#### Volume 11, issue 4 (2011)

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Free degrees of homeomorphisms on compact surfaces

### Jianchun Wu and Xuezhi Zhao

Algebraic & Geometric Topology 11 (2011) 2437–2452
##### Abstract

For a compact surface $M$, the free degree $\mathfrak{f}\mathfrak{r}\left(M\right)$ of homeomorphisms on $M$ is the minimum positive integer $n$ with property that for any self homeomorphism $\xi$ of $M$, at least one of the iterates $\xi ,{\xi }^{2},\dots ,{\xi }^{n}$ has a fixed point. This is to say $\mathfrak{f}\mathfrak{r}\left(M\right)$ is the maximum of least periods among all periodic points of self homeomorphisms on $M$. We prove that $\mathfrak{f}\mathfrak{r}\left({F}_{g,b}\right)\le 24g-24$ for $g\ge 2$ and $\mathfrak{f}\mathfrak{r}\left({N}_{g,b}\right)\le 12g-24$ for $g\ge 3$.

##### Keywords
fixed point, periodic point, surface, homeomorphism
Primary: 55M20
Secondary: 37E30
##### Publication
Received: 30 March 2011
Revised: 6 August 2011
Accepted: 12 August 2011
Published: 5 September 2011
##### Authors
 Jianchun Wu Department of Mathematics Soochow University Suzhou 215006 China Xuezhi Zhao Department of Mathematics & Institute of Mathematics and Interdisciplinary Science, Capital Normal University Beijing 100048 China