Group actions on finite CW-complexes (Q1801382)

A lot of people have worked on some version of the following conjecture: Let \((\mathbb{Z}/2)^ l\) act freely and cellularly on a CW-complex that is homotopy equivalent to a product of \(m\) spheres. Then \(l\leq m\). (The author gets the inequality wrong in his statement in the introduction.) G. Carlsson showed that \(l\geq 3\) implies \(m\geq 3\). The author pushes Carlsson's calculations one step further to show that \(l\geq 4\) implies \(m\geq 4\). In fact, he shows that the dimension of the \(\mathbb{Z}/2\)- cohomology of a finite CW-complex \(X\) on which \((\mathbb{Z}/2)^ l\) acts freely and cellularly, has to be even and at least 10.