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Higher order representation stability and ordered configuration spaces of manifolds
- Source :
- Geom. Topol. 23, no. 5 (2019), 2519-2591
- Publication Year :
- 2016
-
Abstract
- 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.<br />45 pages, 41 figures. Minor corrections and revisions made throughout to clarify and shorten the paper. For reasons for length the appendix has been cut and will be made into a separate paper. Final version; accepted at Geometry & Topology
- Subjects :
- Pure mathematics
media_common.quotation_subject
Stability (learning theory)
Characterization (mathematics)
Homology (mathematics)
Mathematics::Algebraic Topology
FOS: Mathematics
Algebraic Topology (math.AT)
Mathematics - Combinatorics
secondary homological stability
55U10
Mathematics - Algebraic Topology
Representation Theory (math.RT)
arc resolution
Mathematics::Symplectic Geometry
media_common
Mathematics
representation stability
twisted commutative algebras
homological stability
Representation (systemics)
Order (ring theory)
Infinity
Mathematics::Geometric Topology
Injective function
18A25
55R80
secondary representation stability
55R40
Geometry and Topology
Combinatorics (math.CO)
configuration spaces
complex of injective words
Mathematics - Representation Theory
55R40, 55R80, 18A25, 55U10
Singular homology
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Geom. Topol. 23, no. 5 (2019), 2519-2591
- Accession number :
- edsair.doi.dedup.....c9d827598639e9f995c20033dc10b701