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Homotopy of compact symmetric spaces

Published online by Cambridge University Press:  18 May 2009

John M. Burns
John M. Burns
Affiliation:
Mathematics Department, University College, Galway, Ireland
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In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 2 , May 1992 , pp. 221 - 228
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

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