v 1 - and v 2 -Periodicity in Stable Homotopy Theory

Abstract

this paper we construct some self-maps related to the elements v1 E 7r2(BP) and v2 E 76(BP) and use them to obtain families in the 2-primary stable homotopy of spheres. In particular, we obtain nonzero elements in the 48k - 10 stem for all k > 0. Here BP denotes the Brown-Peterson spectrum associated to p 2. Although our results have convenient interpretations in the BP-AdamsNovikov spectral sequence ([251), we make hardly any use of this spectral sequence. Except in the final step of one argument, the only knowledge of BP required here is its homotopy ring. Our methods are those of the classical (K(Z2)) Adams spectral sequence (ASS). As we shall be working entirely in the stable category we will denote by [X, Y] the abelian group of stable homotopy classes of maps X to Y when X is a finite CW complex and Y a spectrum. We will identify [X, Y] with [EiX, 'Y] for any integer i, and we will not distinguish between a map and its (stable) homotopy class. If v E [E'X, X], vk denotes