ContemporaryVMathematicsolume 00, 0000
Mark Mahowald's work on the
homotopy groups of spheres
H. R. MILLER AND D. C. RAVENEL
July 22, 1992
In this paper we attempt to survey some of the ideas Mark Mahowald has
contributed to the study of the homotopy of spheres. Of course, this represents
just a portion of Mahowald's work; some other aspects are described elsewhere
in this volume. Even within the restricted area of the homotopy of spheres, this
survey can only touch on some of Mahowald's most seminal contributions, and
will leave aside many of his ideas on the subject. On the other hand we will
try to set the stage upon which Mahowald has acted, so we give brief reviews of
certain parts of homotopy theory in Sections 1 and 4. This includes the Image
of J, the EHP sequence, and the Adams spectral sequence. Of course we will
not attempt an exhaustive survey of the relevant history of homotopy theory;
for more information, the reader may look at G. W. Whitehead's "Fifty years of
homotopy theory" [88 ] or at Chapter One of the second author's book [76 ].
We will base our account on a discussion of three of Mahowald's most influ
ential papers: The Metastable Homotopy of Sn (1967), A new infinite family in
2ssS*(1977), and The Image of J in the EHP sequence (1982).
One of Mahowald's jokes is that in his world there are only two primes: 2,
and the "infinite prime." We will always work localized at 2, unless obviously
otherwise.
Both authors are happy to have this occasion to thank Mark for the many
exciting and fruitful interactions we have had with him.
This manuscript is missing some arrows, on pages 9, 11, 14, and 23.
____________
1991 Mathematics Subject Classification. Primary 55Q40, 55Q50.
Key words and phrases. Homotopy groups of spheres, EHP sequence, Jhomomorph*
*ism.
Both authors were supported in part by the N. S. F.
cO00000American0Mathematical0Societ*
*y00000/00 $1.00 + $.25 per page
1
2 H. R. MILLER AND D. C. RAVENEL
1. The Context, I
Understanding of the homotopy groups of spheres in 1967 could be gathered
under three rather separate heads: the Image of J, the EHP sequence, and the
Adams spectral sequence.
1.1. Bott periodicity and the Jhomomorphism. Raoul Bott's proof
[12 ] of the periodicity theorem was published in 1959. This is a computation
of the homotopy groups of the classical groups, in a range of dimensions which
increases linearly with the rank. It can be phrased so as to give all the homot*
*opy
groups of the "stable group," which is the evident union. This represented the
first time all the homotopy groups of a space with infinitely many nontrivial
kinvariants had been computed; and it is still the starting point for most such
knowledge.S For example, the homotopy of the stable orthogonal group O =
O(n) is of period 8 and is given by the following table, which may be sung
(from the bottom up) to the tune of a wellknown lullaby [26 ].
_______________
_i_mod_8_ssiO_
 7 Z 
 6 0 
 5 0 
 4 0 
 3 Z 
 2 0 
 1 Z2 
____0____Z2__
Generators for these groups may be constructed explicitly, for example using
Clifford algebras.
The relevance of these groups to the homotopy groups of spheres was well
known from the construction of the "Jhomomorphism" by G. W. Whitehead
[87 ] in 1942. Since O(n) acts by proper maps on Rn, its action extends to Sn
viewed as the onepoint compactification of Rn. This action has the point at
infinity as a fixedpoint, and so we get an inclusion of O(n) into the monoid F*
* (n)
of all pointed selfequivalences of Sn. F (n) embeds into nSn as the components
of degree 1, but we desire a pointed map (with the constant map as basepoint
of nSn) and so instead we map ff 2 O(n) to the loopdifference ff  1 2 nSn.
These maps are compatible up to homotopy as n increases, and in the limit they
give a map j : O ! QS0, which in homotopy induces the Jhomomorphism
J : ssiO ! ssSi:
Great geometric interest attaches to the image of this map; in terms of framed
bordism, for example, the image is the set of elements which can be represented
by framed spheres. It is very unusual that one has at hand explicitly construct*
*ed
homotopy classes, defined without the indeterminacy usually associated with a
MAHOWALD'S WORK ON HOMOTOPY GROUPS 3
spectral sequence. Despite the tremendous importance of these classes, they
seem to have no standard names. Following Mahowald,1 we will label them
so that a specific generator of the jth nonzero homotopy group in O maps to
the element fij. The first few of these elements are the wellknown Hopf maps:
fi1 = j, fi2 = , fi3 = oe.
In 1966 Adams's On the groups J(X)_IV [4] was published. This paper
consolidated and organized a number of ideas, in part due independently to M.
G. Barratt [6] and H. Toda [86 ], which were to become increasingly important
in later years. In it he determined the order of the image of J up to a factor
of 2 in dimensions of the form 8k  1. The remaining factor of 2 led Adams to
propose the celebrated Adams Conjecture. We will return to these ideas in more
modern form in x4.
Adams's method was also important. He constructed a certain KOtheory
analogue of the Hopf invariant, replacing Steenrod operations by Adams oper
ations, and housed it in a suitable Ext1 group. This is the einvariant, and it
detects the image of J. A byproduct of this method was that (if the Adams con
jecture holds) the image of J is in fact a direct summand in ssS*. One may also
consider the "zerothorder" KOtheory invariant, assigning to a map its induced
map in KOtheory. Adams showed that this detects elements as well, in dimen
sions congruent to 1 and 2 mod 8. These are the and j elements, and Adams
constructed them by means of Toda brackets. He also reformulated these con
structions in terms of iterates of a Ktheory selfequivalence of a Moore space;
and this idea of periodic families was to become one of Mahowald's principal
interests.
1.2. Toda's suspensiontheory. In 1962, Hirosi Toda's book Composition
Methods in Homotopy Groups of Spheres [84 ] appeared. This book is based upon
the "EHP sequence" of I. M. James [38 ], which Toda had extended in suitable
form to odd primes. James had obtained this sequence by constructing a map,
the "Hopf invariant" H : Sn ! S2n1, whose homotopy fiber is (localized at
2) Sn1 included into Sn by the suspension map. The long exact sequence in
homotopy has the form
. .H.!ssi+1S2n+1 P!ssi1Sn E!ssiSn+1 H!ssiS2n+1 P!. .:.
Toda wrote for the map P , which Mahowald always calls the "Whitehead
product"; the precise relationship with the Whitehead product is ([87 ], p. 548)
that for fl 2 ssin1Sn1,
P (En+2fl) = [n; Efl]:
In particular, P (2n+1) is the Whitehead square
wn = [n; n] 2 ss2n1Sn:
____________
1This is the convention used in [47] and in [10], but in [50] the numbering *
*differs by 1, so
that there fi1 = , etc.
4 H. R. MILLER AND D. C. RAVENEL
Toda explored this sequence as an inductive mechanism for computing unsta
ble homotopy groups of spheres. Given a nonzero element ff 2 ssiSn+1 one may
inquire after its "EHP history":
o What sphere was ff born on, and what is the Hopf invariant fl of a
maximal desuspension?
o What sphere will ff die on (or is it immortal, that is, stable), and what
element fi has P fi equal to the maximal nonzero suspension?
The class fl is the "birth certificate" of ff, and fi is its "death certificate*
*."
The kstem consists of the groups ssk+nSn together with the suspension maps
between them. It is also convenient to indicate the Hopf invariant of an element
when it is nonzero, and a class fi such that P fi = ff if Eff = 0.
The EHP sequence, together with the connectivity of S2n+1, implies Freuden
thal's theorem that E : ssk+nSn ! ssk+n+1Sn+1 is an isomorphism for n > k+1.
This is the stable range of the kstem. The EHP sequence also shows that when
n > 1_3(k + 1), the obstructions to E : ssk+nSn ! ssk+n+1Sn+1 being an isomor
phism lie in the stable range; this is called the metastable range of the kste*
*m.
In his book Toda computed the kstem for k 19, with some ambiguity about
the behavior of P and H. His method involved the construction of elements
with specified Hopf invariant by means of compositions and Toda brackets, or
by means of an extension of the EHP sequence in the metastable range, in which
the suspension map E is replaced by the iterated suspension. In particular when
E is replaced by the stabilization map
ssk(X) ! lim!ssk+i(iX) = ssSk(X)
i
then one has the Toda stabilization sequence: for k < 3n  3, there is an exact
sequence 1
. ..!ssSk+1Pn T!ssn+kSn E!ssSkI!ssSkPn ! . .:.
Here ssS*is the stable homotopy ring ssS*(S0) and Pn is the stunted projective
space
Pn = RP 1=RP n1:
The map T is a form of the Whitehead product map P , but we refrain from
using this letter yet again. In modern terms, this exact sequence arises from
identifying the homotopy fiber of nSn ! QS0 with QPn through a range of
dimensions. It is related to the EHP sequence in the metastable range by the
commutative ladder ([47 ], p. 51)
. . . ssSk+1Snx ! ssSk+1Pn!? ssSk+1Pn+1!? ssSkSnx . . .
??~= ?yT ?yT ??~=
. .s.sn+k+2S2n+1P!ssn+kSn E! ssn+k+1Sn+1H! ssn+k+1S2n+1. . .
