Brazas, Jeremy The infinitary \(n\)-cube shuffle. (English) Zbl 1457.55009 Topology Appl. 287, Article ID 107446, 13 p. (2021). Higher homotopy groups of a pointed topological space \((X,x_0)\) are commutative, a fact that can be proved by rearranging finitely many cubes on the domain of a map \((S^n, s_0) \to (X,x_0)\). In spaces which are not locally contractible, there is often also a topologically significant operation of concatenation of infinitely many maps \((S^n, s_0) \to (X,x_0)\), as long as the maps grow “ever smaller” in a certain way. A natural question that arises in this setting is whether it would be possible to reshuffle such a concatenation in a reasonable way. This amounts to a rearrangement of infinitely many cubes on the domain of a map \((S^n, s_0) \to (X,x_0)\).Such a rearrangement has been constructed in [K. Eda and K. Kawamura, Fundam. Math. 165, No. 1, 17–28 (2000; Zbl 0959.55010)] and generalized in [K. Kawamura, Colloq. Math. 96, No. 1, 27–39 (2003; Zbl 1037.55003)]. In this paper the author provides a simpler and more general proof, with the generality referring to the fact that a reshuffle remains constant on a predetermined set of finitely many cubes whose complement is path connected. The author concludes by relating his results the higher loop space structure. Reviewer: Ziga Virk (Litija) MSC: 55Q52 Homotopy groups of special spaces 55Q20 Homotopy groups of wedges, joins, and simple spaces 08A65 Infinitary algebras Keywords:higher homotopy group; infinite product; infinitary commutativity; Eckmann-Hilton principle; \(k\)-dimensional Hawaiian earring Citations:Zbl 0959.55010; Zbl 1037.55003 PDFBibTeX XMLCite \textit{J. Brazas}, Topology Appl. 287, Article ID 107446, 13 p. (2021; Zbl 1457.55009) Full Text: DOI arXiv References: [1] Barratt, M. G.; Milnor, J., An example of anomalous singular homology, Proc. Am. Math. Soc., 13, 2, 293-297 (1962) · Zbl 0111.35401 [2] Boardman, J. M.; Vogt, R. M., Homotopy - everything h-spaces, Bull. Am. Math. Soc., 74, 1117-1122 (1968) · Zbl 0165.26204 [3] Eckmann, B.; Hilton, P. J., Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann., 145, 3, 227-255 (1962) · Zbl 0099.02101 [4] Eda, K.; Kawamura, K., Homotopy and homology groups of the n-dimensional Hawaiian earring, Fundam. Math., 165, 17-28 (2000) · Zbl 0959.55010 [5] Kawamura, K., Low dimensional homotopy groups of suspensions of the Hawaiian earring, Colloq. Math., 96, 1, 27-39 (2003) · Zbl 1037.55003 [6] May, J. P., The Geometry of Iterated Loop Spaces, Lecture Notes in Math., vol. 271 (1972), Springer: Springer New York · Zbl 0244.55009 [7] Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Mathematics (1978), Springer Verlag · Zbl 0406.55001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.