论文标题
次要表示稳定性和曾经是符合的圆环的有序配置空间
Secondary representation stability and the ordered configuration space of the once-punctured torus
论文作者
论文摘要
在本文中，我们研究了曾经由仪式圆环的有序配置空间的同源性中的稳定模式。在过去的十年中，教堂和教堂 - 埃伦贝格·法布尔（Church-Ellenberg-Farb）证明，在表示理论上稳定相关的不可能的定向流形的有序配置空间的同源群体，随着配置中的积分数量的增长，相对于添加每个新点的地图“在Infinity”中增加了。米勒（Miller）和威尔逊（Wilson）证明了不稳定同源性类别之间存在次要表示稳定性模式，以添加一对轨道上的“无穷大”。该模式是通过将同源类序列的序列视为fim $^{+} $ - 模块形式化的。我们证明，作为fim $^{+} $ - 模块，在曾经countundund的圆环上的订单配置空间的n-同源物中的“新”同源性生成器的顺序既不是“自由”也不是“稳定为零”。我们还表明，该序列最多是由同源性类别生成的。我们的证明使用Pagaria在圆环的有序配置空间的Betti数字上的工作来计算曾经由启用的圆环的有序配置空间的Betti数字的增长率。我们的计算是第一个证明次级表示稳定性是正发表面中的非平凡现象。
In this paper we study stability patterns in the homology of the ordered configuration space of the once-punctured torus. In the last decade Church and Church-Ellenberg-Farb proved that the homology groups of the ordered configuration space of a connected noncompact orientable manifold stabilize in a representation theoretic sense as the number of points in the configuration grows, with respect to a map that adds each new point "at infinity." Miller and Wilson proved that there is a secondary representation stability pattern among the unstable homology classes, with respect to adding a pair of orbiting points "near infinity." This pattern is formalized by considering sequences of homology classes as FIM$^{+}$-modules. We prove that, as FIM$^{+}$-modules, the sequence of "new" homology generators in the n-th homology of the ordered configuration space of 2n-2 points on the once-punctured torus is neither "free" nor "stably zero." We also show that this sequence is generated by homology classes on at most 4 points. Our proof uses Pagaria's work on the Betti numbers of the ordered configuration space of the torus to calculate the growth rate of the Betti numbers of the ordered configuration space of the once-punctured torus. Our computations are the first to demonstrate that secondary representation stability is a non-trivial phenomenon in positive-genus surfaces.
