Coincidences and secondary Nielsen numbers

Abstract

Abstract. Let f1, f2 : X m −→ Y n be maps between smooth connected manifolds of the indicated dimensions m and n. Can f1, f2 be deformed by homotopies until they are coincidence free (i.e. f1(x) = f2(x) for all x ∈ X )? The main tool for addressing such a problem is tradionally the (primary) Nielsen number N (f1, f2) . E.g. when m < 2n − 2 the question above has a positive answer precisely if N (f1, f2) = 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 (f1, f2) which allow us to answer our question e.g. when m = 2n − 2, n = 2 is even and Y is simply connected.Mathematics Subject Classification (2010). Primary 54H25, 55M20; Secondary 55Q40.