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It has been conjectured that if $G= \mathop{({ \mathbb{Z} }_{p} )}\nolimits ^{r} $ acts freely on a finite $CW$-complex $X$ which is homotopy equivalent to a product of spheres ${S}^{{n}_{1} } \times {S}^{{n}_{2} } \times \cdots \times {S}^{{n}_{k} } $, then $r\leq k$. We address this question with the relaxation that $X$ is finite-dimensional, and show that, to answer the question, it suffices to consider the case where the dimensions of the spheres are greater than or equal to $2$. [ABSTRACT FROM AUTHOR]

A free Γ- complex is a connected complex X together with an action of Γ which permutes freely the cells of X. Let Γ be a group in the class and X be an infinite-dimensional free Γ-complex which is homotopy equivalent to some sphere Sm, m > 1, and let Ω be the Euler class of X/Γ. Then we prove the following main results: Theorem B. Suppose Γ induces a trivial action on H* ( X). Then X/Γ is homotopy equivalent to a finite-dimensional complex if and only if Γ is torsion-free, or else the natural map Hm+1(Γ,ℤ) → Ĥm+1(Γ,ℤ) sends Ω to Ωˆ, which is an invertible element of the generalized Farrell-Tate cohomology ring of Γ, and m is odd. Theorem C. Suppose Γ induces a nontrivial action on H* (X). Then X/Γ is homotopy equivalent to a finite-dimensional complex if and only if either (1) Γ is torsion-free, (2) Γ≅Γ0 ⋊ H where Γ0 is torsion-free and H is isomorphic to ℤ/2, resΓH(Ω) ≠ 0, and m is even, or else (3) all the torsion elements of Γ lie in Γ0, and Ω is mapped to Ωˆ0 for which some power of Ωˆ0 is an invertible element of the generalized Farrell-Tate cohomology ring of Γ0, and m is odd. [ABSTRACT FROM AUTHOR]