# Coincidence Wecken property for nilmanifolds

## TL;DR

This paper proves a Wecken-type property for maps between infra-nilmanifolds and nilmanifolds, showing that vanishing Nielsen number implies deformability to coincidence-free maps, with additional results on Reidemeister coincidence numbers.

## Contribution

It establishes a coincidence Wecken property for maps between infra-nilmanifolds and nilmanifolds, extending known fixed point theory results to this setting.

## Key findings

- If Nielsen number N(f,g) is zero, f and g can be deformed to be coincidence free.
- Infinite Reidemeister number R(f,g) implies the existence of a map homotopic to f with no coincidence points.
- The paper extends Wecken-type coincidence results to infra-nilmanifolds and nilmanifolds.

## Abstract

Let $f,g:X\to Y$ be maps from a compact infra-nilmanifold $X$ to a compact nilmanifold $Y$ with $\dim X\ge \dim Y$. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number $N(f,g)$ vanishes then $f$ and $g$ are deformable to be coincidence free. We also show that if $X$ is a connected finite complex $X$ and the Reidemeister coincidence number $R(f,g)=\infty$ then $f\sim f'$ so that $C(f',g)=\{x\in X \mid f'(x)=g(x)\}$ is empty.