K_g is not finitely generated
K_g is not finitely generated
复制标题
K_g 不是有限生成的
作者：
Daniel K. Biss;Benson Farb
1介绍让Σg是一个封闭的可定向的表面,属g。Σg的映射类Mod g组定义的组合痕类保持定向解析Σg→Σg。回想一下,一个重要的简单闭曲线γΣg称为边界曲线,曲线或分离,如果是null-homologousΣg或者,同样,如果γΣg分为两个连接组件。设K g表示由Σ g中关于边界曲线的Dehn扭转的（无限）集合生成的Mod g的子群。注意k1是平凡的。在曲面组合拓扑中，确定K g群在g≥2时是否有限生成一直是一个长期存在的问题。McCullough-Miller [MM]证明了k2不是有限生成的；然后，Mess证明了k2实际上是一个无限秩自由群。Akita在[Ak]中证明了对于所有g≥2，有理同调H * （K g； Q）作为向量空间在Q上是无限维的。注意，由于K g允许对Σ g的有限维可缩并的Teichmuller空间的自由作用，因此K g具有有限的上同调维数。有一段时间，我们不知道K g是否等于，或者可能是托雷利群I g的一个有限索引子群，它是元素的子群
1 Introduction Let Σ g be a closed orientable surface of genus g. The mapping class group Mod g of Σ g is defined to be the group of isotopy classes of orientation-preserving diffeomorphisms Σ g → Σ g. Recall that an essential simple closed curve γ in Σ g is called a bounding curve, or separating curve, if it is null-homologous in Σ g or, equivalently, if γ separates Σ g into two connected components. Let K g denote the subgroup of Mod g generated by the (infinite) collection of Dehn twists about bounding curves in Σ g. Note that K 1 is trivial. It has been a long-standing problem in the combinatorial topology of surfaces to determine whether or not the group K g is finitely generated for g ≥ 2. McCullough-Miller [MM] proved that K 2 is not finitely generated; Mess then proved that K 2 is in fact an infinite rank free group. Akita proved in [Ak] that for all g ≥ 2, the rational homology H * (K g ; Q) is infinite-dimensional as a vector space over Q. Note that since K g admits a free action on the Teichmuller space of Σ g , which is contractible and finite-dimensional, K g has finite cohomological dimension. For some time it was not known if K g was equal to, or perhaps a finite index subgroup of, the Torelli group I g , which is the subgroup of elements