Free Finite-Group Actions on Products of Spheres

Definitions and mathematical statement

Question. Which additional conditions on a finite group $G$ yield a free action by homeomorphisms or diffeomorphisms on $(S^n)^k$ with $k>1$? Write $X_{n,k}=\prod_{j=1}^k S^n$, with the standard product topology and, for the smooth version, standard product smooth structure. A free topological action is a homomorphism $\rho:G\to\operatorname{Homeo}(X_{n,k})$ such that $\rho(g)x\ne x$ for every $x\in X_{n,k}$ and every $g\ne e$; a free smooth action requires $\rho(g)\in\operatorname{Diff}(X_{n,k})$. The two categories give distinct existence questions and are not identified. The action need not preserve factors, be linear, preserve orientation, or act trivially on homology unless further source hypotheses require it. The displayed product has equal-dimensional sphere factors.

A precise version must specify whether $n$ and $k$ are prescribed, whether $k$ is prescribed and $n$ may depend on $G$, or whether both may vary with $G$; in the positive-dimensional convention $n\ge1$. The word additional refers to preceding conditions not present in this record. Consequently the requested necessary or sufficient group-theoretic conditions cannot be fixed from the excerpt alone. Neither an action on a space merely homotopy equivalent to $X_{n,k}$ nor an action on unequal-dimensional spheres automatically answers this actual-product question.

The supplied literature triage checked 2026-08-21 labels the item open. Its assessment discusses the related realization problem: given a finite free $G$-CW complex homotopy equivalent to a product of two spheres, can it be replaced by a free smooth action on the actual product? It reports many later examples but no general realization theorem. This extra starting hypothesis and two-sphere specialization are part of the supplied assessment, not assumptions stated in the main question. The exact sphere dimensions and meaning of replacement or equivariant realization are not given. The dated assessment requires independent verification and does not establish an equivalence between the two formulations.

Short English statement

Find group-theoretic hypotheses ensuring a free action on an actual product of equal-dimensional spheres, separately in the topological and smooth categories. The excerpt omits the preceding hypotheses and whether the sphere dimensions and number of factors are fixed. The attached review concerns a related two-sphere realization problem.

Sources

• Local UnsolvedMath registry, permanent ID 30000130, code catalogue entry; Free Finite-Group Actions on Products of Spheres, Oberwolfach Reports Open Problems.
• Topologie (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/44, pp. 2311--2348. https://doi.org/10.4171/owr/2004/44. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label Hambleton, free actions on products of spheres: https://arxiv.org/abs/math/0309152.
• Supplied literature label Hambleton--Unlu, further examples: https://arxiv.org/abs/0705.4081.
• Supplied literature label Original OWR report: https://doi.org/10.4171/owr/2004/44.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified label concerns removal of a repeated label and balanced notation. Verify dated claims independently.