The three faces of Nielsen: Coincidences, intersections and preimages

In addition to the many different Nielsen-type numbers that have been introduced to study fixed points, there are Nielsen-type numbers that have been created to study coincidences of maps; intersections of maps; and preimages of maps. These problems, while distinct, are all similar, and the Nielsen theories that have been created to study them display strong structural similarities as well. In this paper, we explore these similarities, and show that the relations between the three theories are closer and more formal than just similarity. There are transformations that allow any of the three Nielsen problems to be converted into either of the other two. Analysis of these transformations allows us to make precise the relationships between the three Nielsen theories.