On the Homology of Elementary Abelian Groups as Modules over the Steenrod Algebra

TL;DR

This paper investigates the structure of elements in homology that are annihilated by certain Steenrod algebra operations, showing they form a free associative algebra, extending previous results on the dual hit problem.

Contribution

It proves that for each non-negative integer k, the set of elements partially annihilated by Steenrod squares forms a free associative algebra, generalizing known results.

Findings

The set of k partially $ ext{A}$-annihilated elements is a free associative algebra.

Extends Anick's result on $ ext{A}$-annihilated elements to partial annihilation.

Provides algebraic structure insights into the dual hit problem.

Abstract

We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$ at the prime 2. The dual problem is to determine the set of $\mathcal {A}$-annihilated elements in homology. The set of $\mathcal{A}$-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each $k \geq 0$, the set of {\it $k$ partially $\mathcal{A}$-annihilateds}, the set of elements that are annihilated by $Sq^i$ for each $i\leq 2^k$, itself forms a free associative algebra.