#### Volume 23, issue 5 (2019)

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Higher-order representation stability and ordered configuration spaces of manifolds

### Jeremy Miller and Jennifer C H Wilson

Geometry & Topology 23 (2019) 2519–2591
##### Abstract

Using the language of twisted skew-commutative algebras, we define secondary representation stability, a stability pattern in the unstable homology of spaces that are representation stable in the sense of Church, Ellenberg and Farb (2015). 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 secondary homological stability in the sense of Galatius, Kupers and Randal-Williams (2018). 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.

##### Keywords
representation stability, secondary representation stability, configuration spaces, twisted commutative algebras, homological stability, secondary homological stability, complex of injective words, arc resolution
##### Mathematical Subject Classification 2010
Primary: 18A25, 55R40, 55R80, 55U10