On the first Betti number of a constant negatively curved manifold
On the first Betti number of a constant negatively curved manifold
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关于恒定负曲流形的第一个贝蒂数
DOI：
10.2307/1971046
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发表时间：
1976
影响因子：
4.9
通讯作者：
J. Millson
中科院分区：
文献类型：
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作者：
J. Millson
具有任意大第一Betti数的常数负曲n维流形。事实上，我们证明，任何常数负曲流形，其基本群是一个算术群，可与二次型的单位群，承认有限覆盖的第一贝蒂数不等于零;特别是，例子博雷尔在年底[4]都承认这样的覆盖。在此之前，有几个例子是已知的低维（参见。Vinberg [14]），在紧凑的情况下，例子高达5维。然而，他的建设使用双曲Coxeter群只存在于低维。而不是攻击的问题代数计算abelianized基本组群论的方法，似乎除了上述低维的例子是无望的，我们采取几何方法，并构造明确的无约束余维1周期，然后呼吁庞加莱对偶。虽然这些例子是明显的几何利益，其主要意义是群论。Kajdan [6]的消失定理的结果是所有秩大于2的紧致局部不可约局部对称空间的第一Betti数为零。这是由S。P. Wang [15]和Kostant [8]证明了Kajdan的判据适用于除了与SO（n，1）和SU（n，1）相关的紧局部不可约局部对称空间之外的所有紧局部对称空间。本文的结果表明，消失定理对SO（n，1）不成立。它是否对SU（n，1）成立仍然没有解决。我们的结果是有趣的同余子群问题。Bass，Milnor，Serre [1]证明了如果一个算术群F满足同余子群性质，即每个有限指数子群都包含一个同余子群，则：
constant negatively curved n-dimensional manifolds with arbitrarily large first Betti number. In fact we show that any constant negatively curved manifold whose fundamental group is an arithmetic group commensurable with the group of units of a quadratic form admits a finite covering with first Betti number not equal to zero; in particular, the examples given by Borel at the end of [4] all admit such coverings. Previous to this a few examples were known in low dimensions (cf. Vinberg [14]) with examples up to dimension 5 in the compact case. However, his construction uses hyperbolic Coxeter groups which exist only in low dimensions. Rather than attack the problem algebraically by computing the abelianized fundamental group by group theory an approach that appears hopeless except for the above low dimensional examples, we take a geometric approach and construct explicit nonbounding codimension 1 cycles and then appeal to Poincare duality. Although these examples are of obvious geometric interest their main significance is group theoretic. The vanishing theorem of Kajdan [6] has as a consequence that the first Betti number of all compact locally irreducible, locally symmetric spaces of rank greater than 2 vanishes. This was extended by S. P. Wang [15] and Kostant [8] who show that Kajdan's criterion applies to all compact locally irreducible locally symmetric spaces except those associated with SO(n, 1) and SU(n, 1). The results of this paper show that the vanishing theorem does not hold for SO(n, 1). Whether or not it holds for SU(n, 1) remains unsolved. Our results are of interest in connection with the congruence subgroup problem. Bass, Milnor, Serre [1] show that if an arithmetic group F satisfies the congruence subgroup property that every subgroup of finite index contains a congruence subgroup, then: