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In [1], Miller, Ravenel and Wilson defined generalized beta elements in the E2-term of the Adams-Novikov spectral sequence converging to the stable homotopy groups of spheres, and in [5], Oka showed that the beta elements of the form βtp2/r for positive integers t and r survives to the stable homotopy groups at a prime p > 3, when r ≤ 2p − 2 and r ≤ 2p if t > 1. In this paper, we expand the condition so that βtp2/r for r ≤ p2 − 3 survives to the stable homotopy groups.

The beta elementsβtp2∕rin the homotopy of spheres

The beta elementsβ_tp²∕rin the homotopy of spheres