Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

Abstract

Abstract. The mod 2 Steenrod algebra A and Dyer-Lashof algebra R have both striking similarities and differences arising from their common origins in "lower-indexed" algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, but quite different from, those of A and R. The exact relationships emerge as "sheared algebra bijections", which also illuminate the role of the cohomology of K. As a bialgebra, K * has a particularly attractive and potentially useful structure, providing a bridge between those of A * and R * , and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras.

The odd-primary Kudo–Araki–May algebra of algebraic Steenrod operations and invariant theory

“…We have also used this structure in work to appear [16] to give a minimal presentation for the mod p cohomology of CP(∞) as a module over the Steenrod algebra, which in turn allows us to give a minimal presentation of the cohomology of the classifying space BU (ie the algebra of symmetric invariants) as an algebra over the Steenrod algebra. Corresponding results at the prime 2, some joint with Peterson, appear in [15,17,18]. In [17] we analyzed the analogous algebra of operations D i at the prime 2 and named it the Kudo-Araki-May algebra K. Where results in this paper are completely analogous to those in [17], we shall omit their proofs; proofs that are not analogous or immediate will be given in subsequent sections.…”

“…Corresponding results at the prime 2, some joint with Peterson, appear in [15,17,18]. In [17] we analyzed the analogous algebra of operations D i at the prime 2 and named it the Kudo-Araki-May algebra K. Where results in this paper are completely analogous to those in [17], we shall omit their proofs; proofs that are not analogous or immediate will be given in subsequent sections.…”

“…Definition 1.1 Let U be the free noncommutative F p -algebra generated by elements e i , for i ≥ 0, of topological degree |e i | = 2i. The topological degree of a product xy in U is given by |xy| = |x| + p |y| (cf [17]). Note that U is bigraded, by topological degree t and by length n. We write U n,t for the component in this bidegree.…”

“…Note that since, as remarked above, the relations are homogeneous in both bidegrees, K, K, and K will inherit these bidegrees as well. By analogy with the prime 2 [17], and because it consists of the algebra of operations acting in topology on even degrees, we shall call K the even topological Kudo-Araki-May algebra. Obviously K and K provide larger, purely algebraic, versions.…”

Unstable modules with only the top k Steenrod operations

“…Remark 3.2.11. The work[PW00] introduced a bigraded bialgebra 𝒦 over F 2 and its sub-bialgebra 𝒦(𝑘). The category 𝒰 is equivalent to the category of unstable modules over 𝒦, and the category 𝒰 𝑘 is equivalent to the category of unstable modules over 𝒦(𝑘).…”