Higher Order Operations

Abstract

Introduction.In a recent paper [14] the author gave a general definition of secondary operations </, g, h > which included as special cases various examples of secondary operations which had been considered previously [1; 3; 9; 11; 18].This provided a unified treatment of these particular operations and their properties.As pointed out in [14], however, not all examples of secondary operations were included in this treatment.Specifically, cohomology operations involving cup products (such as Steenrod's functional cup product [17] and the Massey triple product [6]) were not obtainable as special cases of the operation <f,g,h>.The present paper contains a definition and discussion of operations of all orders.The definition is such that for secondary operations it is more general than the definition in [14] and includes, in addition to the operation (.f,g,h}, examples of secondary operations involving cup products.The definition of a higher order operation we adopt is that it is a higher order obstruction to the existence of a null-homotopy of a map carried by a carrier on a simplicial complex.Thus a higher order operation will be defined only when certain lower order operations vanish and, when defined, will have a certain indeterminacy.Naturality properties of obstructions imply those of operations, and the fact that obstructions are cocycles gives relations among operations.Various examples of operations are obtained by specializing to particular simplicial complexes, particular carriers on them, and particular maps carried by these carriers.We present no applications herein of the operations.Many applications of special operations can be found in the literature (for example in [1; 6; 10; 17; 18]), and it is our intention to present a unification of the theory of operations rather than new applications of them.However, new secondary operations based on the suspension and loop functors are defined (see (4.4)-(4.7)below) which should have interesting applications.Special cases of the one involving suspension have been considered by Toda in connection with his toral construction (see [19]).After a first section devoted to notation and some standard facts, the second section deals with coefficient stacks on simplicial complexes and their corre-