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

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

2011-05-05

arxiv.org/abs/1105.1139v1

Mathematics

Algebraic Topology

6 pages + references

Scientific paper

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.

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