This paper is the first in a series aimed at clarifying and extending of parts of the last chapter of [Rav86], in which we described a method for computing the AdamsNovikov E2-term and used it to determine the stable homotopy groups of spheres through dimension 108 for p = 3 and 999 for p = 5. The latter computation was a substantial improvement over prior knowledge, and neither has been improv...

In this paper we attempt to survey some of the ideas Mark Mahowald has contributed to the study of the homotopy of spheres. Of course, this represents just a portion of Mahowald’s work; some other aspects are described elsewhere in this volume. Even within the restricted area of the homotopy of spheres, this survey can only touch on some of Mahowald’s most seminal contributions, and will leave ...

We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime 2. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the P map, and the Goodwillie differentials to the H map. F...

We give a modern account of Pontryagin’s approach to calculating πn+1(S) and πn+2(S) using techniques from low-dimensional topology.

We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial. 0. Introduction. A group G is said to act semifreely on a space X if every point is e...