Cohomogeneity One Manifolds and Self-Maps of Nontrivial Degree
Cohomogeneity One Manifolds and Self-Maps of Nontrivial Degree
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同齐性一流形与非平庸度自映射
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发表时间：
2007
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通讯作者：
T. Püttmann
中科院分区：
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作者：
T. Püttmann
我们构造了紧致上齐一流形的自然自映射，并计算了它们的度和Lefschetz数。在具有简单上同调环的流形上，这产生了Weyl群的阶与主轨道的欧拉特征之间的关系。作为例子，我们确定了紧型不可约黎曼对称空间上的所有上齐性1作用，这些作用导致次数为<$−1; 0; 1的自映射。我们导出了紧矩阵群SU（3），SU（4）和SO（2n）的新的坐标多项式自映射的显式公式。对于SU（3），我们精确地确定了哪些整数可以被实现为自映射的次数。
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.