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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.

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