An exotic involution of the 5-sphere
A QuickTime movie by Uwe Abresch, Carlos Duran, Thomas Püttmann,
and A. Rigas
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in QuickTime-format (7.6 MByte)
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Additional comments on the movie:
- If one rotates twice as much in the second step one gets a regular involution
(i.e an involution which is conjugate by diffeomorphisms to the antipodal map).
Actually, for rotations by an angle of k |p| 360° one gets regular involutions for even k and exotic involutions for odd k.
- The involution described in the movie is fixed point free since the axis of rotation
in the second step is contained in the plane spanned by w and p. Hence, not both vectors, w and p, can be mapped to their negatives in the second step.
- The exoticity of the involution is deduced from its origin: The Gromoll-Meyer
sphere in dimension 7. It would be interesting to see directly why the construction
presented in the movie leads to something exotic.
All pictures and submovies were created with the shareware program CurvusProX by Simon Bovet.
- U. Abresch, C. Duran, T. Püttmann, A. Rigas:
Wiedersehen metrics and exotic involutions of Euclidean spheres,
J. Reine Angew. Math. 605 (2007), 1-21.
- M. Hirsch, J. Milnor: Some curious involutions of spheres,
Bull. Amer. Math. Soc. 70 (1964), 372-377.
- S. Lopez de Modrano: Involutions on manifolds, Springer-Verlag 1971.
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