Journal of the
Korean Mathematical Society JKMS

We will consider $f, g : S_1 \to S_2$ a pair of maps between two orientable compact surfaces. The purpose of this paper is to decide when the pair can be deformed to a pair $(f',g')$ such that $\#\coin(f',g') = N(f,g)$, the Nielsen coincidence number of $(f,g)$. We derive an equivalent algebraic condition and show that if we compose $(f,g)$ with certain maps $h : S \to S_1$ then the answer is positive. Finally, we analyze the case of roots, i.e., $g$ is the constant map. When $S_2$ is the torus we give a new proof of the converse of the Lefschetz theorem for coincidence.

Keywords: Nielsen coincidence number, Braid group, Wecken property, compact surface

MSC numbers: Primary 55M20; Secondary 55M25, 57N75