Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 44 min ago
PreprintCC BY 4.0arXiv

Stability Patterns for Spherical and Projective Braid Groups

  • Sarah Anderson
Open at sourcePDFDOICite

Abstract

McDuff and Segal proved homological stability for unordered configuration spaces of connected manifolds with non-empty boundary. Later, Church proved representation stability for ordered configuration spaces on compact manifolds. These stability patterns extend to surface braid groups and pure surface braid groups, respectively, whenever the associated configuration space is an Eilenberg--Maclane space. However, these results do not apply to braid groups and pure braid groups on S2 and RP2 since ordered and unordered configuration spaces on these surfaces are not Eilenberg--Maclane spaces. This paper will use results by Cohen--Pakianathan and the theory of FI-homology to extend known stability patterns to these unknown cases. This is a special case of a more general result proven in this paper: the equivariant homology of ordered and unordered configuration spaces exhibit representation stability and stable periodicity respectively.

Related

  1. On unifying Jones representations of braid groups and Thompson's group
    Yangxiao Luo, Shunyu Wan
    arXivPreprintCC0Full textAlgebra & number theoryGeometry & topologyarXiv
  2. The first homology of square-section braid groups
    Minh Pham, Nicholas Wawrykow, Jacob Zweiback
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  3. Hierarchical geometry and right-angled Artin groups in graph braid groups
    Byung Hee An, Sangrok Oh, Jihoon Park
    arXivPreprintarXiv licenceFull textAlgebra & number theoryGeometry & topologyarXiv
  4. Spherical Milnor Spaces II: Projective Quotients and Higher Topological Structures
    Jean-Pierre Magnot
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryGeometry & topologyarXiv
  5. Braid Stability and a Floer theory of Braid Isotopies
    Nicolas Grunder
    arXivPreprintarXiv licenceFull textDynamical systemsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{anderson2026stability,
  title = {{Stability Patterns for Spherical and Projective Braid Groups}},
  author = {Sarah Anderson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05367},
  eprint = {2609.05367},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05367},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stability Patterns for Spherical and Projective Braid Groups
AU  - Sarah Anderson
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05367
UR  - https://arxiv.org/abs/2609.05367
AB  - McDuff and Segal proved homological stability for unordered configuration spaces of connected manifolds with non-empty boundary. Later, Church proved representation stability for ordered configuration spaces on compact manifolds. These stability patterns extend to surface braid groups and pure surface braid groups, respectively, whenever the associated configuration space is an Eilenberg--Maclane space. However, these results do not apply to braid groups and pure braid groups on S^2 and RP^2 since ordered and unordered configuration spaces on these surfaces are not Eilenberg--Maclane spaces. This paper will use results by Cohen--Pakianathan and the theory of FI-homology to extend known stability patterns to these unknown cases. This is a special case of a more general result proven in this paper: the equivariant homology of ordered and unordered configuration spaces exhibit representation stability and stable periodicity respectively.
ER  -