Elementary abelian groups acting on products of spheres (Q1273145)

A classical result in transformation groups says a finite abelian group acting freely on a sphere must be cyclic. This result was extended by G. Carlsson. If \(G\) is an elementary abelian \(p\) group of rank \(r\), and \(G\) acts freely, cellularly, and homologically trivially on a CW-complex \(X\simeq(S^n)^k\), then \(k\geq r\). Adem and Browder verified the same inequality also for actions which are not assumed to be homologically trivial, except in the case \(p=2\) and \(n=1, 3\), or 7. The question was raised whether a stronger inequality holds, namely \(r\leq\dim H_n(X;\mathbb{F}_p)^G\). In their paper, the authors establish this stronger inequality for a large class of actions, namely those which induce a permutation module on \(H^n(X;\mathbb{F}_p)\). If \(p=2\) and \(n\neq 1, 3\), or 7 this is the general case.