Nielsen theory and related topics

Abstract

The International Conference on Nielsen Theory and Related Topics took place from June 28 through July 2, 2004 at Memorial University, St. John’s, Newfoundland, Canada. This was the 13th such conference in a series that began in 1977 with a conference in Oberwolfach, Germany. Nielsen theory is named after Jakob Nielsen who, in the 1920s, turned the focus of fixed point theory from the existence of fixed points (as, e.g., in the famous Lefschetz fixed point theorem) to the problem of estimating the actual number of such points within the homotopy class of a given map. He did this by introducing what is now called the Nielsen number of a self map, a homotopy invariant lower bound for the number of fixed points of the map. After important initial contributions by Reidemeister and Wecken, there was little activity in Nielsen theory until the 1960s when a breakthrough by Boju Jiang allowed for easy calculations of the Nielsen number for maps on Lie groups and some other interesting kinds of spaces. It was these and other important examples that would guide the direction of research. As the present collection of papers illustrates, the frontiers of the subject now involve an impressive variety and interplay of algebraic and geometric techniques, on a wide class of spaces. Consequently, there continue to be an increasing number of interesting areas for future investigations, both in the areas of computation and the development of new invariants. As research in Nielsen theory progressed, its concern with fixed points expanded, on the one hand, into related issues such as coincidences, periodic points, and roots and, on the other, into various refinements for restricted classes of maps and homotopies such as those that occur in the fiber space, relative and equivariant contexts. Just about all of these aspects of Nielsen theory were represented by the talks in Newfoundland. The conference