L2-invariants

L2 不变量

• 批准号: 42819878
• 负责人: Professor Dr. Thomas Schick
• 金额: --
• 依托单位: Mathematisches Institut (MI)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2007
• 资助国家: 德国
• 起止时间: 2006-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/42819878?language=en
• 关键词: L2; invariants

L2-invariants play an important role in geometry and topology. In particular, they provide useful connections between (differential) geometry and questions arising from topology and even algebra, like the computation of the sign of the Euler characteristic for fundamental groups of negatively curved Kahler manifolds. The more refined of these invariants express the geometry of non-compact manifolds with topological invariants of natural compactifications, via the explicit calculation of L2-invariants. It turns out, however, that except for L2-Betti numbers there is a lack of explicitly calculated examples. Our goals in this project are the calculation of Novikov-Shubin invariants, L2-eta invariants, and L2-torsion for the natural compactifications of all locally symmetric spaces of finite volume (relating the topology of the compactification to the geometry in new ways). Moreover, we plan to extend such calculations in two directions: first to spaces which are only asymptotically locally symmetric,secondly to convex cocompact hyperbolic manifolds.

L2-不变量在几何和拓扑中起着重要的作用。特别是，它们提供了（微分）几何和拓扑学甚至代数所产生的问题之间的有用联系，例如计算负弯曲卡勒流形基本群的欧拉特征的符号。这些不变量中更精细的不变量通过L2-不变量的显式计算，用自然紧化的拓扑不变量表达非紧流形的几何。然而，事实证明，除了L2-Betti数之外，缺乏明确计算的例子。我们在这个项目中的目标是计算Novikov-Shubin不变量，L2-eta不变量和L2-挠的所有有限体积的局部对称空间的自然紧化（以新的方式将紧化的拓扑与几何相关联）。此外，我们计划扩展这样的计算在两个方向：首先是空间，只有渐近局部对称，其次是凸余紧双曲流形。