The Role of Commutators in a Non-Cancellation Phenomenon

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 S³- principal bundles over S⁷ and relates the study of this type of noncancellation phenomena to commutators of groups. Our formulas depend only on specifying homotopies of powers of commutators to constants.