in which the top sequence is associated to the evident cofibration sequence
Sn ! Pn ! Pn+1.
MAHOWALD'S WORK ON HOMOTOPY GROUPS 5
1.3. The Adams spectral sequence. In 1958 Frank Adams [1] revolution
ized the way stable homotopy groups were studied, with his introduction of the
Adams spectral sequence. This spectral sequence effectively broke the problem
of computing the pprimary part of these groups into three parts: First, one co*
*n
structs an estimate of the stable homotopy which depends functorially on the
action of Steenrod operations on the mod p cohomology_this is the E2term in
the spectral sequence. It is expressed homologically in terms of the algebra A *
*of
all Steenrod operations:
Es;t2(X) = Exts;tA(H*(X); Fp):
This E2term is an upper bound, and the second step is to cut it down to
the correct size (via differentials in the spectral sequence). Finally, one "so*
*lves
the extension problems," passing from the associated graded group back to the
actual stable homotopy, with as much structure as desired.
This spectral sequence opened the door to a vast computational effort. The
theses of J. P. May [65 ] (1964, under J. C. Moore) and of M. C. Tangora [82 ]
(1966, under Mahowald) resulted in extensive computations of the E2term and
considerable information about higher differentials. Work by Mahowald and
Tangora [59 ] contemporaneous with the Memoir resulted in a rather complete
picture of the Adams spectral sequence through the 45stem. A few oversights
were corrected by Barratt, R. J. Milgram [67 ], Barratt, Mahowald and Tangora
[11 ], G. W. Whitehead, and (in 1983) by R. Bruner [22 ]. One of the great re
sources Mahowald quickly came to be able to draw on was an intimate familiarity
with this large family of homotopy classes_knowing them by their first name,
so to speak: their position in the Adams spectral sequence, their relationships
with each other, and, as we shall presently describe, their unstable behavior.
The Adams spectral sequence offered more than such stembystem calcula
tions, however; and indeed, Adams' motive for constructing it, and his first and
most celebrated application of it, was the resolution in all dimensions of the
Hopf invariant one problem ([2], 1960). The Adams spectral sequence gave a
decreasing filtration of stable homotopy groups, providing some measure of the
complexity of a class. In the stable homotopy of a sphere, only odd multiples
of the fundamental class are not in filtration 1. Adams interpreted the Hopf
invariant as picking out F 1=F 2. At E12there is for each i a nonzero class hi
in thei(2i 1)stem, corresponding the the indecomposable Steenrod operation
Sq2 . The element h0 survives to the element 2; the next three survive to the
Hopf maps j; , and oe; and Adams showed there were nonzero differentials on
all the rest. Besides the wellknown interpretation involving division algebra*
*s,
this proved that the only parallelizable spheres are S1; S3 and S7.
Mahowald has credited this paper with arousing his interest in Homotopy
Theory: it served as his textbook on the subject.
6 H. R. MILLER AND D. C. RAVENEL
2. The Memoir, The Metastable Homotopy of Sn
In his Introduction to this remarkable paper [47 ] Mahowald says, "Our object
is to bring to Toda's theorem [the stabilization sequence described above] the
power of stable methods developed by Adams," and later in the Introduction
he describes his results on the EHP history of a family of classes which he
conjectured (correctly, as it turned out) to be the Image of J. Mahowald's aim
was to unite the three principal strands of contemporary research in Homotopy
Theory. He perceived in Toda's iteratedsuspension EHP sequence a means of
trapping the metastable homotopy groups of a sphere between two stable groups,
each of which could be attacked using the Adams spectral sequence.
Before we examine this work in detail, it is worthwhile pointing out what
Mahowald did not use.
o The mod p lower central series spectral sequence of [15 , 78], with its
attendant algebra model, appeared in 1966, and the MasseyPeterson
theory in 1967 [64 ], but neither were at this point part of Mahowald's
methodology. The Adams spectral sequence was still for him an exclu
sively stable tool.
o The splitting theorems of Barratt and P. J. Eccles [7] and of V. P. Snai*
*th
[81 ] did not appear until 1974.
o The Adams conjecture was not resolved until 1970 [80 , 75 ]. (In fact
in 1970 Mahowald himself [48 ] offered a line of argument resolving the
ambiguous factor of 2 in the Image of J. The details of a variant of this
argument were filled in by Mahowald and D. M. Davis in [34 ].)
The first order of business in the Memoir was the construction of a map
I : Es;tr(S0) ! Es1;tr(Pn)
of spectral sequences (in the range t  s 2n  2) compatible in the abutment
with Toda's "Hopf invariant" I. This map is of course not induced by a stable
map of spaces. Constructions of this sort are today a standard part of the
technique. Mahowald's construction was generalized by Milgram [68 ]. For a
modern setting for results of this sort, see [37 ].
This spectral version of I allowed Mahowald to do computations using the
Adams spectral sequence. The next problem was to discover means of computing
part of E2(Pn) from the known structure of E2(S0). For this Mahowald devised
his "prespectral sequence," associated to the filtered complex arising from the
skeleton filtration of Pn. This is the Adams E2 version of the AtiyahHirzebruch
spectral sequence for ssS*(Pn). A long central section carries out this computa*
*tion,
and displays the result through dimension 29 in sixteen impressively concise
tables. Mahowald asserts that "The use of a large computer was important in
this work," and this seems to be the first such mechanization of a problem in
homological algebra_certainly the first in algebraic topology. Mahowald has
maintained an active interest in computeraided computation; for a recent effort
MAHOWALD'S WORK ON HOMOTOPY GROUPS 7
see [30 ].
From this it was an easy step to use Toda's stabilization sequence (in its sp*
*ec
tral form) and its relationship with the EHP sequence to obtain the metastable
homotopy of spheres through about ss40, and good information on the entire
kstem for k 28.
Another contribution of the Memoir is the construction of classes in the stab*
*le
homotopy ring which behave as the Adams conjecture predicted that the Image
of J should behave. Mahowald produced such elements in a novel way, via a
stable map : RP 1 ! S0 obtained using James periodicity . In modern terms,
he wrote down the transfer associated to the universal double cover over RP 1.
This is a stable map P01 ! S0 (where P01 denotes the suspension spectrum
of RP 1 with a disjoint basepoint adjoined), and he composed this with the
0connected covering map P 1 ! P01. To our knowledge this represents the
first published occurrence of this map. Mahowald was careful to distinguish
from another map from RP 1, namely the Jhomomorphism composed with the
usual inclusion of RP 1 into SO. He conjectured that is surjective to the
2component in positive stems; this is the KahnPriddy theorem ([40 ], 1972).
The virtue of replacing O (which has simple homotopy) by P 1 (which has
complicated stable homotopy) is that one can describe elements of ssS*P 1, and
their images in ssS*, by means of the (stable) Adams spectral sequence_and, as
indicated above, this computation is the heart of the Memoir. Mahowald proved
that certain classes in high filtration survive in the Adams spectral sequence *
*for
the stunted projective space Pn, by an ingenious inductive argument. He was
then able to compute the einvariant of the image of these classes under , to
see that they remain essential in ssS*. This produced summands in ssS*of order
equal to the order of the image of J specified by the Adams conjecture, together
with elements resembling the other elements constructed by Adams. Mahowald
conjectured that these elements were in fact in the image of J, and this was la*
*ter
established in [48 , 34].
In any case, the result was that in the Adams spectral sequence for a sphere,
certain elements along the Adams vanishing line survive. The pattern is dis
played in Figure 1. (The case k = 1 is exceptional and we leave it aside.)
In this portion of the Adams spectral sequence "chart" for the sphere, vertic*
*al
lines denote multiplication by h0 and diagonal lines represent multiplication by
h1. We have denoted classes in E2 by the names of elements to which they survive
in homotopy. Thus for example fi4k1 is a class in the (8k  1)stem of order
2(k)+4, which is the 2primary part of 16k. There is also an "exotic extension,"
jfi4k1 = fi4k. The elements of E2 labeled ffs survive to welldefined classes
in stable homotopy, since they are in the smallest Adams filtration subgroup of
their stable stem which contains any nontrivial 2torsion.
Toda's stabilization sequence, in its spectral form, allowed Mahowald now to
compute many of the Whitehead products of these elements. This work extended
[46 ], which dealt with the Hopf maps. There are two kinds of results at this p*
*oint.
8 H. R. MILLER AND D. C. RAVENEL


