An alternative approach to homotopy operations

Abstract

AbstractWe give a particular choice of the higher Eilenberg-MacLane mapsby a recursive formula. This choice leads to a simple description ofthe homotopy operations for simplicial Z/2-algebras. 1 Introduction This paper is about the ring of homotopy groups of a simplicial ring. Thisring of homotopy classes has a huge amount of additional structure. Thetheory is best worked out for algebras over F 2 , and we will restrict ourselvesto this case. §§2-3 in [3] contain a good survey with references to the originalarticles. We just recall the points which are most important to us.The main observation is that the square of every element in positivedegree is zero. Analyzing this fact gives rise to a divided power structure onthe ring of homotopy groups. There is a refinement of this, which constructsa sequence of homotopy operations δ i . These are defined by Dwyer in [2] andalso by Bousfield.The homotopy of a simplicial algebra R • is isomorphic to the homology ofthe associated chain complex C ∗