For a compact surface , the free degree of homeomorphisms on is the minimum positive integer with property that for any self homeomorphism of , at least one of the iterates has a fixed point. This is to say is the maximum of least periods among all periodic points of self homeomorphisms on . We prove that for and for .
Jianchun Wu. Xuezhi Zhao. "Free degrees of homeomorphisms on compact surfaces." Algebr. Geom. Topol. 11 (4) 2437 - 2452, 2011. https://doi.org/10.2140/agt.2011.11.2437