A vanishing line in the BP〈1〉-Adams spectral sequence

Abstract

Using techniques of relative homological algebra, for an odd prime p, we describe a vanishing line of slope ( p 2− p−1) −1 in the second term of the BP〈1〉-Adams spectral sequence for the sphere spectrum. As a consequence, the E ∞ term of the classical Adams spectral sequence is shown to have a similar line of slope (2p−1)/[(2p−2) (p 2−p−1)] , above which only the image of the stable J-homomorphism lies. This produces upper bounds for the exponent at p of the stable homotopy groups of spheres.