Toda [7] has asked whether Smith's K(rc)-construction for n = 3 yields a nontrivial element yx in the p-component p%% of the stable homotopy of spheres (p a prime, p ^ 5). This question has become a major stumbling block, since yt has stubbornly refused to be detected by most conventional invariants [9]. We can now show that yt is essential; moreover (for p ^ 7) it is only the first of a new fa...

We bound the volume of homotopy groups 2-local Goodwillie approximations a sphere in terms amount 2-torsion stable stems, providing Goodwillie-theoretic refinement result Burklund and Senger. At 2 k $2^k$ -excisive approximation, this is obtained by ‘multiplying answer polynomial degree $k$ ’. The main tool Behrens' Goodwillie–EHP long exact sequence.

We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum LK(2)S as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form EhF 2 where F is a finite subgroup of the Morava stabilizer group and E2 is the second Morava or Lubin-Tat...

A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p) acts freely on a product of k spheres, then r ≤ k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non-simply connected spaces.

This paper constructs a new family in the stable homotopy of spheres πt−6S represented by hng0γ3 ∈ E 2 in the Adams spectral sequence which revisits the bn−1g0γ3-elements ∈ πt−7S constructed in [3], where t = 2pn(p− 1) + 6(p + p + 1)(p− 1) and p ≥ 7 is a prime, n ≥ 4.

This paper is a continuation of [Rav02] and the second in a series of papers intended to clarify and expand results of the last chapter of [Rav04], in which we described a method for computing the Adams-Novikov 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 ...

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 will describe a point of view that has emerged as a result of research on the homotopy groups of spheres in the last decade. This philosophy is difficult to translate into theorems or even into precise conjectures, and it is certainly not apparent in the formal literature on the subject. With the exception of Theorem 10, we will not present any proofs or announcements of new re...