Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)
Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold)
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三流形类场论（非虚 b1 正流形的覆盖同调）
DOI：
10.1007/s000290050015
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
A. Reznikov
中科院分区：
文献类型：
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作者：
A. Reznikov
这是第一次在一系列的文件中，我们探讨了深刻的概念之间的关系，三流形和数域，一个主题，这可能是一个名称的“算术拓扑”。这种关系的一个直接和透明的原因是，Gal（K）的上同调维数|K）对于任意数域K等于3。本文是作者试图解决Thurston覆盖猜想的结果，即任何具有无穷π1的不可约三元流形M都有一个正Betti数的有限覆盖。然而，在研究过程中，很明显，不存在的非虚b1正流形表现出“强烈的生存愿望”，也就是说，人们可以得到关于它们的信息被组织成一个非矛盾性质的和谐画面。因此，我们不试图证明这样的流形不存在，而是采用更积极的方法来研究它们的有限覆盖的同调。这是平行的研究理想类群的有限扩展的一个给定的数域。事实上，我们开发的技术将对这个数论问题非常有用。在三流形类场论中，我们要问的基本问题是：·类塔。设M是一个非虚b1-正三元流形。固定一个素数p。对于M的有限覆盖N，第一个同调群H1（N，Z）的p-分量增长有多快？·理想的类模块。H1（N）（p）作为伽罗瓦模的结构是什么
This is a first in a series of papers in which we explore a deep conceptual relation between three-manifolds and number fields, a subject which may be given a name of “arithmetic topology”. An immediate and transparent reason for such a relation is the fact that the cohomological dimension of Gal (K| K) equals three for any number field K. See [KR].This paper resulted from the author’s attempt to settle the Thurston’s covering conjecture, that is, that any irreducible three-manifold M with infinite π1 has a finite covering with positive Betti number. In the process of study it became clear, however, that conjecturally nonexisting nonvirtually b1-positive manifolds show “a strong wish to survive”, that is, the information that one can derive about them is organized into a harmonious picture of a noncontradictory nature. So instead of trying to show that such manifolds do not exist, we adopt a more positive approach to study the homology of their finite coverings. This is parallel to studying ideal class groups of finite extensions of a given number field. In fact, the techniques we develop will be very useful for this number-theoretic problem. The fundamental questions in three-manifold class field theory that we ask are:• Class towers. Suppose M is a nonvirtually b1-positive three-manifold. Fix a prime p. How fast does the p-component of the first homology group H1 (N, Z) grow for finite coverings N of M?• Ideal class modules. What is the structure of H1 (N)(p) as a Galois module