < 0.
This is "almost" in the literature. When p = 2, this is roughly [BC , Lemma
2.3(i)], though a form of it appears earlier in [M , proof of Lemma 5.6], and i*
*n Brown
and Gitler's original article [BG , Lemma 2.8]. Cohen proves an odd prime versi*
*on
strong enough for our applications in [C , Cor.III.3.6]. See also [H , Prop.5.2*
*.5]. In
all of these references, explicit resolutions are constructed, using quotient s*
*ubcom-
plexes of the Lamda algebra.
The lemma immediately implies
Theorem 5.3. If M is an unstable A-module,
(.Pn)* : Exts;tA(M; J(2pn)) -!Ext s;tA(M; J(2n))
is zero for all t - s < 0.
Remark 5.4.When s = 0, the theorem reduces to the statement that, if M is un-
stable, then Pn : M2n-t -!M2pn-tis zero for all t > 0. This is, of course, (part
of) the unstable condition.
We now continue as did G.Carlsson in [Ca ]. Let K(2n) be the unstable module
defined as the inverse limit
n .Ppn .Pp2n
K(2n) = lim{J(2n) .P--J(2pn) --- J(2p2n) ---- : :}::
By (5.1), for all M 2 U, there is a natural isomorphism
n Ppn Pp2n
(5.2)Hom A(M; K(2n)) ' (colim{M2n P--!M2pn- -! M2p2n---! : :}:)*:
Theorem 5.3 has the following consequence.
Corollary 5.5.If M is an unstable A-module,
Exts;tA(M; K(2n)) = 0; for allt - s < 0:
MAHOWALDEAN CONSTRUCTIONS IN STABLE HOMOTOPY 11
5.2. topological results. The topological input we need is given by the next
lemma.
Lemma 5.6. There exists a map : T (2n) -! T (2pn) such that * = .Pn :
J(2pn) -!J(2n).
When p = 2 this is proved in [CMM ]. A similar method can be used at odd
primes [H , Thm.5.1.1], and we will be outline the construction of below. A odd
prime version sufficient for our purposes appears as [C , Thm.III.4.1]4.
Now define spectra "T(2n) to be the telescopes
"T(2n) = hocolim{T (2n) -!T (2pn) -!T (2p2n) -!: :}::
By construction, H*(T"(2n); Z=p) ' K(2n), as A-modules.
Now suppose X is a space with H*(X; Z=p) of finite type. Consider the Adams
spectral sequence {Es;tr} that computes maps from T"(2n) to 1 X. By Corol-
lary 5.5, we see that
Es;t2= 0 ift - s < 0:
Thus Es;s2will consist of permanent cyles for all s 0. By (5.2),
E0;02' limjH2pjn(X; Z=p):
We conclude
Theorem 5.7. In this situation, the natural map
[T"(2n); 1 X] -!limjH2pjn(X; Z=p)
is onto.
Proof of Proposition 2.2.H*(BZ=p; Z=p) ' (x) Z=p[y], where x is one dimen-
sional and fi(x) = y. From this, one deduces that
limjH2pj(BZ=p; Z=p) ' Z=p:
Choosing a nonzero element of this inverse limit, Theorem 5.7 says that there e*
*xists
a corresponding stable map f : "T(2) -!BZ=p. Such a map will be nonzero in mod
p homology in dimensions 1 and 2. With ffl equal to 1 if p is odd and 3 if p = *
*2, let
fj be the composite
T (2pj-ffl) -!T"(2) f-!BZ=p:
By construction, these maps have the needed property. __|_|
We end this section with a sketch of the construction of the map appearing in
Lemma 5.6.
Recall that, if X is path connected, the Milgram-May model for 22X comes
equipped with a natural filtration, and if Fm (22X) 22X denotes the mth
stage of this, D2;mX = Fm (22X)=Fm-1 (22X).
____________
4However, Cohen's proof relies on [C, Prop.II.1.2] which, following traditio*
*n on this point, he
incorrectly asserts is proved in [CLM ].
12 D.J.HUNTER AND N.J.KUHN
Let j : S2r+1-! S2pr+1be the pth Hopf invariant. Fixing n as in the lemma,
if r is chosen sufficiently large, purely dimension reasons imply that
j : 2S2r+1-! 2S2pr+1
will carry Fp2n(2S2r+1) to Fpn(2S2pr+1), and Fp2n-1(2S2r+1) to Fpn-1(2S2pr+1).
There is thus an induced map
D2;p2nS2r-1-! D2;pnS2pr-1:
Recalling Lemma 4.1, the map of Lemma 5.6 is defined to be the appropriate
S-dual of this.
Remarks 5.8.
(1) Complete proofs of both Lemma 5.2 and Lemma 5.6 will appear in [HK ].
(2) In the p = 2 case, generalizations of the telescopes "T(2n) are constructed*
* in the
second author's paper [K2 ], based on using S-duals of pieces of higher loopspa*
*ces.
Algebraically there appear to be analogues of the maps fj, with BZ=2 replaced by
higher Eilenberg-MacLane spaces K(Z=2; m). A heuristic argument is proposed,
which, if it can be made rigorous, would give an alternative proof of Propositi*
*on 2.2
avoiding all Adams spectral sequence arguments and Brown-Gitler module tech-
nology.
6.The proof of Proposition 2.3
We begin this section with some notation.
If F is a finite complex with S-dual D(F ), we let : S0 -! F ^ D(F ) be the
duality copairing.
We let Mn denote the mod p Moore spectrum Sn-1 [pDn. Note that the S-dual
of Mn is M1-n.
With these conventions, we proceed to define the maps e of Proposition 2.3.
Definition 6.1.Let n = 2 when p = 2, and let n = 2p - 4 if p is odd. Then
e : BZ=p -!M1-n is defined to be the composite
BZ=p 1^---!BZ=p ^ Mn ^ M1-n -^1--!M1-n;
where : BZ=p ^ Mn -! S0 is the composite
tZ=p 0
BZ=p ^ Mn -i^i-!B(Z=p x Z=p)+ -B(add)---!BZ=p+ ---! S :
Here maps labelled "i" are inclusions. We remind the reader that the 2p - 2
skeleton of BZ=p is stably M2 _. ._.M2p-4_M2p-2. (At odd primes, these Moore
spaces correspond to the bottom cells of the spectra Z(j) of Remarks 3.3(3).)
To compute cohomology operations in H*(C(e); Z=p) as needed in Proposi-
tion 2.3, we first observe that, by construction, there is a diagram of cofibra*
*tion
sequences:
MAHOWALDEAN CONSTRUCTIONS IN STABLE HOMOTOPY 13
M1-n ________________C(e)-_______________BZ=p-
| | |
| | |
(6.1) |1 | |
| | |
|? i |? ss |?
M1-n ___________C(tZ=p)-^ M1-n _____(BZ=p+-) ^ M1-n
where the bottom sequence is just the cofibration sequence
S0 -i!C(tZ=p) ss-!(BZ=p+ )
smashed with M1-n, and is the composite
BZ=p 1^---!BZ=p^Mn ^M1-n -i^i^1--!B(Z=pxZ=p)+ ^M1-n -B(add)^1-----!BZ=p+ ^M1-n:
Having made this observation, the A-module structure of H*(C(e); Z=p) can be
easily computed, as the A-module structure of H*(C(tZ=p); Z=p) is well known,
and it is routine to calulate in cohomology. We sketch the details.
First assume p = 2. Then H*(C(tZ=2); Z=2) is the submodule of nonnegative
degree elements in Z=2[x; x-1], and the Steenrod operations act by Sqi(xk) =
k k+i i -1
ix . Let mi2 H (M ; Z=2); i = -2; -1 be generators.
We wish to show that Sq4 acts nonzero on Hi(C(e); Z=2), for i = -2; -1. By
(6.1), it is equivalent to show that *((ss*)-1(Sq4oe(x-1 mi))) 6= 0, where oe
denotes suspension. Since Sq4(x-1 mi) = x3 mi, we just need to check that
*(x3mi) 6= 0. By the next lemma, *(x3m-2) = 32x = x, and *(x3m-1) =
3 2 2
1x = x .
k k
Lemma 6.2. *(xk m-1) = 1xk-1 and *(xk m-2) = 2xk-2.
When p is odd, H*(C(tZ=p); Z=p) is the submodule of nonnegative degree ele-
ments in (x)Z=p[y; y-1], and the Steenrod operations act by fi(yi) = 0, fi(x) =
y, and Pi(xfflyk) = kixfflyk+(p-1)i. Let mi2 Hi(M5-2p; Z=p), i = 4 - 2p; 5 - 2*
*p be
generators chosen to be dual to xyp-3 and yp-2 in H*(M2p-4; Z=p) respectively.
We wish to show that P1 acts nonzero on Hi(C(e); Z=2), for i = 4 - 2p; 5 - 2p.
By (6.1), it is equivalent to show that *((ss*)-1(P1oe(xy-1 mi))) 6= 0, where *
*oe
denotes suspension. Since P1(xy-1 mi) = xyp-2 mi, we just need to check that
*(xyp-2 mi) 6= 0. By the next lemma, *(xyp-2 m4-2p) = p-2p-2x = x, and
*(xyp-2 m5-2p) = - p-2p-3y = 2y.
k k
Lemma 6.3. *(xykm4-2p) = p-2xyk+2-p, *(xykm5-2p) = - p-3 yk+3-p,
k
*(yk m4-2p) = p-2yk+2-p, and *(yk m5-2p) = 0.
Remark 6.4.It is an exercise in the properties of the transfer to show that the
composite
tZ=p 0
B(Z=p x Z=p)+ -B(add)---!BZ=p+ ---! S
agrees with the composite
B(Z=p x Z=p)+ -t! BZ=p+ -c!S0;
14 D.J.HUNTER AND N.J.KUHN
where c is the collapse map and t is the transfer associated to the diagonal i*
*nclusion
: Z=p -! Z=p x Z=p. From this it follows that e : BZ=p -! M1-n as defined
here is the map BZ=p -i!BZ=p+ -s!D(BZ=p+ ) -Di!D(Mn ), where s is the map
arising in the Segal conjecture: it is adjoint to B(Z=p x Z=p)+ -t! BZ=p+ -c!S0:
Related to this, we note that, when p is odd, the S-dual of Z(1) is Z(p - 2).
