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Higher order Nielsen numbers by Saveliev

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Higher order Nielsen numbers by Peter Saveliev

Fixed Point Theory and Applications, 2005:1 (2005) 47-66, also talks at the Joint Mathematics Meeting in January 2002, San Diego and at the Conference on Geometric Topology in August 2002, Shaanxi Normal University, Xi'an, China.

For the Coincidence Problem: "If $X$ and $Y$ are a manifolds of dimension $n+m$ and $n$ respectively and $f,g:X \rightarrow Y$ are maps, what can be said about the set $C=Coin(f,g)$ of coincidences, i.e., the set of $x \in X$ such that $f(x)=g(x)$?" we suggest a generalization of the classical Nielsen fixed point theory. The number $m=\dim X-\dim Y$ is called the codimension of the problem. More general is the preimage problem. For a map $f:X\rightarrow Y$ and a submanifold $Y$ of $Z$, it studies the preimage set $C=\{ x:f(x)\in Y\} $, and the codimension is $m=dimX+dimY-dimZ$.

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 $\Omega _p(C)$ 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 bordism. We let $S'p(f,Y)$ be the quotient group of $\Omega _p(C)$ with respect to this equivalence relation, then the Nielsen group of order $p$ is the part of this group preserved under homotopies of $f$.

The Nielsen number $Np(f,Y)$ of order $p$ is the rank of this group (then $N(f,Y)=N_0(f,Y)$). These numbers are new obstructions to removability of coincidences and preimages. Some examples and computations are provided.

Full text: Higher order Nielsen numbers (18 pages)