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 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.)

Freudenthal's theorem and spherical classes in H*QS0

“…(p > 2) Let f be an element of the family considered by Hunter and Kuhn [12,Corollary 3.8] detected by dh 0 h i in the Adams spectral sequences. Then, h(f…”

“…where D k,n X is the cofibre of C k n−1 (X) → C k n (X) given by F (R k , n) ⋉ Σn X ∧n [21]. By construction, D k,n X is a genuine space for any k and n which has its bottom cell in dimension rn if X has its bottom cell in dimension r. For a prime p and t ∈ {0, 1}, following Hunter and Kuhn [12], at the prime p define…”

“…be an element of the family considered by Hunter and Kuhn [14,Corollary 3.8] detected by dh 0 h i in the ASS whose construction depends on the existence of certain elements δ…”

“…Acknowledgements. I am grateful to Nick Kuhn for his comments and notes on earlier versions of this document, as well as the material of [14,16]. I am grateful to an anonymous referee for his/her helpful comments; in particular, for suggesting a simple proof of Proposition 2.1 as quoted, as well as Theorem 1.2, which allowed the quick proof of some of the results in this paper.…”

Eilenberg-MacLane spectra as equivariant Thom spectra

“…One of Mahowald's motivations for proving the equivalence 2 S 3 q µ » HF 2 is that the left hand side carries a natural filtration due to Milgram and May. This produces a filtration of HF 2 by spectra which turn out to be the Brown-Gitler spectra of [BG73] (see [BP78,Coh79,HK99]). The G-space Ω λ S λ`1 also carries the arity filtration from the E λ -operad, so we could define equivariant Brown-Gitler spectra using this filtration.…”