Odd primary Steenrod operations in first-quadrant spectral sequences

Abstract

This paper defines two kinds of Steenrod operations in the spectral sequence of a bisimplical mod p coalgebra and shows them to be a complete list of all such possible Steenrod operations. These operations are compatible with the differentials and with Steenrod operations on the total complex. A general rule is given for computing the operations on E2. A generalization of the Kudo transgres- sion theorem is also proved, placing it in a larger and more natural setting. 1. Introduction. In 1955, Massey (9) asked if it were possible to introduce Steenrod squares and/ith powers into the spectral sequence of a fibre space. Since that time several people have contributed to the solution of this problem, notably Vazquez (22), Araki (1), Kristensen (5), Kuo (7), Rector (15), Smith (18), Singer (16), Mori ((12) and (13)), and Dwyer (4). This paper will define mod p Steenrod operations on a class of spectral sequences; its sequel will define higher divided powers in certain spectral sequences. These operations are shown to be a complete list of all Steenrod-type operations on spectral sequences. This extends to the odd prime case the work of Singer (16) on Steenrod squares in first-quadrant spectral sequences and the work of Dwyer (4) on higher divided squares in second quadrant spectral sequences. Mori ((12) and (13)) has also obtained most of our results for the odd prime case of the Eilenberg-Moore spectral sequence through different methods. In the course of these papers the ambiguity regarding indeterminacy in the definition of the operations in (16), (12), and (13) is resolved, and the question asked by Singer (16) regarding the existence of horizontal operations in the spectral sequence of a mixed simplicial object is answered. A generalization of the Kudo transgression theorem (6) is also proved, placing the usual Kudo transgression theorem in a larger and more natural setting. The principal applications of these results are to the Eilenberg-Moore and the Serre spectral sequences as well as to the change-of-rings spectral sequence as outlined by Singer in (17). Let A be the category of finite ordered sets and nondecreasing maps (10, p. 4) and let Aop be the opposite category. A bisimplicial coalgebra is a covariant functor from