In the mid 1970s Mark Mahowald constructed a new infinite family of elements in
the 2-component of the stable homotopy groups of spheres,
ηj∈πSj2
(S0)(2) [M]. Using
standard Adams spectral sequence terminology (which will be recalled in Section 3
below), ηj is detected by
h1hj∈Ext2,*[Ascr ]
(Z/2, Z/2). Thus he had found an infinite 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 theory and, on the other
hand, mod2 cohomology making them amenable to Adams Spectral Sequence
analysis and suggesting that they were related to the new discovered Brown–Gitler
spectra [BG].
In the years that followed, various other related 2-primary infinite families were
constructed, perhaps most notably (and correctly) Bruner's family detected by
h2h2j∈
Ext3,*[Ascr ](Z/2, Z/2)
[B]. An odd prime version was studied by Cohen [C],
leading to a family in
πS∗(S0)(p) detected by
h0bj∈
Ext3,*[Ascr ]
(Z/p, Z/p) and a filtration 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 1980s,
using a different approach, by Goerss [G], and given the ultimate ‘modern’
treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers
in the late 1970s and early 1980s, 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 work on
Brown–Gitler spectra led to clarification of the original constructions. The other was
to see if taking advantage of post Segal Conjecture knowledge of the stable
cohomotopy of the classifying space BZ/p would help in
constructing new families at odd primes, in particular a conjectural family detected by
h0hj∈
Ext2,*[Ascr ]
(Z/p, Z/p).
(This followed a paper [K1] by one of us on 2 primary families from this point of
view.)