ContemporaryMathematicsVolume 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
contributedto 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 ofhomotopy 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 homotopytheory;
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 ona discussion of three of Mahowald's most influ-
entialpapers: 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 thatin his world there are only two primes: 2,
and the "infiniteprime." 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, J-homomorphis*
*m.
Both authors were supported in part by the N. S. F.
fcl00000American0Mathematical0Society*
*0-0000/00 $1.00 + $.25 per page
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 J-homomorphism. Raoul Bott's proof
[12] of the periodicity theorem was published in 1959. This is acomputation
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
k-invariants had been computed;!and!it!is!still the starting point for most such
knowledge.SFor example, the!homotopy!of!thestable orthogonal group O =
O(n) is of period 8 and isgiven!by!the!following table, which may be sung
(from the bottom up)to the tune!of!a!well-known!lullaby[26].
!!!
___!i mod 8s___siO ___
___ 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
Cliffordalgebras.
The relevance of these groups to thehomotopy groups of spheres was well-
known from the construction of the "J-homomorphism" by G. W. Whitehead
[87] in 1942. Since O(n) acts by proper maps on Rn,its action extends to Sn
viewed as the one-point compactification of Rn. This action has the point at
infinity asa fixed-point, and so we get an inclusionof O(n) into the monoid F(n)
of all pointed self-equivalences 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 ff2 O(n) to the lo op-difference 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 J-homomorphism
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 explicitlyconstructed