Constructing free actions of \(p\)-groups on products of spheres. (English) Zbl 1237.57037

The algebraic rank of a group \(G\), \(rk(G)\), is the largest integer \(k\) such that there is a prime \(p\) such that \((\mathbb{Z}/p)^k< G\). The homotopy rank of \(G\), \(hrk(G)\), is the smallest integer \(k\) such that \(G\) acts freely on a finite CW-complex in the homotopy type of a product of \(k\) spheres. If \(G\) is a finite group, then \(rk(G)= 1\) if and only if \(hrk(G)= 1\) [R. G. Swan, Ann. Math. (2) 72, 267–291 (1960; Zbl 0096.01701)]. It has been conjectured [D. J. Benson and J. F. Carlson, Math. Z. 195, 221–238 (1987; Zbl 0593.20062)] that \(rk(G) = hrk(G)\) if \(G\) is finite. If \(G\) has odd order and \(rk(G)= 2\), then \(hrk(G)= 2\) [A. Adem, Lectures on the cohomology of finite groups. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 436, 317–334 (2007; Zbl 1132.20031)].
The main result in this paper is that if \(p\) is an odd prime, then every finite \(p\)-group of \(G\) of rank 3 acts freely on a finite CW-complex in the homotopy of type of a product of 3 spheres, that is \(hrk(G)\leq 3\). It follows that \(hrk(G)= 3\) for such groups because \(hrk(\mathbb{Z}/p)^3)> 2\) [A. Heller, Ill. J. Math. 3, 98–100 (1959; Zbl 0084.38803)].


57S17 Finite transformation groups
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
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