Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 84093, 15 pages

Geometric and homotopy theoretic methods in Nielsen coincidence theory

Ulrich Koschorke

Universität Siegen, Emmy Noether Campus, Walter-Flex Street 3, Siegen D-57068, Germany

Received 30 November 2004; Accepted 21 July 2005

Copyright © 2006 Ulrich Koschorke. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In classical fixed point and coincidence theory, the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f1,f2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC (f1,f2) (and MC (f1,f2), resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are (f1,f2). Furthermore we deduce finiteness conditions for MC (f1,f2). As an application, we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f1,f2) into path components. Its higher-dimensional topology captures further crucial geometric coincidence data. An analoguous approach can be used to define also Nielsen numbers of certain link maps.