Stabilizers of fixed point classes and Nielsen numbers of $n$-valued maps

Abstract

The stabilizer of a fixed point class of a map is the fixed subgroup of the induced fundamental group homomorphism based at a point in the class. A theorem of Jiang, Wang and Zhang is used to prove that if a map of a graph satisfies a strong remnant condition, then the stabilizers of all its fixed point classes are trivial. Consequently, if $\phi_{p, f}$ is the $n$-valued lift to a covering space $p$ of a map $f$ with strong remnant of a graph, then the Nielsen numbers are related by the equation $N(\phi_{p, f}) = n \cdot N(f)$. Additional information concerning Nielsen numbers is obtained for $n$-valued lifts of maps of graphs with positive Lefschetz numbers and of maps of spaces with abelian fundamental groups and for extensions of $n$-valued maps.