Nielsen coincidence theory of (n,m)-valued pairs of maps

@misc{graff2026nielsen,
  title = {{Nielsen coincidence theory of $(n,m)$-valued pairs of maps}},
  author = {Grzegorz Graff and P. Christopher Staecker and Alan Żeromski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.07015},
  eprint = {2605.07015},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.07015},
}

TY  - UNPB
TI  - Nielsen coincidence theory of (n,m)-valued pairs of maps
AU  - Grzegorz Graff
AU  - P. Christopher Staecker
AU  - Alan Żeromski
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2605.07015
UR  - https://arxiv.org/abs/2605.07015
AB  - We consider pairs of maps (f,g), where f is an n-valued map and g is an m-valued map, defined on connected finite polyhedra. A point x such that f(x)∩ g(x)≠ ∅ is called a coincidence point of f and g. A useful device for studying coincidence points would be a Nielsen-type invariant which provides a lower bound for the number of coincidence points of all $(n, m)$-valued pairs of maps homotopic to (f,g). The construction of such an invariant N(f:g) was proposed in [J. Fixed Point Theory Appl. 14, 309--324 (2013)]. Unfortunately, this approach has some flaws. In this paper, we present a modified construction that yields a corrected form of the invariant, defined in terms of the intersection points of the graphs of f and g. In the case of $(n, m)$-valued pairs of maps of the circle our invariant provides a sharp lower bound, which we precisely determine.
ER  -