Peter Saveliev

Copyright © 2005 Peter Saveliev. 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.

Abstract

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−dim Y 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=dim X+dim Y−dim Z. 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 Sp'(f,Y)=Ωp(C)/∼N, then the Nielsen group of order p is the part of Sp'(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.