##### Tools

This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Higher order Nielsen numbers by Saveliev

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.