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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Blakers-Massey elements and exotic diffeomorphisms of $S^6$ and $S^{14}$ via geodesics


Authors: C. E. Durán, A. Mendoza and A. Rigas
Journal: Trans. Amer. Math. Soc. 356 (2004), 5025-5043
MSC (1991): Primary 53C22; Secondary 58F07, 57R15
Published electronically: June 29, 2004
MathSciNet review: 2084409
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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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Additional Information

C. E. Durán
Affiliation: Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado 21827, Caracas 1020A, Venezuela
Address at time of publication: IMECC-UNICAMP, C.P. 6065, 13083-970, Campinas, SP, Brazil
Email: cduran@cauchy.ivic.ve, cduran@ime.unicamp.br

A. Mendoza
Affiliation: Departamento de Matemáticas Puras y Aplicadas, Universidad Simón Bolívar, Apartado 28000, Caracas 1080, Venezuela
Email: jacob@usb.ve

A. Rigas
Affiliation: IMECC-UNICAMP, c.p. 6065, 13083-970, Campinas, SP, Brazil
Email: rigas@ime.unicamp.br

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03469-5
PII: S 0002-9947(04)03469-5
Keywords: Geometry of geodesics, Blakers-Massey element, exotic spheres
Received by editor(s): March 21, 2003
Received by editor(s) in revised form: July 29, 2003
Published electronically: June 29, 2004
Article copyright: © Copyright 2004 American Mathematical Society