The Adams-Novikov spectral sequence for the spheres

Abstract

The Adams spectral sequence for the stable homotopy of the spheres has been extensively studied since its introduction in 1958 [1]. Variants of this sequence using extraordinary cohomology theories have exciting possibilities, but are not well understood. In this paper we consider the MU-cobordism version first studied by Novikov [13]. We find that for odd primes this sequence is drastically simpler than the classical Adams spectral sequence; while it does not collapse, as Novikov first thought it would, we have been able to find only one family of nonzero differentials. For the prime 2 the sequence is more complicated and harder to compute in low dimensions, but still displays several interesting new patterns (see Tables 2 and 3). Our results also comprise vanishing lines for the E2-term of the MUspectral sequence and periodicity phenomena near this edge. We have not yet determined the full extent of this periodicity, but in a range of dimensions the diagonal parallel to the line t = 2s eventually stabilize for p = 2; for odd primes these leaning towers form jagged lines at smaller angles. We also find recurring families of elements in E,' *. For p = 2 one of these families corresponds to the classical Arf-invariant elements h2 [8]. Our basic method is simply construction of economical resolutions over the algebra of operations BP*(BP), which is the analogue of the classical Steenrod algebra in the cohomology theory given by the p-primary BrownPeterson spectrum pBP. Using Quillen's results [14] about the structure of BP*(BP), we calculate E2 in the range t - s ? 17 for p = 2 and t - s < 45 for p = 3, as well as the multiplicative structure in a somewhat smaller range. We can then independently determine all differentials in this range, with the exception of the 3-primary differential needed to kill the ephemeral element a/. In this way we can recover the corresponding 2-primary and 3-primary stable stems up to group extension. In the course of the work we