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Partition of R^4 into two-dimensional spheres

Determine whether the four-dimensional Euclidean space R^4 can be partitioned into isomorphic copies of the two-dimensional sphere S^2.

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Background

This is highlighted as the simplest specific instance of the broader unresolved question about partitioning Rm into copies of Sn in intermediate dimensions.

Resolving the R4 versus S2 case would shed light on techniques needed for higher-dimensional sphere partitions and clarify the necessity (or not) of choice principles in such decompositions.

References

As the simplest example, it is not known whether $R4$ can be partitioned into two dimensional spheres Question 3.1.iv.

— Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals (2501.03131 - Fatalini, 6 Jan 2025) in Subsection “Overview of PUCs” (Subsection \ref{subsection: PUC lit review})