s = 4k + _3 ff4k+q3
 
 
2_ ff4k+q2 q
 
_ ff q q
1 4k+1 fi4k+2

_ ff4kq qfi
0  4k+1
 
1 _ q qfi4k
 
 
2 _ q
 
._ .
.. ..

_ q
(k)  3  fi4k1

_______________________________________
t  s = 8k+ 2 1 0 1 2 3 4 5 6
Figure 1. Mahowald's "Im J"
First, on most spheres, the elements in Im J have as Whitehead product (with the
fundamental class) other elements in Im J (possibly the zero element!). Second,
there is the beautiful theorem of Toda [85 ] to the effect that the Hopf invari*
*ant
of the RadonHurwitz desuspension of the Whitehead square is an appropriate
generator of the image of J. That generator therefore has trivial Whitehead
product. Mahowald was able to verify this for his elements. His approach to
this, via the Adams spectral sequence, led him to make the following conjecture,
which describes the behavior of most of the remaining elements. We assume we
are in the metastable range.
Conjecture: Assume n + fij 2( mod 2j+1).
o If n + fij 6= 2j+1  2, [n; fij] 6= 0.
o Otherwise, [n; fij] = 0 if and only if h2jis a permanent cycle in the
Adams spectral sequence. Moreover, if j 2 ssS2j+12is represented by
h2j, then I(j) = i*(fij) in the diagram
ssS2j+12I!ssS2j+12Pn i*ssS2j+12Sn;
where i : Sn ! Pn denotes the inclusion of the bottom cell.
The way in which this conjecture fits into the EHP sequence will become clearer
in x5.
After Browder's work [18 ], we would paraphrase the second part of this con
jecture by saying that the stable Hopf invariant of a class of Kervaire invaria*
*nt
one is a generator of the image of J. A passion for the Kervaire invariant [49 *
*, 8]
is another of Mahowald's distinguishing characteristics; and one may trace the
origin of this fixation to this point. Notice that he remained neutral about
whether such elements j exist or not.
In conversation Mahowald will often refer to this paper as a reference for
MAHOWALD'S WORK ON HOMOTOPY GROUPS 9
the source of some idea: "It's all in the Memoir." When we turn to his later
work in the sections below, this will be amply borne out: many of the ideas are
refinements or "recasts" of ones discussed above. As a final example of this, we
note the rather obscurepdefinition_(on p. 24) of the "root" of a stable homotopy
class ff, written ff. This was the original mention of the Mahowald invariant,
which we discuss further in x5.2.
3. An infinite family in 2ssS*
The solution to the Hopf invariant one problem showed that there are only
four nonzero permanent cycles in the "oneline" Ext1A(F2; F2) of the Adams spec
tral sequence for the sphere. In homological dimension 2, the most interesting
classes are the h2j's; they represent framed manifolds of Kervaire invariant 1.*
* The
construction of such classes seems a very casebycase affair [9], and the Hopf
invariant one story would seem to support the belief that only finitely many h2*
*j's
survive. More strongly, there was the "Doomsday Conjecture": In every homo
logical degree there are only finitely many survivors. This conjecture had its
adherents in the early 1970's, but the paper we review in this section demolish*
*ed
it.
Work of Mahowald and Tangora [59 ] put an upper bound on the survivors
in Ext2A(F2; F2); besides the h2j's, there are h0h2, h0h3, h2h4 (which survive
respectively to 2, 2oe, and a class called * by Toda), and the classes h1hj for
j 3. These were known to survive for i 5 by explicit computation. (h1h4
survives to a class named j* by Toda.) Mahowald's theorem is that they all
survive, to classes jj 2 ssS2j.
The class jj is constructed as a composite
S2j f! Xj?  g! S0
j ?yp
S2j1
where Xj is a suitably chosen finite spectrum. The map f gives us the h1, and
j
Sq2 is nonzero on the bottom class of the cofiber of g, so the composite will be
represented in Ext by h1hj. The two maps are constructed by entirely different
procedures.
The construction of g begins by regarding the Hopf class oe as a map S7 ! SG
and using the BoardmanVogt infinite loop space structure on SG_actually only
the double loop space structure_to extend to a map
2S9 ! SG ! QS0;
where the second factor is the shift map used in the construction of the J
homomorphism. This is adjoint to a stable map
^oe: 2S9 ! S0:
10 H. R. MILLER AND D. C. RAVENEL
Mahowald used the iterated loop structure to blow the lowdimensional class oe
up to highdimensional homotopy on the cells of the infinite complex 2S9.
The relevant part of this homotopy is picked out using the Snaith splitting of
2S9. This splitting has the form
_
2S9 ~= 7kBk;
k0
where Bk is a certain finite spectrum with bottom cell in degree 0 and top
cell in degree k  ff(k). Mahowald could compute the cohomology of Bk, and
on the basis of this and other evidence he conjectured that it was precisely the
spectrum Bk constructed by E. H. Brown and S. Gitler [19 ]. Those authors used
an elaborate Postnikov system approach in their construction, and had in mind
entirely different applications of the Bk's: they play a key role in the Brown
Peterson approach to the immersion conjecture, which was finally implemented
by R. L. Cohen.
Mahowald's guess was verified soon after by Brown and F. P. Peterson [20 ],
and although Mahowald found ways to avoid it we will use this fact as a matter
of convenience.
Let us write k = 2j. Then Mahowald took for Xj the spectrum 7kBk; for
g, the restriction of ^oeto that factor of 2S9; and for p, the collapse map to *
*the
top cell.
Mahowald then analyzed the Adams spectral sequence for Bk, to find the class
f 2 sskBk = ss1(1kBk). The E2term could be computed in this stem by the
algebra machinery developed by Brown and Gitler in their construction of the
spectrum. Keeping the notation k = 2j3, one finds for each j 3 a class
uj 2 Ext1;2A(H*(1kBk); F2)
lifting h1 2 Ext1;2A(F2; F2). The problem is that there are also lots of classe*
*s in
the zerostem ready to accept differentials starting on uj.
Mahowald's ingenious solution to this uses the fact that the Bk's fit into an
inverse system
. .q.!4B8 q!2B4 q!1B2 q!B1
in such a way that the uj's match up. Again using the algebra machinery he
was able to show that in the zerostem, each of these maps induces the zero map
in Ext. This implies that these maps also induce the zero map in Es;srfor all
r and s, and this forces all the uj's to be permanent cycles: d2uj = d2quj+1 =
qd2uj+1 = 0, so uj survives to E3(Bk). In this computation j was arbitrary, so
uj+1 survives to E3 as well and uj 2 E3(Bk) is hit by uj+1 2 E3(B2k). We can
then run the same argument over again to show that each uj survives to E4(Bk)!
By induction, they are all permanent cycles.
This approach to constructing homotopy classes has had widespread fallout
in homotopy theory. The BrownGitler spectra and this sort of application of
MAHOWALD'S WORK ON HOMOTOPY GROUPS 11
them was extended to odd primes by R. L. Cohen [27 ] in his thesis. Other class*
*es
have been constructed by similar methods by Mahowald [52 ], Bruner [21 ], and
W. H. Lin [45 ].
More generally, the inverse system used above, with its nice mapping prop
erties, was used by G. Carlsson [23 ] in his work_on the Segal conjecture for
elementary abelian 2groups. He showed that H *RP 1 splits off of the direct
limit of the cohomologies of the BrownGitler spectra. The first author formu
lated Carlsson's work as saying that H*RP 1 is an injective unstable Amodule
(actually the work was done in homology), and used it in his proof of the Sulli*
*van
conjecture [71 ].
4. The Context, II
In this section we will briefly review some of the developments in the study
of homotopy groups of spheres in the years between 1967 and 1982. We update
the three areas discussed in x1.
4.1. The Adams Conjecture and the Space J. We briefly describe the
standard picture (see [66 ]) associated with a solution of the Adams conjecture.
Recall that we always work locally at 2.
The Adams operation 3 is a map BO ! BO. In homotopy it behaves as
follows: In dimension 4k it multiplies by 32k, and in other dimensions it is the
identity. Thus (using the Hspace structure on BO to form the difference), 31
lifts to a map
' : BO ! BSpin:
The fiber of this map is a space J, whose homotopy can easily be computed
using the fact that (32k 1) = 3 + (k) (where 2(k) is the largest power of 2
dividing k). The generators of ss*Spin map by the boundary map to elements
fij 2 ss*J. There are in addition distinguished elements of order 2 in ss*J, de*
*noted
ffj. ff4k1 and ff4k denote respectively the elements of order 2 in ss8k5J and
ss8k1J; these are multiples of fi4k2 and fi4k1, respectively. ff4k+1 and ff4*
*k+2
map to generators of ss8k+1BO and ss8k+2BO respectively. The low dimensional
behavior of this fibration sequence is set out in Figure 2.
The Adams conjecture asserts that the composite of 3  1 with the map
Bi : BO ! BG is trivial, where Bi is induced by the inclusion of O into the
compositionmonoid G of stable selfmaps of spheres. A solution to the Adams
conjecture is a nullhomotopy; and this gives us the following diagram:
Spin ! J? ! BO? '! BSpin?
?y ?y ?yk
SG? ! * ! BSG
?y*[1]
Q0S0
12 H. R. MILLER AND D. C. RAVENEL
 ...... ...  ... ...  ... 
      
