Coarse cohomology and index theory on complete Riemannian manifolds
Coarse cohomology and index theory on complete Riemannian manifolds
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DOI：
10.1090/memo/0497
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
J. Roe
中科院分区：
文献类型：
--
作者：
J. Roe
粗几何是从渐近的角度研究度量空间：从很远的距离看起来相同的两个度量空间(如整数和实数)被认为是等价的。这本书发展了一个适合于粗几何的上同调理论。然后利用该理论构造了非紧完备黎曼流形上椭圆算子的高指数。这样的椭圆算子在某种算子代数的理论中有一个指数，它自然地与粗结构相联系，然后这个理论与粗上同调配对。由于Connes和Moscovici的工作，更高的指数可以用拓扑学的术语计算出来。它们也可以用与粗糙结构自然相关的理想边界的同调来解释。这本书给出了几何学的应用，最后讨论了诺维科夫猜想的粗略相似之处。
Coarse geometry''is the study of metric spaces from the asymptotic point of view: two metric spaces (such as the integers and the real numbers) which look the same from a great distance''are considered to be equivalent. This book develops a cohomology theory appropriate to coarse geometry. The theory is then used to construct higher indices''for elliptic operators on noncompact complete Riemannian manifolds. Such an elliptic operator has an index in the-theory of a certain operator algebra naturally associated to the coarse structure, and this-theory then pairs with the coarse cohomology. The higher indices can be calculated in topological terms thanks to the work of Connes and Moscovici. They can also be interpreted in terms of the-homology of an ideal boundary naturally associated to the coarse structure. Applications to geometry are given, and the book concludes with a discussion of the coarse analog of the Novikov conjecture.