Free $\mathcal{A}_4$-actions on products of spheres

Abstract

We summarize what is known about free actions of $\mathcal{A}_4$, the alternating group on four letters, on products of spheres. New results are also included: in particular, we prove that $\mathcal{A}_4$ acts freely on $S_n \times S_n \times S_n$ if and only if $n = 1, 3, 7$.