A note on free actions of groups on products of spheres (Q2865139)

The paper under review considers the following long-standing conjecture in algebraic topology, saying that if \(({\mathbb Z}_p)^r\) acts freely on a finite CW-complex which is homotopic to a product of spheres \(S^{n_1}\times\cdots \times S^{n_k}\), then \(r\leq k\). Note that it has been shown that the conjecture holds in many cases. In the above statement, replacing `a finite CW-complex' by `a finite-dimensional CW-complex', the authors of the paper show that the conjecture can be induced by the following conjecture:NEWLINENEWLINE\noindent { Conjecture.} If \(({\mathbb Z}_p)^r\) acts freely on a finite-dimensional CW-complex which is homotopic to a product of spheres \(S^{n_1}\times\cdots \times S^{n_k}\) with \(n_i\geq 2\), then \(r\leq k\).NEWLINENEWLINEThe proof is based upon the following main theorem of the paper: Let \(M\) be a compact solvmanifold and \(L\) a finite-dimensional CW-complex with a finite fundamental group. Then NEWLINE\[NEWLINE\text{hfrk}_p(M\times L)\leq \dim M+\text{hfrk}_p(L)NEWLINE\]NEWLINE where \(p\) is a prime, and \(\text{hfrk}_p(X)\) denotes the maximal \(r\) such that \(({\mathbb Z}_p)^r\) acts freely on a finite-dimensional CW-complex homotopic to \(X\).