Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: GEOMETRIC AND HOMOTOPY THEORETIC METHODS IN NIELSEN COINCIDENCE THEORY | GEOMETRIC AND HOMOTOPY THEORETIC METHODS IN NIELSEN COINCIDENCE THEORY ULRICH KOSCHORKE Received 30 November 2004 Accepted 21 July 2005 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. 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. 1. Introduction and discussion of results Throughout this paper f1 f2 M - N denote two continuous maps between the smooth connected manifolds M and N without boundary of strictly positive dimensions m and n respectively M being compact. We would like to measure how small or simple in some sense the coincidence locus Cf f2 x e M I f1 x f2 x can be made by deforming f1 and f2 via homotopies. Classically one considers the minimum number of coincidence points MC f f2 min C f f2 I f - f 1 f2 - f2 cf. 1 . It coincides with the minimum number min C f1 f2 I f1 - f1 where only f1 is modified by a homotopy cf. 2 . In particular in topological fixed point theory Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006 Article ID 84093
