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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.
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Saveliev, P. Higher-order Nielsen numbers. Fixed Point Theory Appl 2005, 290956 (2005). https://doi.org/10.1155/FPTA.2005.47
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DOI: https://doi.org/10.1155/FPTA.2005.47