Abstract
We construct natural self-maps of compact cohomogeneity one manifolds and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields relations between the order of the Weyl group and the Euler characteristic of a principal orbit. As examples we determine all cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type that lead to self-maps of degree ≠ −1; 0; 1. We derive explicit formulas for new coordinate polynomial self-maps of the compact matrix groups SU(3), SU(4), and SO(2n). For SU(3) we determine precisely which integers can be realized as degrees of self-maps.
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Supported by a DFG Heisenberg scholarship and DFG priority program SPP 1154.
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Püttmann, T. Cohomogeneity One Manifolds and Self-Maps of Nontrivial Degree. Transformation Groups 14, 225–247 (2009). https://doi.org/10.1007/s00031-008-9037-6
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DOI: https://doi.org/10.1007/s00031-008-9037-6