The Nielsen number N(f) of a self-map \(f: X\to X\) of a finite polyhedron has the property that every map homotopic to f has at least N(f) fixed points. If X is an n-manifold with \(n\geq 3\), then N(f) is the best possible such lower bound for the number of fixed points because there is a map g homotopic to f with exactly N(f) fixed points. The long-unsolved problem for 2-manifolds was settled in the negative by B. Jiang who showed that any surface with a negative Euler characteristic supports a map f such that \(N(f)=0\) yet every map homotopic to f has a fixed point [Math. Ann. 272, 249-256 (1985)].
In the present paper, the author constructs a class of mappings \(f_ n: X\to X\), where X is a disc with two or more holes, such that \(N(f_ n)=1\) but any map homotopic to \(f_ n\) has at least n fixed points. Thus he shows that there is no bound on the ”gap” between the Nielsen number and the minimum number of fixed points for maps in the same homotopy class at least for these surfaces.
The author considerably extends Jiang’s use of the braid group of the surface as the principal tool in analyzing the maps. He establishes a correspondence between the maps with finite fixed point sets homotopic to a given \(f: X\to X\) and the solutions to a specific equation in the braid group. The equation is determined by the homomorphism of the fundamental group of X induced by f. The analysis of the equation makes use of a clever combinatorial ”index” for words in a free group that may find other applications in combinatorial group theory.
As the author notes, a different class of examples \(g_ n: X\to X\), where X is the disc with two holes, was constructed independently by M. R. Kelly [Pac. J. Math. 126, 81-123 (1987; Zbl 0571.55003)] with the property that \(N(g_ n)=0\) but every map homotopic to \(g_ n\) has at least 2n fixed points. Kelly’s methods are very geometric in nature and entirely different from those of the present paper.