Abstract
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
