Fixed points theory on closed 2-dimensional manifolds (Q2764922)

For \(f:M\to M\) a map on a closed manifold, the Nielsen number \(N(f)\) is a homotopy invariant lower bound for the number of fixed points of \(f\) that is difficult to calculate in general. A map is said to be homotopically periodic if some iterate of it is homotopic to the identity map. \textit{S. Kwasik} and \textit{K. B. Lee} [J. Lond. Math. Soc., II. Ser. 38, No. 3, 544-554 (1988; Zbl 0675.55004)] proved that, for the maps described in the title of their paper, \(N(f)= L(f)\), the Lefschetz number, which is relatively easy to calculate. This paper presents a proof that the equality \(N(f)= L(f)\) holds for all homotopically periodic maps \(f\) on any closed surface other than the sphere \(S^2\) (where it is false for the identity map). The idea of the proof is to show first that if \(M\) is a hyperbolic surface then there is a hyperbolic surface \(M'\), a diffeomorphism \(g:M\to M'\) and an isometry \(f'\) of \(M'\) such that \(g\circ f= f'\circ g\). Then, for almost all surfaces, the proof of the equality needs only to be established for isometries.