- Research Article
- Open access
- Published:
Higher-order Nielsen numbers
Fixed Point Theory and Applications volume 2005, Article number: 290956 (2005)
Abstract
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Saveliev, P. Higher-order Nielsen numbers. Fixed Point Theory Appl 2005, 290956 (2005). https://doi.org/10.1155/FPTA.2005.47
Received:
Revised:
Published:
DOI: https://doi.org/10.1155/FPTA.2005.47