Exceptional Dehn filling
Exceptional Dehn filling
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卓越的 Dehn 填充
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发表时间：
2009
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影响因子：
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通讯作者：
S. Boyer
中科院分区：
文献类型：
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作者：
S. Boyer
团队工作坊的研究集中在三维拓扑中例外Dehn填充理论的几个问题上。Dehn填充是这样一种构造：取一个具有独特环面边界分量T的3-流形M，并通过从∂V到T的某种同胚将一个实体环面V粘合到M上。由此得到的流形仅依赖于T上的等度类(斜率)α，它被标识为V的子午盘的边界，因此我们将其记为M(α)。这种构造可以追溯到1910年的Dehn，他在M是S中的纽结的外部的特殊情况下引入了它。1960年初S的Lickorish-Wallace定理表明，任何闭的、连通的、可定向的三维流形都可以通过Dehn填充S中某一环的边界环面来获得。因此，三维流形中的许多基本问题都可以用运算来分析。随着瑟斯顿在20世纪70年代的开创性工作，S用它来研究三维流形上的双曲几何结构，重新引起了人们对这种构造的兴趣。特别地，瑟斯顿证明了如果M是双曲的，则M(α)对于T上的有限多个斜率α以外的所有斜率也是双曲的。当M是双曲线而M(α)不是时，有人说(M；α)是例外的。尽管很明显，人们不能期望将所有的例外(M；α)S归类，但事实证明，对于一个双曲线三维流形来说，在T上具有两个截然不同的例外斜率α和β是相对罕见的，尝试将所有这样的(M；α，β)S归类并不是太不合理。这通常是通过考虑M(α)和M(β)不是双曲的各种不同的方法来实现的，并且沿着这些方向已经有了很多进展。最不为人所知的情况是，当M的边界是环面，其中一个填充物，比如M(β)，是一个小的塞弗特纤维空间。这是研讨会的重点。
This Research in Team workshop focused on several problems in the theory of exceptional Dehn fillings in 3dimensional topology. Dehn filling is the construction in which you take a 3-manifoldM , with a distinguished torus boundary component T , and glue a solid torus V to M via some homeomorphism from ∂V to T . The resulting manifold depends only on the isotopy class (slope) α on T that is identified with the boundary of a meridian disk of V , so we denote it by M(α). The construction goes back to Dehn in 1910, who introduced it in the special case where M is the exterior of a knot in S. The Lickorish-Wallace theorem of the early 1960’s showed that any closed, connected, orientable 3-manifold can be obtained by Dehn filling the boundary tori of the exterior of some link in S. Consequently, many of the basic problems in 3-manifold topology can been analysed in terms of the operation. Renewed interest in the construction arose with the ground-breaking work of Thurston in the 1970’s, who used it to study hyperbolic geometric structures on 3-manifolds. In particular, Thurston showed that if M is hyperbolic then M(α) is also hyperbolic for all but finitely many slopes α on T . When M is hyperbolic but M(α) is not, one says that (M ;α) is exceptional. Although it is clear that one cannot hope to classify all exceptional (M ;α)’s, it turns out that it is relatively rare for a hyperbolic 3-manifold to have two distinct exceptional slopes α and β on T , and it is not too unreasonable to try to classify all such (M ;α, β)’s. This has usually been approached by considering the various different ways in which M(α) and M(β) can fail to be hyperbolic, and there has been a lot of progress along these lines. The cases about which least is known is when the boundary of M is a torus and one of the fillings, say M(β), is a small Seifert fiber space. This was the focus of the workshop