Computation of Nielsen numbers for maps of closed surfaces

Let $X$ be a closed surface, and let $f: X \rightarrow X$ be a map. We would like to determine $\text {Min}(f):= \mathrm {min} \{ | \mathrm {Fix}g| : g \simeq f\}.$ Nielsen fixed point theory provides a lower bound $N(f)$ for $\text {Min}(f)$, called the Nielsen number, which is easy to define geometrically and is difficult to compute.

We improve upon an algebraic method of calculating $N(f)$ developed by Fadell and Husseini, so that the method becomes algorithmic for orientable closed surfaces up to the distinguishing of Reidemeister orbits. Our improvement makes tractable calculations of Nielsen numbers for many maps on surfaces of negative Euler characteristic. We apply the improved method to self-maps on the connected sum of two tori including classes of examples for which no other method is known. We also include the application of this algebraic method to maps on the Klein bottle $K$. Nielsen numbers for maps on $K$ were first calculated (geometrically) by Halpern. We include a sketch of Halpern's never published proof that $N(f)= \text {Min}(f)$ for all maps $f$ on $K$.