  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.
