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Archiv der Mathematik

, Volume 82, Issue 2, pp 185–192 | Cite as

Isometric actions on the projective planes and embedded generators of homotopy groups

  • T. Püttmann
  • A. Rigas
Original paper
  • 56 Downloads

Abstract.

We determine all orbits of isometric actions on \( \mathbb{CP}^2 \) , \( \mathbb{HP}^2 \) , and \( \mathbb{OP}^2 \) that are covered by spheres and represent nontrivial classes in the corresponding homotopy groups. As new examples we find orbits that represent generators of \( \pi_{5}(\mathbb{HP}^2) \approx \mathbb{Z}_2 \) and \( \pi_{11}(\mathbb{OP}^2) \approx \mathbb{Z}_{24} \) . These orbits are the quaternionic and octonionic analogues of the quadric \( z_1^2 + z_2^2 + z_3^2 = 0 \) in \( \mathbb{CP}^2 \) .

Mathematics Subject Classification (2000):

Primary 57T20 Secondary 53C35, 53C42, 57S15. 

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Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.IMECCUNICAMPCampinasBrazil

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