Hakenness和b_1
Hakenness and b_1
摘要：
Thurston的三个伟大定理是：哈肯流形是双曲的;大分歧覆盖是双曲的;大手术是双曲的。最近的发展表明，后两个定理基本上是第一个的推论，即有更多的哈肯流形比预期的瑟斯顿。事实上，弗里德曼最近表明，大分枝覆盖物是哈肯。一个版本，他的证明与各种改进是由库珀龙和库珀龙里德。本文件的两个主要贡献如下。给出了Freedman结果的一个解析证明和Cooper-Long-Reid的改进形式。这个证明是基于根本不同的方法比弗里德曼的，是10倍短，但使用了双曲化定理的全部力量。其次，证明了紧纽结的任何分歧覆盖都是Haken的。此外，任何结在大手术后都会变得很紧。所以任何大手术的枝节报道都是哈肯的。本文的许多其他结果从引言中可以更好地看到。这篇论文使用了许多不同的技术，对于初学者来说可能很难阅读。
英文摘要：
Three great theorems of Thurston read: Haken manifolds are hyperbolic; big ramified coverings are hyperbolic; big surgeries are hyperbolic. Recent developments indicate that the later two theorems are essentially a corollary of the first, that is there are much more Haken manifolds than expected by Thurston. In fact Freedman showed very recently that big ramified coverings are Haken. A version of his proof with various improvements was obtained by Cooper-Long and Cooper-Long-Reid. The two main contributions of the present paper are the following. I give an analytic proof of Freedman's result and the improved version of Cooper-Long-Reid. This proof is based on fundamentally different approach than Freedman's and is 10 times shorter, but uses the full forse of the hyperbolization theorem. Secondly, I prove that ANY ramified covering of a tight knot is Haken. Morover any knot becomes tight after a big surgery. So any ramified covering of a big surgery is Haken. Many other results of the paper are better seen from the introduction. The paper uses many different techniques and may be difficult to read for a beginner.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