 11 ff7  Z8 fi6  0  0 
 10 ff6Z2   Z2  Z2 
 9 ff5Z2 Z2 fi5  Z2  Z2 
 8   Z2 fi4  Z  Z 
 7 ff4  Z16 fi3  0  0 
 6   0   0  0 
 5   0   0  0 
 4   0   Z  Z 
 3 ff3  Z8 fi2  0  0 
 2 ff2Z2   Z2  0 
__1_ff1Z2_____________Z2______0_____
__n________ssnJ_________ssnBO_ssnBSpin_
Figure 2. The Homotopy Long Exact Sequence for ss*J
in which k is the composite BSpin ! BSO ! BSG. The diagonal composite is
the Jhomomorphism composed with the covering map from Spin.
This puts an upper bound on the size of the image of J: the Jhomomorphism
(restricted to Spin) factors through ss*J. To obtain a lower bound, one works
on the level of spectra. The map ' is the restriction to the zerocomponent of
an infiniteloop map Z x BO ! BSpin, which is associated to a map of spectra.
The spectrum associated to Z x BO represents connective real Ktheory, and
Mahowald denotes it by bo. Its threeconnected cover is bo<4>, with zerospace
BSpin. (Mahowald often writes 4bsp for this spectrum.) Thus
' : bo ! bo<4>:
Write j for the fiber of this map; it is a spectrum with zerospace Z x J. The
unit map S0 ! bo clearly lifts uniquely to a map
e : S0 ! j;
which induces on the zerocomponents of zerospaces a map e : Q0S0 ! J. This
is a geometric version of the (real) einvariant. The beautiful fact is that t*
*he
composite
J ! Q0S0 e!J
is a homotopy equivalence. This gives a product splitting
Q0S0 = J x C
in which C is a space called "Coker J". In particular, the homotopy of J sits in
the stable homotopy groups of spheres as a direct summand. It is the "linear"
part of the stable homotopy ring, and we will denote this subgroup by ss*J. C
is 5connected. j is in ss*J, though it declines "for personal reasons" (to quo*
*te
MAHOWALD'S WORK ON HOMOTOPY GROUPS 13
Mahowald) to be in the image of the version of the Jhomomorphism adopted
here, since Spin is 2connected.
In [48 ] Mahowald announced a resolution of the conjecture made in the Mem
oir, identifying the high Adams filtration classes he studies there (Figure 1 a*
*bove)
with the summand ss*J. For more detail, see [34 ].
4.2. The EHP spectral sequence. This is merely a more sophisticated
point of view on the classical EHP sequence.
The EHP sequence is an exact couple, so there is an associated spectral
sequence. We can regardSthis as the homotopy spectral sequence associated to
a filtration of QS0 = nSn:
* ! S1? ! 2S2? ! 3S3?  ! . ..!QS0
?yH ?yH ?yH
S1 2S3 3S5
Let us agree that FnQS0 = n1Sn1. Then if we use the usual Serre indexing
for the spectral sequence,
E1s;t= sss+t(s+1S2s+1) = ss2s+1+tS2s+1
_that is, the tstem on the (2s + 1)sphere. The spectral sequence converges to
ss*(QS0) = ssS*. If we arrest the filtration at nSn, we get a spectral sequence
converging instead to ss*Sn, whose differentials are simply the differentials i*
*n this
range of the full spectral sequence; this is the "restriction principle" for sp*
*ectral
sequences. This offers the prospect of an inductive computation of the homotopy
groups of spheres; notice that the portion of the E1term contributing to the n
stem_where s + t = n_involves the homotopy of (odd) spheres in stems less
than n.
From the general theory of spectral sequences we can make the following
comments about the EHP spectral sequence:
o Elements of the abutment are represented at E1 by the Hopf invariant
of a maximal desuspension.
o Each differential represents a Whitehead product (with the fundamental
class), and gives the EHP history of an unstable homotopy class.
In more detail, to compute a differential on a class whose representative at *
*E1
is x 2 E1s;t= ss2s+1+tS2s+1, one applies P to get to ss2s1+tSs, pulls this cla*
*ss
back as far as possible under suspension_say to z 2 ss2sr+tSsr+1_and then
applies the Hopf invariant: Hz 2 ss2(sr)+t+rS2(sr)+1= E1sr;t+r1:
r1
z 2 ss2sr+tSsr+1? E! ss2s1+tSsx 3 P x
?yH ??P
drz 2 ss2(sr)+t+rS2(sr)+1 ss2s+1+tS2s+13 x
14 H. R. MILLER AND D. C. RAVENEL


