Fixed Point Theory and Applications
Volume 2005 (2005), Issue 1, Pages 47-66
Higher-order Nielsen numbers
Department of Mathematics, Marshall University, Huntington 25755, WV, USA
Received 24 March 2004; Revised 10 September 2004
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.
Suppose , are manifolds, are maps. The well-known coincidence problem studies the coincidence set . The number is called the codimension of the problem. More general is the preimage problem. For a map and a submanifold of , it studies the preimage set , and the codimension is . In case of codimension , the classical Nielsen number is a lower estimate of the number of points in changing under homotopies of , and for an arbitrary codimension, of the number of components of . We extend this theory to take into account other topological characteristics of . The goal is to find a “lower estimate” of the bordism group of . The answer is the Nielsen group defined as follows. In the classical definition, the Nielsen equivalence of points of based on paths is replaced with an equivalence of singular submanifolds of based on bordisms. We let , then the Nielsen group of order is the part of preserved under homotopies of . The Nielsen number of order is the rank of this group (then ). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided.