Rank 3 finite \(p\)-group actions on products of spheres (Q2801727)

Classifying all finite groups that can act freely on a product of spheres is a classical problem in the theory of transformation groups. \textit{J. W. Milnor} has shown that, in general, the rank condition \(\text{rk}(G)\leq k\) is not sufficient for the existence of a free smooth action of a finite group \(G\) on a product of \(k\) spheres [Am. J. Math. 79, 623--630 (1957; Zbl 0078.16304)]. However, for constructing free smooth actions of \(p\)-groups, there are no known necessary conditions on the group other than the rank condition. The main result of this paper states that, for an odd prime \(p\), every rank \(3\) \(p\)-group acts freely and smoothly on a product of three spheres. To prove the result, the author first generalizes a result of \textit{W. Lück} and \textit{B. Oliver} on constructions of \(G\)-equivariant vector bundles over finite-dimensional \(G\)-CW-complexes [Topology 40, No. 3, 585--616 (2001; Zbl 0981.55002)]. He also gives some other applications of this generalization.