The Three Faces of Nielsen: Coincidences, Intersections and Preimages
نویسنده
چکیده
In addition to the many diierent Nielsen-type numbers that have been introduced to study xed 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.
منابع مشابه
Higher-order Nielsen Numbers
Suppose X , Y are manifolds, f ,g : X → Y are maps. The well-known coincidence problem studies the coincidence set C = {x : f (x) = g(x)}. The number m= dimX −dimY is called the codimension of the problem. More general is the preimage problem. For a map f : X → Z and a submanifold Y of Z, it studies the preimage set C = {x : f (x) ∈ Y}, and the codimension is m = dimX + dimY − dimZ. In case of ...
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