Equivariant cohomology of K-contact manifolds
Equivariant cohomology of K-contact manifolds
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DOI：
10.1007/s00208-011-0767-8
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发表时间：
2011-02
影响因子：
1.4
通讯作者：
Oliver Goertsches;Hiraku Nozawa;Dirk Töben
中科院分区：
文献类型：
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作者：
Oliver Goertsches;Hiraku Nozawa;Dirk Töben
研究了K-切触流形上自然环面作用的等变上同调及其与Reeb流拓扑的关系。利用接触矩映射，我们证明了该作用量的等变上同调是Cohen-Macaulay，它是无不动点环面作用量等变形式的自然替代。因此，接触矩映射的一般分量是Reeb流的轨道面理的基本上同调的完美Morse-Bott函数。假设闭Reeb轨道是孤立的，我们证明了的基本上同调在奇数阶为零，并且它的维数等于闭Reeb轨道的个数.我们将具有最少闭Reeb轨道数的K-切触流形刻画为真实的上同调球面。我们还证明了一个GKM型定理的K-接触流形，它允许计算的等变上同调代数下的非孤立GKM条件。
We investigate the equivariant cohomology of the natural torus action on aK-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliationof the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology ofvanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterizeK-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem forK-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.