Higher-order Nielsen numbers
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Suppose X, Y are manifolds, f ,g : X → Y are maps. The well-known coincidence problem studies the coincidence set C = {x : f (x) = g(x)}. The number m = dim X − dimY is called the codimension of the problem. More general is the preimage problem. For a map f : X → Z and a submanifold Y of Z, it studies the preimage set C = {x : f (x) ∈ Y }, and the codimension is m = dimX + dimY − dimZ. In case of codimension 0, the classical Nielsen number N( f ,Y ) is a lower estimate of the number of points in C changing under homotopies of f , and for an arbitrary codimension, of the number of components of C. We extend this theory to take into account other topological characteristics of C. The goal is to find a “lower estimate” of the bordism group Ω p (C) of C. The answer is the Nielsen group S p ( f ,Y ) defined as follows. In the classical definition, the Nielsen equivalence of points of C based on paths is replaced with an equivalence of singular submanifolds of C based on bordisms. We let Sp ( f ,Y ) = Ω p (C)/ ∼N , then the Nielsen group of order p is the part of Sp ( f ,Y ) preserved under homotopies of f . The Nielsen number N p (F,Y ) of order p is the rank of this group (then N( f ,Y ) = N0 ( f ,Y )). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided. 1. Introduction Suppose X, Y are smooth orientable compact manifolds, dimX = n + m, dimY = n, m ≥ 0 the codimension, f ,g : X → Y are maps, the coincidence set
C = Coin( f ,g) = x ∈ X : f (x) = g(x)
(1.1)
is a compact subset of X \∂X. Consider the coincidence problem: “what can be said about the coincidence set C of ( f ,g)?” One of the main tools is the Lefschetz number L( f ,g) defined as the alternating sum of traces of a certain endomorphism on the homology group of Y . The famous Lefschetz coincidence theorem provides a sufficient condition for the existence of Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 47–66 DOI: 10.1155/FPTA.2005.47
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Higher-order Nielsen numbers
coincidences (codimension m = 0): L( f ,g) = 0 ⇒ C = Coin( f ,g) = ∅, see [1, Section VI.14], and [31, Chapter 7]. Now, what else can be said about the coincidence set? As C changes under homotopies of f and g, a reasonable approach is to try to minimize the “size” of C. In case of zero codimension, C is discrete and we simply minimize the number of points in C. The result is the Nielsen number. It is defined as follows. Two points p, q ∈ C belong to the same Nielsen class if (1) there is a path s in X between p and q; (2) f s and gs are homotopic relative to the endpoints. A Nielsen class is called essential if it cannot be removed by a homotopy of f , g (alternatively, a Nielsen class is algebraically essential if its coincidence index is nonzero [2]). Then the Nielsen number N( f ,g) is the number of essential Nielsen classes. It is a lower estimate of the number of points in C. In case of positive codimension N( f ,g) still makes sense as a lower estimate o
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