Blakers-Massey elements and exotic diffeomorphisms of 𝑆⁶ and 𝑆¹⁴ via geodesics

Abstract

We use the geometry of the geodesics of a certain left-invariant metric on the Lie group S p ( 2 ) Sp(2) to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of π 6 ( S 3 ) \pi _6(S^3) ) and an exotic (i.e. not isotopic to the identity) diffeomorphism of S 6 S^6 (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of π 14 ( S 7 ) \pi _{14}(S^{7}) and exotic diffeomorphisms of S 14 S^{14} , thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.