Note on beta elements in homotopy, and an application to the prime three case

Abstract

Abstract. Let S 0 (p) denote the sphere spectrum localized at an odd prime p. Then we have the first beta element β 1 ∈ π 2p 2 −2p−2 (S 0 (p) ), whose cofiber we denote by W . We also consider a generalized Smith-Toda spectrum V r characterized by BP * (V r ) = BP * /(p, v r 1 ). In this note, we show that an element of π * (V r ∧ W ) gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of β 9t+3 at the prime three to complete a conjecture of Ravenel's: β s ∈ π 16s−6 (S 0 (3) ) exists if and only if s is not congruent to 4, 7 or 8 mod 9.

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

“…Behrens and Pemmaraju [3] show there is a v 9 2 self-map on S=.3; v 1 / and use this to prove the existence of nonzero homotopy classes represented by ˇ9tCs for s D 1; 2; 5; 6; 9 and t 0. The second author [18] proves the existence of ˇ9tC3 . By comparison to L 2 -local homotopy, he shows in [15] that the elements These families are interesting in part because the Hurewicz map .S/ !…”