Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds (Q1186155)

Let \(S_ 1\) and \(S_ 2\) be compact connected orientable solvmanifolds with \(\dim(S_ 1)=\dim(S_ 2)\) and \(f,g: S_ 1\to S_ 2\) a pair of maps. The main result of this paper is to show that each essential Nielsen coincidence class of \((f,g)\) has index \(+1\) or \(-1\). This is Theorem 1. It was known that for nilmanifolds this index is the same for all classes and it is \(+1\) or \(-1\). The proof is by induction on the dimension of the manifold. The author uses the derived series of the fundamental group of \(S_ 2\) to construct fibrations with total spaces \(S_ 2\) in order to apply the induction procedure. The paper contains a short but very good review of the basic Nielsen coincidence theory.