The Adams-novikov Spectral Sequence for the Spheres

The Adams spectral sequence has been an important tool in research on the stable homotopy of the spheres. In this note we outline new information about a variant of the Adams sequence which was introduced by Novikov [7]. We develop simplified techniques of computation which allow us to discover vanishing lines and periodicity near the edge of the E2-term, interesting elements in E^'*, and a counterexample to one of Novikov's conjectures. In this way we obtain independently the values of many low-dimensional stems up to group extension. The new methods stem from a deeper understanding of the Brown-Peterson cohomology theory, due largely to Quillen [8]; see also [4]. Details will appear elsewhere; or see [ l l ] . When p is odd, the p-primary part of the Novikov sequence behaves nicely in comparison with the ordinary Adams sequence. Computing the £2-term seems to be as easy, and the Novikov sequence has many fewer nonzero differentials (in stems ^ 4 5 , at least, if p = 3), and periodicity near the edge. The case p = 2 is sharply different. Computing E2 is more difficult. There are also hordes of nonzero differentials dz, but they form a regular pattern, and no nonzero differentials outside the pattern have been found. Thus the diagram of £4 ( =£oo in dimensions ^17) suggests a vanishing line for Ew much lower than that of £2 of the classical Adams spectral sequence [3]. I t is a pleasure to thank Arunas Liulevicius, my thesis adviser, for his help. In particular, parts of the proofs of Proposition 1 and Theorem 7 are due to him. I am also grateful to many others for their suggestions, and especially to Frank Adams.