Mahowaldean families of elements in stable homotopy groups revisited

Abstract

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 h₂h²_j∈ Ext^3,*_{[Ascr ]}(Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in π^S_∗(S⁰)_(p) detected by h₀b_j∈ Ext^3,*_{[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 h₀h_j∈ Ext^2,*_{[Ascr ]} (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)