Model solvmanifolds for Lefschetz and Nielsen theories

Abstract

In this paper we construct a class of solvmanifolds and certain (diagonal type) self maps on them. These solvmanifolds and their maps serve firstly as a rich source of examples. Secondly they serve as models for Nielsen theory in the sense that any map f : S → S of an arbitrary compact solvmanifold S, has the same Lefschetz and Nielsen theory(ordinary and periodic) as a "diagonal" map f′ on a "model solvmanifold" S′. Thus Nielsen theory calculations never get more complicated than they do on these model maps and spaces. The models also have the property (unlike arbitrary solvmanifolds) that their diagonal maps are characterized by a single simple purely matrix theoretical condition which is merely necessary for arbitrary solvmanifolds. As a result, the models S′ often exhibit many more maps than the original solvmanifolds S from which they are derived. Moreover, it is often on these "extra" maps are where the more interesting Nielsen theory occurs.