On the non-existence of free \(\mathcal A_d\)-actions on products of spheres (Q2883882)

The paper investigates the well known problem of the existence of a free group action on a product of spheres. The main result states that the alternating group \(A_p\), where \(p\geq 7\) is a prime number, does not act on a space whose integral homology is isomorphic to that of the product of \(p\) copies of \(S^n\). The main technique of the proof is to investigate integral representations of cyclic groups of prime order and to use Bob Oliver's result on the action of \(A_4\) on the product of two spheres.