计算群和群胚的L2-贝蒂数的谱序列
A spectral sequence to compute L2-Betti numbers of groups and groupoids
摘要：
我们构造了离散可测群胚的L2型上同调群的谱序列。基于该谱序列，我们证明了具有多面基本群的非球面流形的Hopf-Singer猜想.在更一般的情形下，我们得到了Hopf-Singer猜想在底空间为具有多面基本群的非球面流形的纤维丛中的持久性结果.作为谱序列的进一步应用，我们得到了群和流形的L2-贝蒂数的消灭定理和显式计算，以及在度量等价关系中存在正规子关系的障碍.
英文摘要：
We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations.
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