The v₁-periodic part of the Adams spectral sequence at an odd prime/

Abstract

We tell the story of the stable homotopy groups of spheres for odd primes at chromatic height 1 through the lens of the Adams spectral sequence. We find the "dancers to a discordant system." We calculate a Bockstein spectral sequence which converges to the 1-line of the chromatic spectral sequence for the odd primary Adams E₂-page. Furthermore, we calculate the associated algebraic Novikov spectral sequence converging to the 1-line of the BP chromatic spectral sequence. This result is also viewed as the calculation of a direct limit of localized modified Adams spectral sequences converging to the homotopy of the v1 -periodic sphere spectrum. As a consequence of this work, we obtain a thorough understanding of a collection of q₀-towers on the Adams E₂-page and we obtain information about the differentials between these towers. Moreover, above a line of slope 1/(p²-p-1) we can completely describe the E₂ and E₃ -pages of the mod p Adams spectral sequence, which accounts for almost all the spectral sequence in this range.