Coincidences and secondary Nielsen numbers

Abstract

Let $f_1,f_2:X^m⟶Y^n$ be maps between smooth connected manifolds of the indicated dimensions $m$ and $n$. Can $f_1,f_2$ be deformed by homotopies until they are coincidence free (i.e. $f_1(x)\ne f_2(x)$ for all $x\in X$)? The main tool for addressing such a problem is tradionally the (primary) Nielsen number $N(f_1,f_2)$. E.g. when $m<2n-2$ the question above has a positive answer precisely if $N(f_1,f_2)=0$. However, when $m=2n-2$ this can be dramatically wrong, e.g. in the fixed point case when $m=n=2$. Also, in a very specific setting the Kervaire invariant appears as a (full) additional obstruction. In this paper we start exploring a fairly general new approach. This leads to secondary Nielsen numbers $SecN(f_1,f_2)$ which allow us to answer our question e.g. when $m=2n-2,n\ne 2$ is even and $Y$ is simply connected.

Coincidence, Nielsen number, minimum number, Wecken theorem, Kervaire invariant