Homotopy type and volume of locally symmetric manifolds
Homotopy type and volume of locally symmetric manifolds
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DOI：
10.1215/s0012-7094-04-12432-7
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发表时间：
2001-11
影响因子：
2.5
通讯作者：
T. Gelander
中科院分区：
文献类型：
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作者：
T. Gelander
我们考虑具有固定泛覆盖的局部对称流形，并为每个这样的流形M构造一个单纯复形R，其大小与M的体积成正比。当M是非紧的时，R同伦等价于M，而当M是紧的时，R同伦等价于M\N，其中N是一个有限维数的子流形。这表达了体积如何控制M的拓扑结构，并为以前没有定量证明的各种有限性陈述提供了具体的界限。例如，它给出了体积以v>0为界的局部对称流形的可能个数的显式上界，并给出了流形的基本群的极小表示的大小的估计。它也产生了一些新的有限性结果。
We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold M a simplicial complex R whose size is proportional to the volume of M. When M is non-compact, R is homotopically equivalent to M, while when M is compact, R is homotopically equivalent to M\N, where N is a finite union of submanifolds of fairly smaller dimensions. This expresses how the volume controls the topological structure of M, and yields concrete bounds for various finiteness statements which previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by v>0, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results.