Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 84093, 15 pages
doi:10.1155/FPTA/2006/84093
Geometric and homotopy theoretic methods in Nielsen coincidence theory
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.
Abstract
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.