The Anosov Relation for Nielsen Numbers of Maps of Infra-nilmanifolds

In this book we examine the 'Anosov relation' for selfmaps of infra-nilmanifolds. A map satisfies the Anosov relation if the Nielsen number and the Lefschetz number are equal up to sign. These numbers find their origin in fixed point theory and provide information on the numbers of fixed points of the given map. In 1985 D. Anosov proved that maps of nilmanifolds always satisfy the Anosov relation. Since this result is only valid for a specific class of manifolds, natural questions are then (i)wether this theorem can be generalized towards other classes of manifolds and (ii)to classes of maps for a given manifold. In the first part, we introduce the theory, concerning infra-nilmanifolds and fixed point theory, needed to understand our results. In the second part we show that we can indeed find classes of infra-nilmanifolds for which we can generalize Anosov's result. We also find some classes of maps of infra-nilmanifolds that always satisfy the Anosov relation. In the last part, we present for each infra-nilmanifold up to dimension 4 a proof of, or a counterexample to, the Anosov relation for thisspecific manifold. Examining these results can be fruitful for future research.