Coincidence Wecken property for 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 $f\sim f^{\prime }$ so that $C(f^{\prime },g)=\{x\in X\mid f^{\prime }(x)=g(x)\}$ is empty.

7 pages