 _____
 
9 _ 22 23 24___24oe 23 23 23 23 23 23 23
 HHYHHYHHYHHY  HHYHHHHYHHYY
 HH QkHHQ Z" Z" HH QkQHH
 HHHH HHHHQZ Z HHHH QHHHH
8 _ 2 2 23___23oeQ22Z 22Z 22 22Q 22 22 22
 HHYQk Q Z Z HHYQk Q
 QHH__ Q Z Z QH H Q
_ ___oeQH ___oe2QZ Z___oe2QH ___oe2Q
7  2 8 16 Q 16 16Z 16 Z 16Q 16 16 16
  Q Z Z Q
  Q Z Z Q
6 _ 2 2  2 2 Q 2 2 Z 2 Z 2 Q 2 2
 
 _____ Z" Z"
  Z Z
5 _ 2 ___o2e  Z Z
  Z Z
  Z Z
_ ___oe  Z Z
4  2 2  Z Z
 ______ Z Z
  Z Z
_ ___oe ___oe2 ___oe2 ___oe2 Z ___oe2Z
3  4  8 HHY8 8 8 8 HHY 8 8 8 8 8
  H H Z"
  H H H HZ
2 _ 2  2 2 2 2 2 2 2Z 2 2 2
  HHY HHY HHY HHY Z
 _____ H H H H H H H H Z
  H H H H Z
1 _  2 2 2 2 2 2 2 2 2 Z 2 2
  HHY HHY Z
Stem of   H H H H Z
_  ___oe2 ___oe2 H ___oe2 ___oe2 H ___oe2Z
input 0   1 1 1 1 1 1 1 1 1 1 1
 

________________________________________________________________*
*
1 2 3 4 5 6 7 8 9 10 11
Sphere of Origin:S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
Input Sphere: S3 S5 S7 S9 S11 S13 S15 S17 S19 S21 S23
Total degree = Stem of output
Figure 3. The EHP spectral sequence
MAHOWALD'S WORK ON HOMOTOPY GROUPS 15
Figure 3 displays a portion of the EHP spectral sequence. This is easily
gleaned from Toda's book. We print it here because we will want to compare
the EHP spectral sequence to other spectral sequences. We have used Toda's
convention, writing 23 for example for the group (Z=2)3. We indicate only the
differentials whose source and target are both in the range displayed. A double
arrow indicates a rank 2 map (between elementary abelian 2groups); an arrow
with a 2 over it indicates that the image has index 2. We have also indicated
the edge of the metastable range, by a staircase.
The Whitehead square wn is related to the issue of finding independent vector
fields on a sphere by the fact that it is J composed with the clutching function
o : Sn1 ! SO(n) of the tangent bundle of Sn. Giving k independent vector
fields is equivalent to giving a compression of o through a map __o: Sn1 !
SO(n  k). J composed with such a compression is a desuspension of wn. When
n = 1; 3 or 7, wn desuspends all the way to the zerosphere: it is null, and the
sphere is parallelizable. That this never happens again is the content of the H*
*opf
invariant one problem.
Radon and Hurwitz gave one such "linear" desuspension, which we will call
__w __
n . Toda [85 ] proved that the Hopf invariant of wn is a generator fi(n+1) of *
*the
image of J. (Thus for the RadonHurwitz desuspension, k = fi(n+1). A check
of the numbers shows that this Hopf invariant is in the stable range, so we don*
*'t
have to worry about desuspending elements of Im J.)
This shows that the fundamental class n1 2 E1n1;0survives to Ek+1 in the
EHP spectral sequence, and it computes the differential dk+1n1; but it is not
yet clear that the proposed target has not been hit by an earlier differential *
*in
the spectral sequence. This is in effect what Adams showed in his solution [3] *
*of
the vector field problem; this differential is nonzero.
An alternative and completely stable proof of the vector field problem, not
using Adams operations per se, is given in [55 ] by Mahowald and Milgram.
4.3. The Generalized Adams Spectral Sequence. In 1967 S. P. Novikov
published a long and intriguing paper [74 ] which initiated a study of homotopy
theory "from the viewpoint of [complex] cobordism." He analyzed the spec
tral sequence obtained from the complex cobordism spectrum MU in a manner
analogous to the construction of the Adams spectral sequence. Mahowald inde
pendently considered the case in which the ground spectrum is bo. Very simply,
the stable Adams spectral sequence for the homotopy of X, based on the spec
trum E with unit S0 ! E, is the homotopy spectral sequence associated to the
diagram
__ __ __
X?  E ^?X  E ^?E  . . .
?y ?y ?y
__ __ __
E ^ X E ^ E E ^ E ^ E
16 H. R. MILLER AND D. C. RAVENEL
which is made by splicing and smashing the cofibration sequence
__ 0
E ! S ! E
__
defining E.
The Adams spectral sequence based on MU has had a profound impact on
stable homotopy theory. Localized at a prime p, MU splits into a wedge of
suspensions of the BrownPeterson spectrum BP , with coefficient ring
BP* = Z(p)[v1; v2; : :]:; vn = 2(pn  1):
The generator v1 encodes Bott periodicity, and the other generators provide
analogous "higher periodicity" operators. See the second author's book [76 ] for
more information about the generalized Adams spectral sequence and BP point
of view.
Much of Mahowald's work has focused on v1periodic phenomena; but by the
mid 1970's he was a strong advocate of the BP viewpoint, and has contributed
much to our understanding of higher periodicity (see [32 ], for example).
5. The Image of J in the EHP sequence
This paper [54 ] represents the culmination of Mahowald's work continuing
the line of inquiry started in the Memoir. The principal unstable results had
been announced in [50 ], and the stable methodology was set out in [53 ]. Our
account of the Annals paper includes a partial description of that work, and is
influenced also by later refinements and clarifications [36 , 43, 31, 60, 34, 6*
*1].
5.1. Stable v1periodic homotopy. Let M denote the Moore spectrum
S0 [2 e1. As described in x1.1, Adams had constructed a map A : 8M ! M
which induces an isomorphism in Ktheory. One may formally invert A in the
homotopy of M; the result is the homotopy of the mapping telescope of the
system
8A 16A
M  ! 8M ! 16M ! . . .
which we denote by A1M. The main stable result of this work is a complete
computation of this "v1periodic homotopy" of the Moore spectrum. Following
the setup used in [54 ], we will describe the answer for a different spectrum,
namely
Y = M ^ (S0 [j e2):
There is now a Ktheory isomorphism v1 : 2Y ! Y (a "fourth root of A"), and
one of the principal results of the Annals paper is that the localized homotopy
