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Báo cáo hóa học: " TWO TOPOLOGICAL DEFINITIONS OF A NIELSEN NUMBER FOR COINCIDENCES OF NONCOMPACT MAPS"

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: TWO TOPOLOGICAL DEFINITIONS OF A NIELSEN NUMBER FOR COINCIDENCES OF NONCOMPACT MAPS | TWO TOPOLOGICAL DEFINITIONS OF A NIELSEN NUMBER FOR COINCIDENCES OF NONCOMPACT MAPS JAN ANDRES AND MARTIN VATH Received 25 August 2003 The Nielsen number is a homotopic invariant and a lower bound for the number of coincidences of a pair of continuous functions. We give two homotopic topological definitions of this number in general situations based on the approaches of Wecken and Nielsen respectively and we discuss why these definitions do not coincide and correspond to two completely different approaches to coincidence theory. 1. Introduction The Nielsen number in its original form is a homotopic invariant which provides a lower bound for the number of fixed points of a map under homotopies. Many definitions have been suggested in the literature and in topologically good situations all these definitions turn out to be equivalent. Having the above property in mind it might appear most reasonable to define the Nielsen number simply as the minimal number of fixed points of all maps of a given homotopy class. We call this the Wecken property definition of the Nielsen number the reason for this name will soon become clear . However although this abstract definition has certainly some nice topological aspects it is almost useless for applications because there is hardly a chance to calculate this number even in simple situations. Moreover in most typical infinite-dimensional situations the homotopy classes are often too large to provide any useful information. The latter problem is not so severe instead of considering all homotopies one could restrict attention only to certain classes of homotopies like compact or so-called condensing homotopies. But the difficulty about the calculation or at least estimation of the Nielsen number remains. Therefore the taken approach is usually different one divides the fixed point set into several possibly empty classes induced by the map and proves that certain essential classes remain stable under homotopies in the sense that the .