The main result of this nicely written paper is the following. Let \(f\) be a nontrivial orientation preserving homeomorphism of a connected surface \(M\) with finitely generated homology such that \(Fix(f)\) is compact and has a finite number of components. Assume moreover that \(M -Fix(f)\) has no contractible precompact components. Then the Euler characteristic of \(Fix(f)\) with respect to the Čech cohomology is not less than the Euler characteristic \(\chi(M)\). Moreover, denoting with \(k_n\) the number of components of \(Fix(f)\) whose Euler characteristic is \(n\) the following Morse type inequality holds: \[ k_1\geq \chi(M)+ \sum_{n>0} nk_{-n}. \] It is also shown that the last assumption of not having contractible precompact components holds in a number of interesting cases. For example when \(f\) is area preserving or when the set of nonwandering points is dense or when \(f\) has a nowhere dense attractor. Applications are given to fixed points of area preserving homeomorphisms, analytic homeomorphisms, common fixed points of commuting homeomorphisms, periodic points, acyclic components of fixed points in homoclinic cells and attractors.