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

Abstract

Let S ( p ) 0 S^0_{(p)} denote the sphere spectrum localized at an odd prime p p . Then we have the first beta element β 1 ∈ π 2 p 2 − 2 p − 2 ( S ( p ) 0 ) \beta _1\in \pi _{2p^2-2p-2}(S^0_{(p)}) , whose cofiber we denote by W W . We also consider a generalized Smith-Toda spectrum V r V_r characterized by B P ∗ ( V r ) = B P ∗ / ( p , v 1 r ) BP_*(V_r)=BP_*/(p,v_1^r) . In this note, we show that an element of π ∗ ( V r ∧ W ) \pi _*(V_r\wedge W) gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of β 9 t + 3 \beta _{9t+3} at the prime three to complete a conjecture of Ravenel’s: β s ∈ π 16 s − 6 ( S ( 3 ) 0 ) \beta _{s}\in \pi _{16s-6}(S^0_{(3)}) exists if and only if s s is not congruent to 4 4 , 7 7 or 8 8 mod 9 9 .