On the non‐existence of free \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$ \mathcal {A}_d$\end{document} ‐actions on products of spheres

We investigate the problem of existence of free alternating group actions on products of equidimensional spheres. Most notably, we prove that for a prime number p ≥ 7, the alternating group \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {A}_{p+1}$\end{document} cannot act freely on a finite‐dimensional CW‐complex with the integral cohomology ring isomorphic to that of ( S n ) p for any positive integer n . This extends previous results due to L. P. Plakhta.