PreprintCC BY 4.0arXiv
Loop Space Models and Integral Homology of Connected Quandles
Abstract
We construct a marked integral delooping of a connected quandle's covering monoid, realizing its rack space as a homotopy fiber. For finite connected quandles at primes p not dividing a point-stabilizer order, we determine the local A∞ type and the ordinary rack cohomology shuffle algebra over fields of characteristic p. We compute integral quandle homology in all positive degrees for finite connected regular cyclic Alexander quandles and the tetrahedral quandle. At such primes dividing the quandle order, rack-space homotopy groups have exponentially many cyclic direct summands of fixed p-power order in each sufficiently large degree.
Related
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- Khovanov Homology in Connected Sums, Properties and Applications
- Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology
- The integral homology of SL2(ℤ[1/n])
- Splice formulae for Poincaré series of integral homology spheres
BibTeX
@misc{nosaka2026loop,
title = {{Loop Space Models and Integral Homology of Connected Quandles}},
author = {Takefumi Nosaka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38066},
eprint = {2610.38066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38066},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Loop Space Models and Integral Homology of Connected Quandles
AU - Takefumi Nosaka
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.38066
UR - https://arxiv.org/abs/2610.38066
AB - We construct a marked integral delooping of a connected quandle's covering monoid, realizing its rack space as a homotopy fiber. For finite connected quandles at primes p not dividing a point-stabilizer order, we determine the local A_∞ type and the ordinary rack cohomology shuffle algebra over fields of characteristic p. We compute integral quandle homology in all positive degrees for finite connected regular cyclic Alexander quandles and the tetrahedral quandle. At such primes dividing the quandle order, rack-space homotopy groups have exponentially many cyclic direct summands of fixed p-power order in each sufficiently large degree.
ER -