Given a surface with non-positive Euler characteristic and non-empty boundary, and a map which has the least number of fixed points possible within its homotopy class there are known bounds (both upper and lower) regarding the fixed point indices of the map. This paper gives a new proof of this result. In addition, a relative version of the method is developed, which is then used to establish the same index bounds for the case of a closed surface of negative Euler characteristic.
"Fixed point index bounds for self-maps on closed surfaces." Bull. Belg. Math. Soc. Simon Stevin 24 (4) 673 - 688, december 2017. https://doi.org/10.36045/bbms/1515035016