Mixed configuration spaces, fixed points and Loop braid groups

Abstract

We introduce the notion of mixed configuration spaces of the $3$-ball. That is, we consider both distinct circles and distinct points in the interior of the $3$-ball and we prove that the projection onto the configuration space of circles of the $3$-ball is a locally trivial fibration. The key idea for defining the mixed configuration spaces is to study fixed points of orientation-preserving homeomorphism of the $3$-ball that leave invariant a trivial link of n components in the interior of the $3$-ball. In particular, we prove that two fixed points are Nielsen equivalent if and only if the associated loop braids are conjugate by an element of a distinguished free subgroup of rank n. This result stands as a $3$-dimensional counterpart of the $2$-dimensional result where fixed points of homeomorphisms of the punctured disc are characterized in terms of braid elements of the classical Artin braid group.

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BibTeX RIS

BibTeX

@misc{makri2026mixed,
  title = {{Mixed configuration spaces, fixed points and Loop braid groups}},
  author = {Stavroula Makri},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04382},
  eprint = {2610.04382},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04382},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Mixed configuration spaces, fixed points and Loop braid groups
AU  - Stavroula Makri
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04382
UR  - https://arxiv.org/abs/2610.04382
AB  - We introduce the notion of mixed configuration spaces of the $3$-ball. That is, we consider both distinct circles and distinct points in the interior of the $3$-ball and we prove that the projection onto the configuration space of circles of the $3$-ball is a locally trivial fibration. The key idea for defining the mixed configuration spaces is to study fixed points of orientation-preserving homeomorphism of the $3$-ball that leave invariant a trivial link of n components in the interior of the $3$-ball. In particular, we prove that two fixed points are Nielsen equivalent if and only if the associated loop braids are conjugate by an element of a distinguished free subgroup of rank n. This result stands as a $3$-dimensional counterpart of the $2$-dimensional result where fixed points of homeomorphisms of the punctured disc are characterized in terms of braid elements of the classical Artin braid group.
ER  -

Abstract

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