Beta families arising from a v29 self-map on S∕(3,v18)

Abstract

We show that v 9 2 is a permanent cycle in the 3-primary Adams-Novikov spectral sequence computing .S =.3; v 8 1 //, and use this to conclude that the families ˇ9tC3=i for i D 1; 2, ˇ9tC6=i for i D 1; 2; 3, ˇ9tC9=i for i D 1; : : : ; 8, ˛1ˇ9 t C3=3 , and ˛1ˇ9 t C7 are permanent cycles in the 3-primary Adams-Novikov spectral sequence for the sphere for all t 0. We use a computer program by Wang to determine the additive and partial multiplicative structure of the Adams-Novikov E 2 page for the sphere in relevant degrees. The i D 1 cases recover previously known results of Behrens and Pemmaraju and the second author. The results about ˇ9tC3=3 , ˇ9tC6=3 and ˇ9tC9=8 were previously claimed by the second author; the computer calculations allow us to give a more direct proof. As an application, we determine the image of the Hurewicz map S ! tmf at p D 3. 55Q45, 55Q51, 55T25ˇ9tC4 ; ˇ9tC7 ; ˇ9tC8 ; ˇ9tC3=3 ; ˇ9t=3;2 ; ˇ3i s=3 i are not permanent cycles for t 1, s 6 Á 0 .mod 3/, and i > 1. The main goal of this paper is to construct a v 9 2 self-map on S =.3; v 8 1 / and show that the remaining ˇelements in s .S / for s Ä jv 9 2 j D 144 also give rise to infinite families.Theorem 5.1 For all t 0, the classes ˇ9tC3=j for j D 1; 2; ˇ9tC6=j for j D 1; 2; 3; ˇ9tC9=j for j D 1; : : : ; 8; ˛1ˇ9 t C3=3 and ˛1ˇ9 t C7 are permanent cycles in the Adams-Novikov spectral sequence for the sphere.