Explorations in Nielsen periodic point theory for the double torus

Abstract

For f:X→X, with X a compact manifold, Nielsen periodic point theory involves the calculation of f-homotopy invariant lower bounds for | fix(f n)| and for the number of periodic points of minimal period n. In this paper we combine the covering space approach to Nielsen periodic point theory with an algebraic method of Fadell and Husseini to study the behavior of the Nielsen periodic classes of maps on T 2#T 2 , the surface of genus two. Nil and solvmanifolds have basic properties for Nielsen periodic classes that make the calculation of these lower bounds possible. In this paper we accomplish two objectives. We show firstly that virtually all of these basic properties for the periodic classes fail in general on T 2#T 2 as well as on a collection of manifolds of arbitrarily high dimension. Secondly, despite these difficulties, we develop and apply techniques involving linear algebra, combinatorial group theory, number theory, and geometric facts from the theory of surface homeomorphisms, to make some calculations of the Nielsen periodic numbers. In our final example the combinatorial structure of the essential Nielsen periodic classes is fully displayed in a manner which relies on some of the classic identities involving the Fibonacci and Lucas numbers.