is given by
v11ss*Y = F2[v11] E[a]
where a is a class in dimension 1. A free F2[v1]module of rank 2 can be found
inside ss*Y by direct construction; the force of the result is that any class d*
*iffers
from one of these by v1torsion.
MAHOWALD'S WORK ON HOMOTOPY GROUPS 17
This theorem was interpreted by A. K. Bousfield [13 ] as implying that the map
M ! A1M is the terminal Ktheory equivalence out of M: A1M is the K
theory localization of M. As a corollary he obtains an effective characterizati*
*on
of the Klocal spectra in terms of their mod p homotopy groups: the Adams
map A must act isomorphically on them for each prime p. These results form
the cornerstone and guiding example for much contemporary work in stable
homotopy theory.
The method of proof may be described roughly as follows. Mahowald considers
the generalized Adams spectral sequence based upon the spectrum bo, converging
to the homotopy (localized at 2) of a spectrum X. If X is the spectrum Y , the
selfmap v1 acts on the spectral sequence. Since v1 is detected by Ktheory, it
has filtration 0 and acts "horizontally" in the spectral sequence. Mahowald now
hopes to compute the localization of the E2term and prove that the localized
spectral sequence converges to the localized homotopy of Y . This convergence is
not automatic; it may fail in either of two ways: there may be an infinite stri*
*ng
of v1torsion summands in E1 , in higher and higher filtration, which become
linked together by extensions to produce a torsionfree summand in ss*(v1Y );
or there may be a torsionfree summand in E2 which is eaten away bite by
bite by infinitely many v1torsionvalued differentials, and which thus does not
contribute to ss*(v1Y ) despite consisting entirely of permanent cycles in the
localized spectral sequence.
Mahowald proves three things about the (unlocalized) bospectral sequence,
which together prove convergence and give the computation. We state them in
very weak form, so as not to become embroiled in detailed numerology. We state
them for X = Y .
o Lowdegree computation: E02= F2[v1] and E12is free of rank one over
F2[v1].
o Vanishing line: Es;t2= 0 for s > m(t  s) + b, for certain numbers m and
b. In fact, the slope m can be taken to be 1=5 [31 ].
o Bounded torsion: In filtration 2 and larger, multiplication by v1 raises
filtration enough so that the vanishing line is eventually intersected.
The result clearly follows from these facts. The method used is to analyze
the stages in the bo Adams resolution, and the maps between them, by means of
the classical Adams spectral sequence. One must thus understand the homotopy
type of smashpowers of bo. One of the main tools Mahowald relies on is a
splitting theorem for bo ^ bo, first announced in [48 ]; for proofs see [69 ] a*
*nd [53 ].
At an odd prime it is possible [70 ] to analyze the classical Adams spectral
sequence directly; there the map v1 acts parallel with the vanishing line, so
convergence of the localized spectral sequence is automatic. In the case of the
mod p Moore spectrum there is one nontrivial differential, which can be computed
by comparing the classical Adams spectral sequence with the Adams spectral
sequence based on BP . The result is that the localized homotopy of a Moore
spectrum is exterior on one generator over Fp[v11].
18 H. R. MILLER AND D. C. RAVENEL
Recent work of the second author [77 ] extends this method to study the
homotopy groups of mapping telescopes of v2selfmaps. Surprisingly, it turns
out that the potential failures of convergence mentioned above do in fact occur
in that context.
Another approach to the computation of the localized homotopy of a Moore
space, using the Adams conjecture, has been given by M. C. Crabb and K. Knapp
[28 ].
5.2. The stable EHP sequence and the Mahowald invariant. In the
Annals paper Mahowald replaces the Toda suspension sequence with the "Snaith
maps." In [81 ] Snaith had displayed a stable splitting of nnX, for any con
nected space X. By projecting to the factors, one obtains maps to the zerospac*
*es
of the factors. In the case X = S1, the first of these factors is P n, where P n
denotes the real projective space (or its suspension spectrum). Looping this
map, we obtain s : n+1Sn+1 ! QP n. Later work by N. J. Kuhn [41 ] and by F.
R. Cohen, L. R. Taylor, and May [25 , 24], shows that these maps can be chosen
to be compatible under suspension, so that the following diagram commutes:
nSn? ! n+1Sn+1? ! n+1S2n+1?
?ys ?ys ?yE1
QP n1 ! QP n ! QSn
This gives a relation between the EHP sequence and the stable cellstructure
of projective space in all dimensions, not just in the metastable range. In fac*
*t,
one can assemble the homotopy long exact sequences of the bottom fibration
sequences into an exact couple, whose associated spectral sequence is the Atiya*
*h
Hirzebruch spectral sequence for the stable homotopy of P = P 1; so one obtains
a map
i j i AtiyahHirzebruch j
s : EHP Spectral Sequence ! spectral sequence for ssS:
*P
This map is more convenient than the Toda's Hopf invariant because it provides
stable invariants for unstable classes even outside the metastable range.
Notice that in the metastable range, the map is an isomorphism at E1. In
fact, the map in E1sis the suspension map ss*S2s+1 ! ssS*, and the target spect*
*ral
sequence is sometimes called the stable EHP spectral sequence. It is a certain
localization of the EHP spectral sequence.
We interject here some comments on another idea of Mahowald's (see [5]).
Another way to construct the AtiyahHirzebruch spectral sequence for P = P 1
is via the sequence of collapse maps to stunted projective spaces:
P+ = P0 ! P1 ! P2 ! . .:.
This sequence may be continued backwards as well_
. ..!P2 ! P1 ! P0 ! P1 ! . . .
MAHOWALD'S WORK ON HOMOTOPY GROUPS 19
t
A 
_______A_ 

