On the topology of double coset manifolds
On the topology of double coset manifolds
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关于双陪集流形的拓扑
DOI：
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发表时间：
1993
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通讯作者：
W. Singhof
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文献类型：
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作者：
W. Singhof
我们考虑一个紧李群G和G的两个闭子群H和K。抽象积K × H通过g对G进行运算。（k，h）= k lgh.如果这个运算恰好是自由的，则商K G / H是一个紧流形，我们称之为双陪集流形。这些流形吸引了几个微分几何学家的注意：Gromol]和Meyer [GrM]例如将奇异的7-球面描述为双陪集流形。因此，类的双陪集流形严格大于齐性空间，因为Borel观察到很久以前，一个齐性空间，这是同胚的一个领域实际上是同胚的。一个认真的研究双陪集流形是由Eschenburg [E1-E4]。他表明，他们中的一些承认黎曼度量严格积极的部分曲率，他获得了分类的某些类型的双重陪集流形和计算上同调的一些。本文的目的是表明，在许多方面的拓扑结构的双陪集流形是很容易处理的齐性空间。为了计算G/H的上同调，最好的方法是观察纤维化
We consider a compact Lie group G and two closed subgroups H and K of G. The abstract product K x H operates on G by g. (k, h) = k lgh. If this operation happens to be free, the quotient K G / H is a compact manifold, which we call a double coset manifold. These manifolds have attracted the attention of several differential geometers: Gromol] and Meyer [GrM] for instance described an exotic 7-sphere as a double coset manifold. Therefore the class of double coset manifolds is strictly larger than that of homogeneous spaces since Borel observed long ago that a homogeneous space which is homeomorphic to a sphere is actually diffeomorphic to it. A serious study of double coset manifolds was made by Eschenburg [E1-E4]. He showed that some of them admit Riemannian metrics with strictly positive sectional curvature; he obtained a classification of certain types of double coset manifolds and computed the cohomology of some of them. The objective of the present paper is to show that in many respects the topology of double coset manifolds is as easy to handle as that of homogeneous spaces. To compute the cohomology of G/H, the best way is to look at the fibration