Higher-order Nielsen Numbers

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= dimX −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 Ωp(C) of C. The answer is the Nielsen group Sp( 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 S′p( f ,Y) =Ωp(C)/ ∼N , then the Nielsen group of order p is the part of S′p( f ,Y) preserved under homotopies of f . The Nielsen number Np(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.