Coincidences and secondary Nielsen numbers

Abstract

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.