The root invariant in homotopy theory

Abstract

FOR THE last thirty years the EHP sequence has been a major conceptual tool in the attempt to understand the homotopy groups of spheres. It is a collection of long exact sequences of homotopy groups induced by certain fibrations in which all three spaces are loop spaces of spheres. These fibrations are due originally to James, G. W. Whitehead, and Toda. The Freudenthal suspension theorem and the Adams vector field theorem (which is a strengthened form of the Hopf invariant one theorem) can each be interpreted as statements about the EHP sequence. James periodicity, the Hopf invariant and the Whitehead product all fit into the EHP framework in a very simple way. An expository survey of this material is given in the last section of the first chapter of [36]. More recently the work of Morava led the second author and various collaborators to formulate the chromatic approach to stable homotopy theory and the notion of a v,-periodic family (see [32], [35], [29] and the last three chapters of 11361). The recent spectacular work of Devinatz, Hopkins and Smith [l l] is a vindication of this point of view. The purpose of this paper is to describe the partial understanding we have reached on how the chromatic and EHP points of view interact. The central concept here is the root invariant, which is defined in 1.10 using Lin’s theorem. This assigns to each element in the stable homotopy of a finite complex a nonzero coset in a higher stem. The main conjecture (still unproved) in the subject is that this root invariant converts v,-periodic families to v,+ 1-periodic families. The full implications of this are still not understood. In the first section we will recall the relevant properties of the EHP sequence including James periodicity and define the root invariant in the homotopy of the sphere spectrum. Regular and anomalous elements in the EHP sequence will be defined (1.11). In the second section we will generalize the definition to finite complexes, describe Jones’ connection between the root invariant and the quadratic construction and develop various computational tools. In the third section we will indicate the relation between the root invariant and the Greek letter construction. In the fourth section we will consider unstable homotopy and define the progeny (4.1) and the target set (4.3) of an element in the EHP sequence. In the fifth section we will prove that an element is anomalous if and only if it is a root invariant. In the last section we will give a method of improving James periodicity in many cases. Finally we will construct some similar spectral sequences in which the theorem connecting anomalous elements and root invariants (1.12) does not hold; these will be parametrized by the p-adic integers.