Action of Reduced Powers on the Homology of Products of Complex Projective Spaces and Products of Lens Spaces

The so-called ‘hit problem’ initiated by Peterson in [1] as an attempt at better understanding the \(E_2\)-page of the Adams spectral sequence \(\operatorname{mod} 2\) (that is the cohomology of the Steenrod algebra \( {\mathcal{A}_2} \)) turned out to be very difficult. The hit problem is to determine a minimal generating set for the cohomology of products of infinite projective spaces \({\mathbb R} P^\infty\) as a module over the Steenrod algebra \( {\mathcal{A}_2} \) at the prime 2. The dual problem is to determine the set of \( {\mathcal{A}_2} \)-annihilated elements in the homology of the same spaces. Anick showed that the set of \( {\mathcal{A}_2} \)-annihilated elements in the products of infinite projective spaces \({\mathbb R} P^\infty\) forms a free associative algebra [6]. Ault and Singer proved that, for every \(k \ge 0\), the set of \(k\)-partially \( {\mathcal{A}_2} \)-annihilated elements in homology of products of \({\mathbb R} P^\infty\) (that is a set of elements that are annihilated by \(Sq^{2^i}\) for all \(i \le k\)) also forms a free associative algebra.

In this note, we investigate the dual problem at a prime \(p>2\). In this case, \({\mathbb R} P^\infty\) should be replaced by \({\mathbb C} P^\infty\) if one wants to ignore the action of the Bockstein operation \(\beta\) or by the infinite \(p\)-lens space \(L^\infty\) to take \(\beta\) into consideration. We prove that, for any \(k\ge 0\), a collection of elements in \({\mathbb Z}/p\)-homology of products of \({\mathbb C} P^\infty\) (or \(L^\infty\)) annihilated by all \(P^{p^i}\), \(i\le k\), forms a free algebra. The same holds for the collection of elements annihilated by \(\beta\) and all \(P^{p^i}\), \(i\le k\). We also construct an explicit basis in the subspace \(\bar\Delta(0)_{m,*}\subset H_*(({\mathbb C} P^\infty)^{\wedge m},{\mathbb Z}/p)\), \(m=1, 2\), annihilated by \(P^1\).

DOI 10.1134/S1061920825600230