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

2026/05/07 by Grzegorz Graff, P. Christopher Staecker, Alan Żeromski · 1 voice
Mathematics · #math.GN

paper · pdf

Abstract

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.

Citations

• Averaging formulas for the Reidemeister trace, Lefschetz and Nielsen numbers of n-valued maps
• Nielsen numbers of affine n-valued maps on nilmanifolds
• Axioms for the fixed point index of n-valued maps, and some applications
• Theory of intersecting loops on a torus
• Homotopy Methods in Topological Fixed and Periodic Points Theory

Discussions

• Grzegorz Graff, P. Christopher Staecker, Alan \.Zeromski: Nielsen coincidence theory of $(n,m)$-valued pairs of maps https://arxiv.org/abs/2605.07015 https://arxiv.org/pdf/2605.07015 https://arxiv.org [bsky, 1 points, 0 comments]