Two topological definitions of a Nielsen number for coincidences of noncompact maps
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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 classes remain nonempty and different. The number of essential classes thus remains stable and this is what is usually called the Nielsen number. In “topologically good” Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:1 (2004) 49–69 2000 Mathematics Subject Classification: 47H11, 47H09, 47H10, 47H04, 47J05, 54H25 URL: http://dx.doi.org/10.1155/S1687182004308119
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Topological definition of a Nielsen number
situations, this Nielsen number has the so-called Wecken property, that is, it gives exactly the same number as the above “Wecken property definition” (see, e.g., [12]). The various approaches to the Nielsen number in literature differ in the way how the classes and “essentiality” are defined. In most approaches, “essentiality” is defined in a homologic way (e.g., with respect to some fixed point index or Lefschetz number). However, in view of the above-described Wecken property definition, and since the existence of a fixed point index or Lefschetz number requires certain additional assumptions on the involved maps, we take in this paper the position that “essentiality” should be defined in a homotopic way instead. The
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