On the homology of elementary Abelian groups as modules over the Steenrod algebra (Q649849)

The hit problem of Peterson is concerned with finding an expression such as \(x=\sum_{i=1}^n Sq^i x_i\) for given \(x\in H^*(X;\mathbb F_2)\) where \(x_i\in H^*(X;\mathbb F_2)\), \(i>0\), and \(Sq^i\) denotes the \(i\)-th Steenrod square. Geometrically, the answer to this problem determines how, at the prime 2, cells of a given \(CW\)-complex are connected to the cells below themselves. Algebraically, this is the same problem as determining a minimal generating set for \(H^*(X;\mathbb F_2)\) over the mod \(2\) Steenrod algebra \(\mathcal{A}\). The dual approach to this problem is to find all \(\mathcal{A}\)-annihilated classes in \(H_*(X;\mathbb F_2)\). Here \(x\) is called \(\mathcal{A}\)-annihilated if \(Sq^ix_*=0\) for all \(i>0\) where \(Sq^i_*:H_*(X;\mathbb F_2)\longrightarrow H_{*-i}(X;\mathbb F_2)\) is the operation dual to \(Sq^i\). By its nature, this problem is connected to, and interacts with, other important problems in algebra and topology in the study of the Dickson algebra and its invariants, and (modular) representation theory over general linear groups over fields of characteristic \(2\), etc. From the point of view of stable homotopy, it reveals the stable structure of a given \(CW\)-complex, and possible stable splittings of a given topological space. In spite of its geometric importance it mostly has been studied by algebraic methods. Working at the prime \(2\), it is of special interest to solve the problem for the spaces \(X=\mathbb R P^{\times n}\), the \(n\)-fold products of \(\mathbb R P\). The case \(n=1\) is straightforward as only classes in \(H_{2^i-1}(\mathbb R P;\mathbb F_2)\) are \(\mathcal{A}\)-annihilated. For \(n=2\) the first solution seems to appear in work of Alghamdi, Crabb and Hubbuck. For \(n=3\) the solution appeared in Kameko's thesis. For arbitrary \(n\) the problem in generic degrees was solved by Nguyen Sum. Therefore, a complete solution for the hit problem when \(n>5\) is still not available. By considering \(\mathbb R P\) as \(B\mathbb F_2\) the authors take \(H_*(\mathbb R P^{\times n};\mathbb F_2)\) as the homology of an elementary Abelian \(2\)-group of rank \(n\), namely \((\mathbb F_2)^{\oplus n}\). A result of Anick says that \(\mathcal{A}\)-annihilated elements in \(\bigoplus_n H_*(\mathbb R P^{\times n};\mathbb Z/2)\) form an associative algebra where the algebra structure is induced by the obvious maps \(\mathbb R P^{\times n}\times \mathbb R P^{\times m}\longrightarrow \mathbb R P^{\times (n+m)}\). Working with reduced homology, the authors build upon Anick's result and show that the \(\mathcal{A}\)-annihilated classes in \(\bigoplus_n H_*(\mathbb R P^{\wedge n};\mathbb Z/2)\) also form an associative algebra where in the latter the algebra structure is induced by pairings \(\mathbb R P^{\wedge n}\times \mathbb R P^{\wedge m}\longrightarrow \mathbb R P^{\wedge (n+m)}\). A class \(x\) is referred to as \(k\) partially \(\mathcal{A}\)-annihilated if \(Sq^i_*x=0\) for all \(i\leqslant2^k\). The main result of this paper shows that the set of all \(k\) partially \(\mathcal{A}\)-annihilated classes in \(\bigoplus_n H_*(\mathbb R P^{\wedge n};\mathbb Z/2)\), denoted by \(\Delta(k)\), is a free subalgebra. The algebras \(\Delta(k)\) filter the subalgebra of \(\mathcal{A}\)-annihilated classes in \(\bigoplus_n H_*(\mathbb R P^{\wedge n};\mathbb Z/2)\). Hence, determining the \(\Delta(k)\)'s can lead to a solution to the hit problem. The paper is written in a very algebraic language like many other papers on the problem. The reviewer wishes to thank Reg Wood for helpful notes on the history of the `hit problem'.