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The role of commutators in a non-cancellation phenomenon. (English) Zbl 1034.57029
Author’s abstract: We construct an explicit (from the transition function point of view) diffeomorphism between the cartesian products with the 3-sphere of two 10-dimensional loop spaces that are not homotopy equivalent to each other. Our method employs specific models for some \(S^3\)-principal bundles over \(S^7\) and relates the study of this type of non cancellation phenomena to commutators of groups. Our formulas depend only on specifying homotopies of powers of commutators to constants.

57R50 Differential topological aspects of diffeomorphisms
55P15 Classification of homotopy type
57R19 Algebraic topology on manifolds and differential topology
55Q15 Whitehead products and generalizations