@ A 
@ _____A_ 
@ A 

@ ___A_ 
@ A 
@ _A_ 

@ A 
________________________________________@A
1 s
Figure 4. E1 of the ssS*P1 spectral sequence
_where for any n we define Pn as the Thom spectrum of n times the Hopf
bundle over RP 1 (which is a virtual bundle for n < 0). This inverse system of
spectra goes by the name, "P1 ." There results a spectral sequence occupying
the upper halfplane. James periodicity shows that for each r, Ersis periodic in
s, of period which increases with r; and that the periodicity isomorphisms are
compatible as r changes. Thus the halfplane spectral sequence may be regarded
as a localization of the firstquadrant spectral sequence.
This localized spectral sequence converges to the homotopy groups of the
homotopy inverse limit of the tower of stunted projective spaces. Mahowald
conjectured that this inverse limit is none other than the 2adic completion of
the spectrum S1. This is a strong form of a conjecture made later by Graeme
Segal about the stable cohomotopy of the classifying space of an arbitrary fini*
*te
group, and it was proved by W. H. Lin [44 ].
This shows that the E1 term of the spectral sequence is concentrated on and
above the line s + t = 1. A theorem of J. D. S. Jones [39 ] asserts that E1s;t*
*= 0
for 2s+t > 1. (Jones's result, and proof, were given earlier in different lang*
*uage
by G. E. Bredon [16 ],2 and again later in [72 ].) The E1 term thus has the fo*
*rm
shown in Figure 4.
Any nonzero stable homotopy class ff 2 ssSn(S0^2) = ssSn1(S1 ^2) has a coset
of representatives in E1st= ssStin this spectral sequence, whereps_+ t = n  1.
In the Memoir, Mahowald called this coset the root invariant, ff; we prefer to
call it the Mahowald invariant, R(ff). R(ff) is a subset of ssSt, and the theor*
*em
of Bredon and Jones shows that t 2n. R(ff) is nonempty and never contains
zero, so this gives an interesting way of creating higher and higher dimensional
homotopy classes. A consequence of the work in the Annals paper, for example,
is [56 , 72] that for each s
ffs 2 R(2s);
where ffs is the element of ss*J considered above. This exhibits the tendency of
R to increase the level of periodicity_from 2 = v0 to v1 in this case; and an
____________
2See formula (12) on page 291. We thank N. Minami for bringing this paper to*
* our attention.
For related ideas, and a conjecture closely related to the Sullivan conjecture,*
* see also [17].
20 H. R. MILLER AND D. C. RAVENEL
important current conjecture [56 ] in homotopy theory asks for a general result
like this.
5.3. Unstable v1periodic homotopy. Many of Mahowald's ideas found
better expression as the subject developed. One, which may be traced back to
the Memoir, is that the metastable projective space invariants should carry the
"Ktheoretic" part of the unstable homotopy of spheres in all dimensions. In
The image of J in the EHP sequence Mahowald expresses this in the following
terms.
Instead of applying the functor ss* to the Snaith map s : nSn ! QP n1, he
applies the functor [nY; ]. Here Y denotes the fourcell spectrum C(2)^C(j)
described above; its bottom cell is in degree 0. 3Y is the suspension spectrum
of RP 2^ CP 2, and by nY we mean the space n3RP 2^ CP 2; so n 3.
The square brackets denote pointed homotopy classes. This is a "homotopy
theory"; it converts fibration sequences to long exact sequences. For n > 4 it *
*is
an abelian group. The Ktheory isomorphism v1 desuspends also, and we may
form a localization by inverting it. This gives a module over the graded field
F2[v11]. One should think of this as an analogue of the rational homotopy of a
space; Khomology (or mod 2 Khomology) plays the role of rational homology.
Mahowald proves that s : 2n+1S2n+1 ! QP 2ninduces an isomorphism in
this theory. This can be thought of as an analogue of the theorem of Serre,
that the rational homotopy of an odd sphere maps isomorphically to the stable
homotopy of QS0; the "zeroth" Snaith map is the stabilization map nSn !
QS0.
Notice that inverting v1 in [*Y; ] is not a stabilization; quite to the con
trary, it brings highdimensional_so very unstable_classes down to low degrees.
Mahowald's result is that nonetheless the result can be computed in stable term*
*s.
This result follows from considering the map of fibrations:
2
2n1S2n1? E! 2n+1S2n+1? ! 2n2C(n)?
?y ?ys ?ys
QP 2n2 ! QP 2n ! QP22nn1
Here C(n) (which Mahowald always denotes by W (n)) is the fiber of the double
suspension map, and of course P22nn1is a mod 2 Moore space. Then in fact
the righthand vertical arrow induces an isomorphism in v11[*Y; ]: from the
point of view of v1periodic homotopy the fiber of the double suspension is a
stable mod 2 Moore space. This result is proven by considering certain modified
unstable Adams towers, and showing that the localization of the induced map is
an isomorphism.
These two sequences of fibrations link together over n to give exact couples
in a homotopy theory, and so spectral sequences. We have just said that the
map induces an isomorphism in E1, so by the fivelemma the Snaith maps also
induce isomorphisms.
MAHOWALD'S WORK ON HOMOTOPY GROUPS 21
The issue of the relationship between the condition that a map induce an
isomorphim in v1periodic homotopy and that it induce an isomorphism in mod
p Khomology is an interesting one. Snaith and the first author [73 ] showed
that the limiting Snaith map s : Q0S0 ! QP is a Khomology isomorphism.
In very recent work, A. K. Bousfield [14 ] shows that the theorem of Mahowald
mentioned above, to the effect that s : 2n+1S2n+1 ! QP 2ninduces an isomor
phism in v11[*; ], implies that it is a Ktheoryisomorphism also. This gives
a calculation of K*(2n+1S2n+1). These methods have been extended further
by L. Langsetmo in [42 ], using work of Mahowald and R. D. Thompson [63 ].
5.4. The j*P spectral sequence. The Annals paper considers two further
issues. One wants to know about the image of the true homotopy in the local
izations computed here; and one really is primarily interested in the ordinary
homotopy groups, not in the Y homotopy groups. These questions may be ad
dressed in the following way, which brings us back again to ideas nascent in the
Memoir.
We noted above that the unit of the spectrum j is a version of the Adams
einvariant. It induces a map ssS*X ! j*X, and a corresponding map of Atiyah
Hirzebruch spectral sequences. Mahowald applies this in particular to the case
of P = RP 1:
i AtiyahHirzebruch j i AtiyahHirzebruch j
e : spectral sequence for ssS! :
*P spectral sequence for j*P
Mahowald computes the latter spectral sequence in detail. The result is quite
complex; but after all, via the Snaith map s from the EHP spectral sequence
(described above in Section 5.2), it is the most accurate image of the global
behavior of unstable homotopy theory we have, and we believe that it illuminates
and unifies many of Mahowald's ideas and insights. We describe it here, and then
comment on the behavior of the composite es.
In the spectral sequence, E1s= j* for s 1. (We apologize for the reuse of
the letter s.) The differential d1 comes from the cellular chain complex for P *
*, so
is alternately zero and multiplication by 2. E2sis thus ker(2j*) for s even and
coker(2j*) for s odd. We will use certain graphical conventions to depict these
two graded vector spaces. In Figure 5, the vertical lines denote jmultiplicati*
*ons.
The generators are represented by dots, which are offset to the left or right in
accordance with their origin in the long exact sequence depicted in Figure 2.
Figure 6 depicts the spectral sequence from E2 on. Here we have omitted
the vertical j multiplications for clarity, but preserved the horizontal offset*
*. We
have also omitted the heads of the arrows depicting the differentials; they poi*
*nt
northwest.
Further analysis of the cellstructure of P leads to further "short" differen*
*tials,
d2's and d3's, which are more easily drawn than described. Generally it may be
said that ff's hit ff's and fi's hit fi's. What remains after they have been ex*
*ecuted
is for the most part concentrated along diagonals where s + t 1; 2 mod 4.
22 H. R. MILLER AND D. C. RAVENEL
  
  
11  q fi6 ff7q 
10 ff6 q ff6q 
9 ff5 qfi5qff5qfi5q
8  fi4q fi4q
7  qfi3 ff4q 
6   
5   
4   
3  q fi2 ff3q 
2  ff2 q ff2 q 
   
1  ff1 q ff1 q 
0 ff0 q  
________________________
  
 coker(2j*) ker(2j*) 
________________________
Figure 5. The columns of E2 of the j*P spectral sequence
Here there are longer differentials, starting with the "vector field differenti*
*al"
dr4k1 = fi(k)+24k1r;
where r = fi(k)+2 + 1. The rest of the differentials along this total degree *
*line
slide up one notch at a time; the general formula is
d*(ffi4kffi1) = fi(k)+i+2*;
where each * denotes a number whose value can be determined from the rest
of the formula. In the interest of clarity we have drawn only the differentials
starting on the fundamental classes 4k1.
This completes the description of the spectral sequence depicted in Figure 6.
The composite es sends the EHP spectral sequence (Figure 3) into this one, and
we invite the reader to make the comparison at this point. Remember that we
have drawn the EHP spectral sequence from E1, and the j*P spectral sequence
from E2.
There is a clear periodicity in this spectral sequence. It may be extended
periodically to the left, by replacing the spectrum j with the Khomology local
ization SK of the sphere spectrum, of which j is the (1)connected cover. It
may also be continued downwards, by replacing P by the inverse system P1 . If
both of these changes are made, the only survivors occur along the total degree
lines s+t = 1; 2, where the length of the "long differentials" becomes infini*
*te.
This leads to the conclusion that
holimn!1(Pn ^ SK ) ' 1SQp _ 2SQp:
MAHOWALD'S WORK ON HOMOTOPY GROUPS 23





_ q qHH q qd q qHH q q q qHH q qd q qHH q q q qHH q q q qH q
 Q Q H Q Q H QQ H QQ H Q Q H Q Q HH
18 _ qd qHQqHHq qd qHQqHHq q qHQ qHHq qd qHQqHH q q qHQqHHq q qHQqHH
 H HQH H H HHQH H HHQH H HHQH H HHQH HQ
_ qqdqqdqqHqqHQqq qqdqqHqqHqqQqq qqHqqHqqQqq qqHQqqHqq qq qqHqqHqqQ qq qqH
 HHHHHHQQ HH HHHHQQ HHHHHQHQ HHHHHHQQ HHHHHHQQ H HH
16 _ qd q qH qHQq q qHQqH q q qH qQH q q qH qHQq q qH qHQq q q
 HHH Q HHH Q HH H Q HHH Q HHH Q
_ qd qd q qdH qQ q q qdH qQq q q H qQq q qH qQ q q q H qQ q q

_
14 
_

_
12 

_ qQ qHH q qd qQ qHH q q qQqHH q q qQqHH q q qQ qHH q q qQ qH q
 Q H Q H Q H Q H Q H Q HH
10 _ qd qHHqHHqHQ qd qHHqHHqHQq qHHQqHHqH q qHHqHHHqQ q qHHqHHqHQq qHQqHH
 HQ H HQ H HQ H HQ H HQ H HHQ
_ qqdqqHqqHqqHHqqQqqHqqHqqHqqHqqHQqqHqqHqqHqqHqqHHqqHqqQqqHqqHqqHqqHQqqHqq*
*Qdd
 H HHQHQ H HHQHQ H HQHQH H HHQHQ H HHQHQ HH
8 _ qd q qH qHQq q qHQqH q q qH qQH q q qH qHQq q qH qHQq4 q_q_
 HHH Q HHH Q HH H Q HHH Q HHH Q
_ qd qd q qdH qQ q _qqdHqQq_ q q H qQq q qH qQ q q q H qQ q q_*
*_
 3
_
6 
_

