Coincidences and secondary Nielsen numbers

TL;DR

This paper introduces secondary Nielsen numbers as a new tool to determine when two maps between manifolds can be deformed to avoid coincidences, especially in cases where primary Nielsen numbers are insufficient.

Contribution

It develops a new approach with secondary Nielsen numbers to address coincidence problems in manifold maps, extending the classical Nielsen theory.

Findings

Secondary Nielsen numbers can detect coincidences when primary numbers fail.

The method applies to maps where dimensions satisfy specific relations, e.g., m=2n-2, n even.

The approach works for simply connected target manifolds in certain dimension cases.

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.