7.Two more maps out of (BZ=2)=S1
Theorem 3.2 suggests hunting for maps out of BZ=p which are nonzero when
restricted to S1, and hunting for maps out of (BZ=p)=S1 which are nonzero when
restricted to S2. Using the solution of the Segal Conjecture for the group Z=p *
*(as
in [L1], [R1 ]), one can systematically study such maps. When p = 2, this was d*
*one
in [K1 ], and a similar analysis is possible when p is odd [H ].
Proposition 2.3, of course, gives examples of such maps, constructed by "elem*
*en-
tary" means. Here we note that there are two more 2 primary maps that can be
constructed easily. Like t : BZ=2 -!S0, these have no odd prime analogues.
The first should be compared with Proposition 2.3(1) (or Proposition 3.5(1)).
Proposition 7.1.There exists a map eC : (BZ=2)=S1 -!-7CP 2with Sq8 act-
ing nonzero on H-5(C(eC ); Z=2). Equivalently, there exists a commutative diagr*
*am
i2
S2 _____-(BZ=2)=S1
| |
| |
|oe |eC
| |
| |
|? i |?
S-5 _____--7CP 2:
The map eC is defined using the (desuspended) S1-transfer tS1 : BS1+-! S-1.
Definition 7.2.eC : (BZ=2)=S1 -!-7CP 2is defined to be the composite
(BZ=2)=S1 1^---!(BZ=2)=S1 ^ CP 2^ -7CP 2-^1--!-1DCP 2= -7CP 2;
where : (BZ=2)=S1 ^ CP 2-! S-1 is the composite
tS1 -1
(BZ=2)=S1 ^ CP 2-j^i-!B(S1 x S1)+ -B(add)---!BS1+--! S :
Here i is the obvious inclusion, and j : (BZ=2)=S1 -! BS1+is any extension of
the composite of obvious maps BZ=2 -!BS1 -!BS1+.
To prove the proposition, one computes the action of Sq8 on H-5(C(eC ); Z=2)
using the method of the last section. With i : S2 -!CP 2denoting the inclusion *
*of
the bottom cell, a consequence is
Theorem 7.3. The composite (eC O ss O fj O gj) 2 ssS2j+7(CP 2) is represented*
* by
i*(h3h2j-1) 2 Ext3;*A(H*(CP 2); Z=2).
MAHOWALDEAN CONSTRUCTIONS IN STABLE HOMOTOPY 15
This theorem is essentially due to W.H.Lin [L3, last paragraph of p.136], and
leads to an easy to check criterion for deducing that a family of the form dh3h*
*2j-12
Ext*;*A(Z=2; Z=2) is a permanent cycle [K1 , Thm.5.3].
Our last map arises in the following way. Let ffi : (BZ=p)=S1 -! S2 be defined
by the cofibration sequence
S1 i1-!BZ=p -!(BZ=p)=S1 ffi-!S2:
Then the composite S2 -i2!(BZ=p)=S1 -ffi!S2 has degree p, and, thanks to Theo-
rem 3.2, we can conclude that, when p = 2, h0h2j-12 Ext3;*A(Z=2; Z=2) is a perm*
*a-
nent cycle, and, when p is odd, a0bj-1 2 Ext3;*A(Z=p; Z=p) is a permanent cycle*
*. Due
to the Hopf invariant one differential (see x3), neither of these facts is new.*
* However,
the next lemma shows that, when p = 2, ffi lifts through ss : -2CP 2-! S2.
Lemma 7.4. At the prime 2, j O ffi = 0. At odd primes, ff O ffi 6= 0.
Proof.By diagram chasing, it is easily seen that j Offi = 0 if and only if j : *
*S1 -!S0
extends to a map BZ=2 -! S0, and it does. Similarly, at odd primes, ff O ffi = *
*0 if
and only if ff : S1 -!S4-2p extends to a map BZ=p -!S4-2p, and it doesn't. __|*
*_|
Thanks to the lemma, there is an interesting map ffi0 : (BZ=2)=S1 -! -2CP 2
(and no odd prime analogue). [K1 , Thm.5.5] has a criterion for using the compo*
*sites
ffi0O ss O fjO gj : S2j-! -2CP 2, together with Toda bracket methods, to constr*
*uct
infinite families of permanent cycles in Ext*;*A(Z=2; Z=2). For example, (P h2)*
*h2j-12
j+16
Ext7;2A (Z=2; Z=2) is a permanent cycle [K1 , Ex.5.6].
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Department of Mathematics, University of Virginia, Charlottesville, VA 22903
Current address, D.Hunter: Department of Mathematics, North Central College, *
*Naperville,
IL 60540
E-mail address, N.Kuhn: njk4x@virginia.edu