For a self-map f of a compact connected polyhedron X the Nielsen number N(f) of f is defined to be the number of essential fixed point classes of f. N(f) is a lower bound for the number of fixed points of every map homotopic to f. This lower bound is known to be realizable if X has no local separating points and X is not a surface (closed or with boundary) of negative Euler characteristic [the author, Am. J. Math. 102, 749-763 (1980; Zbl 0455.55001)]. The realization problem of N(f) for self-maps of surfaces is a long- standing question, and this is exactly the question the author answers. Making an essential use of braid theory the author gives an example of a self-map f of a surface (here a disc with two holes) such that \(N(f)=0\) while every map homotopic to f has at least two fixed points, and thus shows that N(f) is not realizable in general for self-maps of surfaces.