Abstract for
"Mahowaldean families of elements in stable homotopy groups revisited"
by
David J. Hunter and Nicholas J. Kuhn
Beginning with Mahowald's work in his 1977 Topology "eta_j" paper, various infinite families of elements in the stable homotopy groups of spheres have been constructed in which all elements have a fixed Adams filtration.
We revisited these constructions, figuring that the hindsight of 20 years and a "modern" understanding of both the Segal conjecture and unstable A-module technology might lead to clarification of this work.
What resulted is the following. We isolate the two crucial results from the older literature, and present these stripped of extraneous detours. We then reorganize how these are used, together with the idea the BZ/p should be an intermdiate space in the constructions. This leads to streamlined and unified constructions of Mahowald's 2-primary filtration 2 family, Bruner's 2-primary fitration 3 family, and R.Cohen's odd primary filtration 3 families. In the odd prime case improvements are most dramatic: generously counting, we recover the main results of Cohen's Memoir with a saving of about 45 pages. All of our techniques were available before 1980.
Our unified presentation reveals that Cohen's odd prime families are the exact analogue of Bruner's 2-primary family, and hint that the odd prime families h_0h_j in Ext_A^2(Z/p,Z/p) may behave more like the 2-primary NONpermanent cycles h_2h_j, than the permanent cycles h_1h_j.
A new example shown is that, at odd primes, k_0h_0h_j in Ext_A^4(Z/p,Z/p) is a permanent cycle.