Higher-order representation stability and ordered configuration spaces of manifolds
Higher-order representation stability and ordered configuration spaces of manifolds
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DOI：
10.2140/gt.2019.23.2519
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发表时间：
2016-11
影响因子：
2
通讯作者：
Jeremy Miller;Jenny Wilson
中科院分区：
文献类型：
--
作者：
Jeremy Miller;Jenny Wilson
利用扭曲斜交换代数的语言，我们定义了\emph{次表示稳定}，这是在Church， Ellenberg和Farb意义上表示稳定的空间的{\it不}稳定同调中的一种稳定模式。证明了非紧流形中有序点的位形空间的有理同调满足二次表示稳定性。当构型空间的同调的表示稳定性涉及到通过引入一个“近无穷”点来实现稳定时，次表示稳定性涉及到通过引入一对轨道点来实现稳定——这是一种将不同同调度的同调群联系起来的操作。这个结果可以被认为是Galatius， Kupers和Randal-Williams意义上的\emph{二次同调稳定性}的表示理论模拟。在证明过程中，我们得到了一些额外的结果：我们给出了内射词复调的一个新的表征，我们给出了非紧流形组形空间的积分表示稳定性的一个新的证明，将之前的结果推广到非定向流形。
Using the language of twisted skew-commutative algebras, we define \emph{secondary representation stability}, a stability pattern in the {\it unstable} homology of spaces that are representation stable in the sense of Church, Ellenberg, and Farb. We show that the rational homology of configuration spaces of ordered points in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing a point ``near infinity,'' secondary representation stability involves stabilizing by introducing a pair of orbiting points -- an operation that relates homology groups in different homological degrees. This result can be thought of as a representation-theoretic analogue of \emph{secondary homological stability} in the sense of Galatius, Kupers, and Randal-Williams. In the course of the proof we establish some additional results: we give a new characterization of the homology of the complex of injective words, and we give a new proof of integral representation stability for configuration spaces of noncompact manifolds, extending previous results to nonorientable manifolds.