Coincidences and secondary Nielsen numbers

Let $f_1, f_2 \colon X^m \longrightarrow Y^n$ 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. $f_1(x) \neq f_2(x)$ 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 $m = 2n - 2,\ \; n \neq 2$ is even and $Y$ is simply connected.

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