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.
Technical comment:
All pictures and submovies were created with the shareware program CurvusProX by Simon Bovet.
References:
- 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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Thomas Püttmann