PreprintCC BY-NC-ND 4.0arXiv
Nielsen coincidence theory for (n,m)-valued pairs
Abstract
We define a Nielsen coincidence number for pairs of maps (f,g) where f is n-valued and g is m-valued. The classical definition of the Nielsen number does not work in this setting, but we propose a theory where coincidence classes are allowed to overlap. We invoke a notion of multiplicity of a coincidence point, which indicates to how many classes it can belong, and we show that our Nielsen number is a homotopy-invariant lower bound for the number of coincidence points with multiplicity. We show a computational result in terms of single-valued lift factors, and using this, a Wecken property for self-maps of the circle.
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BibTeX
@misc{dekimpe2026nielsen,
title = {{Nielsen coincidence theory for $(n,m)$-valued pairs}},
author = {Karel Dekimpe and Lore De Weerdt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29274},
eprint = {2610.29274},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29274},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Nielsen coincidence theory for (n,m)-valued pairs
AU - Karel Dekimpe
AU - Lore De Weerdt
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29274
UR - https://arxiv.org/abs/2610.29274
AB - We define a Nielsen coincidence number for pairs of maps (f,g) where f is n-valued and g is m-valued. The classical definition of the Nielsen number does not work in this setting, but we propose a theory where coincidence classes are allowed to overlap. We invoke a notion of multiplicity of a coincidence point, which indicates to how many classes it can belong, and we show that our Nielsen number is a homotopy-invariant lower bound for the number of coincidence points with multiplicity. We show a computational result in terms of single-valued lift factors, and using this, a Wecken property for self-maps of the circle.
ER -