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: HIGHER-ORDER NIELSEN NUMBERS | HIGHER-ORDER NIELSEN NUMBERS PETER SAVELIEV Received 24 March 2004 and in revised form 10 September 2004 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 - 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 e Y and the codimension is m dimX dim Y - 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 Qp 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 Qp 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. 1. Introduction Suppose X Y are smooth orientable compact manifolds dimX n m dim Y n m 0 the codimension f g X Y are maps the coincidence set C Coin f g x e X f x g x is a compact subset of X dX. 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 .
