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

Durán, C.; Mendoza, A.; Rigas, A.

doi:10.1090/S0002-9947-04-03469-5

Blakers-Massey elements and exotic diffeomorphisms of $S^6$ and $S^{14}$ via geodesics
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by C. E. Durán, A. Mendoza and A. Rigas PDF

Trans. Amer. Math. Soc. 356 (2004), 5025-5043 Request permission

Abstract:

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

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