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The infinitary \(n\)-cube shuffle. (English) Zbl 1457.55009

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
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References:

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