_
4 
 _ _ _
_ q qHH2qqdj3q_qHH q q q qHH q_qj4dqqHH q q q qHHqq_ q qH q
 H H H H H HH
2 _ qd qHHqHHqH qd qHHqHHqH q qHH qHHqH q qHHqHHHq q qHHqHHqH q qH qHH
 H H H H H H H H H H HH
_ qd qd qHHq H q qd qHHqH q q qHHqH q q qHH qH q q qHHqH q q qH
 H H H H H H H H H H H
0 _ qd qd H q qd H q q H q q H q q H q q

___________________________________________________________________________*
*
2 4 6 8 10 12 14 16 18 20 22
Figure 6. The j*P spectral sequence
24 H. R. MILLER AND D. C. RAVENEL
where Qp denotes the padic rational numbers. For us p = 2; but the same
result holds at odd primes, despite the rather different way j* and the spectral
sequence look in the two cases [83 ].
We mention this extended spectral sequence because it gives an interpretation
of the survivors in the original spectral sequence: they are all "edgeeffects.*
*" This
occurs in the AtiyahHirzebruch spectral sequence for ssS*P as well; by Jones's
theorem, all this homotopy supports differentials if we allow the negative cells
to play their part.
The survivors are of three types. First there are the classes which would be *
*the
sources of long differentials, except that these differentials end in s 0. The*
*se
are circled. Next, there are the classes which would be hit by long differentia*
*ls,
except that these differentials would start in t < 0. They are boxed. Finally,
there are classes along t = 3 which occur because the homotopy of the spectrum
j is irregular in low dimensions. They are enclosed in diamonds.
These permanent cycles compute j*(P ). The circled classes in degrees congru
ent to 1 mod 4 assemble to cyclic groups, as do the boxed classes. Mahowald
analyzes the composite
ssS*~=ss*(Q0S0) s!ssS*P e!j*P:
The summand of ss*J in ssS*maps isomorphically to the lifts of the circled sur
vivors. Mahowald proved that the only diamond classes in the image are those
in dimension which is a power of 2 and at least 8. They are the images of the
classes jj whose construction was reviewed in x3.
Of the boxed classes, only the classes of order two (that is, of lowest filtr*
*ation)
and in dimension of the form 2j  2 can be in the image; and they are hit by
elements of Kervaire invariant one and only by such elements. This is the first
way these classes enter the story. The reader should reconcile this with the
conjecture recalled in x2 from the Memoir.
The next question to address is: to what extent are the Snaith map and
the einvariant splitepimorphisms, as maps of spectral sequences? This might
be regarded as a sort of unstable Adams conjecture, if we think of the Adams
conjecture as asserting that the einvariant is split by the Jhomomorphism.
As a start, consider the permanent cycles. The map es puts a lower bound
on the sphere of origin of the elements in ss*J ssS*; these elements cannot be
born before their images on RP 1. These lower bounds are usually attained
[29 , 34], and the Hopf invariants are the elements of ss*J whose einvariant is
the representative in the j*P spectral sequence. The only exception is that fi4k
is not born on S2 with Hopf invariant fi4k1 (which is itself not born till much
later), but rather on S3 with an unstable Hopf invariant (except for fi4 = joe,
which is born on S6 with Hopf invariant ).
The elements of ss*J are thus born very early, with correspondingly high
dimensional Hopf invariant. (On their sphere of birth, they are far from being
in the "unstable image of J": the elements of ss*SO are born rather late [33 ].)
MAHOWALD'S WORK ON HOMOTOPY GROUPS 25
The elements jj, on the other hand, are born very late, with Hopf invariant , as
one expects from the position of their einvariant in the j*P spectral sequence.
The Kervaire classes should also be born "as late as possible," and one hopes
that there is a maximal desuspension of j 2 ssS2j+12with Hopf invariant fij,
since this is the behavior of their einvariants. A theorem of Barratt, Jones, *
*and
Mahowald [10 ] asserts that if 2 ssS2j+12has nonzero image in j*P , then it h*
*as
Kervaire invariant 1; and conversely if it has Kervaire invariant 1 and order 2,
then its image is nonzero.
Next turn to the differentials. The most important one is the "vectorfield
differential,"
dr4k1 = fi(k)+24kr1;
where r = fi(k)+2 + 1. This differential of course holds in the EHP spectral
sequence and the ssS*P spectral sequence just as in the j*P spectral sequence. *
*In
the ssS*P spectral sequence, it can be interpreted as giving the "stable relati*
*ve
attaching map" of the top cell in RP 1; and of course this is the context in
which Adams solved the vector field problem. A glance at Figure 6 shows the
relationship between the maximal desuspension of the Whitehead square (which
is recorded by this differential) and the Kervaire class. On its sphere of bir*
*th
the RadonHurwitz desuspension __w4kof the Whitehead square has rather high
order if (k) is large. If k is a power of 2, then one, two, or four suspensions*
* later
(depending upon k), it should become divisible by 2, and equal to a maximal
desuspension of a Kervaire class. The order is then decreased as one suspends
further, until S4k is reached and the Whitehead square dies. The Kervaire class
then lives on as a stable class.
The best general information Mahowald gives us about the other unstable
classes is Theorem 1.5 in the Annals paper (cf. (4.7) of [50 ]). This asserts t*
*hat
in dimensions bigger than 2n, a class in j*P 2nis in the image of ss*(2n+1S2n+1)
provided only that its image in j*P is in the image of ssS*_an image specified
above, up to the Kervaire invariant problem. This result gives a large family of
unstable homotopy classes with tightly controlled behavior in the EHP sequence.
These classes are constructed by first manufacturing suitable homotopy classes
in projective space, and then using the map P 2n! 2n+1S2n+1 to convert these
to classes in the sphere. This method harks back to the Memoir, and indeed the
proof uses the calculations of the Memoir.
It is interesting to ask the fates of the remaining elements of the image of J
in the ssS*spectral sequence and in the EHP sequence. These are the survivors
in the j*P spectral sequence which are not in the image of e or es. In [50 ]
(Proposition 4.8) a relation is given between the Whitehead products of certain
of these classes and the Kervaire invariant classes. This has not appeared with
detailed proof.
The announcement [50 ] gives certain compositions among these unstable
classes, and details can be found in [61 ].
26 H. R. MILLER AND D. C. RAVENEL
6. Conclusion
We have ended this account of Mahowald's contributions to the study of the
homotopy groups of spheres with a description of a paper published some ten
years ago. One should not infer from this that he has in any way reduced the
intensity of his work. On the contrary, as this or almost any current conference
proceedings shows, he is involved in many aspects of contemporary homotopy
theory. We mention for example:
o The comparison of unstable v1 and Ktheory localization. This work
[63 ] was mentioned above in Section 5.3. It is an integral part of curr*
*ent
foundational work by Bousfield [14 ] and E. Dror Farjoun [35 ], and of
computational work by Langsetmo [42 ] and others.
o The Mahowald invariant (Section 5.2). Its relationship to periodicity is
a guiding idea in contemporary stable homotopy theory. See [56 ], [79 ].
o The homotopy groups of localizations and telescopes. With the failure
of the telescope conjecture [77 ], these computations take on new signif
icance, and Mahowald has been involved in both ends: with the second
author and P. Shick [57 ] and with N. Shimomura [58 ].
o The development of M. J. Hopkins's ideas about generalized dimensions
in localizations of the stable homotopy category.
o The analysis of higher analogues of the EHP sequence [62 ].
We could extend the list further, but we prefer to wish Mark a very happy
birthday and many more years of mathematical leadership!
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Massachusetts Institute of Technology
University of Rochester